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# Dynamic Resource Allocation and Optimization in Wireless Networks

## Material Information

Title: Dynamic Resource Allocation and Optimization in Wireless Networks
Physical Description: 1 online resource (198 p.)
Language: english
Creator: Song, Yang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

## Subjects

Subjects / Keywords: control, learning, optimization, pricing, routing, schedulinig
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Due to the hostile wireless medium and the limited resources in wireless networks, how well wireless networks can perform and how to make wireless networks provide better service are critical and challenging problems. These motivate our research in both theoretical analysis and protocol designs in time-varying wireless networks. In this dissertation, we aim to address the dynamic resource allocation and optimization problems in wireless networks, spanning wireless ad hoc networks, wireless mesh networks, wireless sensor networks, wireless local area networks, and cognitive radio networks. Our contributions can be summarized as follows. First, we propose an online dynamic pricing scheme which maximizes the network revenue subject to stability in multi-hop wireless networks with multiple QoS-specific flows. Secondly, we design a novel green scheduling algorithm in multi-hop wireless networks with stochastic arrivals and time-varying channel conditions which minimizes the energy expenditure subject to network stability. Thirdly, we develop a negotiation-based algorithm which attains an $\epsilon$-optimal solution to the non-convex joint power and frequency allocation problem in cooperative wireless mesh access networks. In addition, we analyze the existence and the inefficiency of the Nash equilibrium in non-cooperative wireless mesh access networks and proposed pricing schemes to improve the equilibrium efficiency. Fourthly, we propose a queueing based model to capture the cross layer interactions in multi-hop wireless sensor networks and designed a joint rate admission control, traffic engineering, dynamic routing and scheduling scheme to maximize the overall network utility. Fifthly, we analyze the thresholds-based rate adaptation algorithms in IEEE 802.11 WLANs from a reverse engineering perspective and propose a threshold optimization algorithm to enhance the performance of IEEE 802.11 WLANs. Finally, we investigate the stochastic traffic engineering problem in multi-hop cognitive radio networks and derive a distributed algorithm based on the stochastic primal-dual approach for convex scenarios as well as a general solution based on the learning automata techniques for non-convex scenarios.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yang Song.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041955:00001

## Material Information

Title: Dynamic Resource Allocation and Optimization in Wireless Networks
Physical Description: 1 online resource (198 p.)
Language: english
Creator: Song, Yang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

## Subjects

Subjects / Keywords: control, learning, optimization, pricing, routing, schedulinig
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Due to the hostile wireless medium and the limited resources in wireless networks, how well wireless networks can perform and how to make wireless networks provide better service are critical and challenging problems. These motivate our research in both theoretical analysis and protocol designs in time-varying wireless networks. In this dissertation, we aim to address the dynamic resource allocation and optimization problems in wireless networks, spanning wireless ad hoc networks, wireless mesh networks, wireless sensor networks, wireless local area networks, and cognitive radio networks. Our contributions can be summarized as follows. First, we propose an online dynamic pricing scheme which maximizes the network revenue subject to stability in multi-hop wireless networks with multiple QoS-specific flows. Secondly, we design a novel green scheduling algorithm in multi-hop wireless networks with stochastic arrivals and time-varying channel conditions which minimizes the energy expenditure subject to network stability. Thirdly, we develop a negotiation-based algorithm which attains an $\epsilon$-optimal solution to the non-convex joint power and frequency allocation problem in cooperative wireless mesh access networks. In addition, we analyze the existence and the inefficiency of the Nash equilibrium in non-cooperative wireless mesh access networks and proposed pricing schemes to improve the equilibrium efficiency. Fourthly, we propose a queueing based model to capture the cross layer interactions in multi-hop wireless sensor networks and designed a joint rate admission control, traffic engineering, dynamic routing and scheduling scheme to maximize the overall network utility. Fifthly, we analyze the thresholds-based rate adaptation algorithms in IEEE 802.11 WLANs from a reverse engineering perspective and propose a threshold optimization algorithm to enhance the performance of IEEE 802.11 WLANs. Finally, we investigate the stochastic traffic engineering problem in multi-hop cognitive radio networks and derive a distributed algorithm based on the stochastic primal-dual approach for convex scenarios as well as a general solution based on the learning automata techniques for non-convex scenarios.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yang Song.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041955:00001

Full Text

DYNAMIC RESOURCE ALLOCATION AND OPTIMIZATION IN WIRELESS
NETWORKS

By

YANG SONG

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

@ 2010 Yang Song

To my beloved parents and friends

ACKNOWLEDGMENTS

I would like to gratefully and sincerely thank my Ph.D. advisor Prof. Yuguang Fang

for his invaluable guidance, understanding, patience, and most importantly, his continual

faith and confidence in me during my Ph.D. studies at the University of Florida. I feel

extremely fortunate to have him as my advisor, who is always willing to help me both

academically and personally. Thanks for everything. I also owe my wholehearted

gratitude to my Ph.D. committee members, Prof. Pramod P. Khargonekar, Prof. Sartaj

Sahni, Prof. Tan Wong, and Prof. Shigang Chen, for their constructive suggestions and

valuable comments on my Ph.D. research and dissertation.

I have been fortunate to have many friends in WINET. I specially thank Chi Zhang,

Xiaoxia Huang, Yun Zhou, Shushan Wen, Jianfeng Wang, Hongqiang Zhai, Yanchao

Zhang, Feng Chen, Pan Li, Jinyuan Sun, Miao Pan, Rongsheng Huang, Yue Hao for

many valuable discussions and good memories.

Last but definitely not least, this work would not have been achieved without the

support and understanding of my parents. They have always supported me in every

choice I have chosen in my life.

page

ACKNOWLEDGMENTS ................. ................. 4

LIST O FTABLES ..................... ................. 8

LIST OF FIGURES .................... ................. 9

ABSTRACT .................... ..................... 11

CHAPTER

1 INTRODUCTION .................... ............... 13

2 REVENUE MAXIMIZATION IN MULTI-HOP WIRELESS NETWORKS ..... 17

2.1 Introduction ................... ................ 17
2.2 Related Work .................... .............. 19
2.3 Revenue Maximization with QoS (Quality of Service) Requirements 22
2.3.1 System Model .................... .......... 22
2.3.2 Problem Formulation .......................... 25
2.4 QoS-Aware Dynamic Pricing (QADP) Algorithm 27
2.5 Performance Analysis ............................. 30
2.5.1 Proof of Revenue Maximization .. .. 31
2.5.2 Proof of Network Stability ..... .... 39
2.5.3 Proof of QoS Provisioning ..... ... ... 40
2.6 S im ulations . .. 41
2.6.1 Single-Hop Wireless Cellular Networks .... 41
2.6.2 Multi-Hop Wireless Networks .. ... 44
2.7 C conclusions . . 47

3 ENERGY-CONSERVING SCHEDULING IN WIRELESS NETWORKS 48

3.1 Introduction . .. 48
3.2 System Model ............ .. ................. 50
3.3 Minimum Energy Scheduling Algorithm ... 54
3.3.1 Algorithm Description ......................... 54
3.3.2 Throughput-Optimality ............ .. .... ... .. 57
3.3.3 Asymptotic Energy-Optimality . ... 62
3.4 Sim ulations ... .. .... .. 64
3.5 Conclusions . . 67

4 CHANNEL AND POWER ALLOCATION IN WIRELESS MESH NETWORKS 70

4.1 Introduction ... .. .... .. 70
4.2 System Model .................. ............. 72
4.3 Cooperative Access Networks ... 74

4.3.1 Cooperative Throughput Maximization Game ...........
4.3.2 NETMA- Negotiation-Based Throughput Maximization Algorithm
4.4 Non-Cooperative Access Networks .....................
4.5 An Extension to Adaptive Coding and Modulation Capable Devices .
4.6 Performance Evaluation .. .......................
4.6.1 Legacy IEEE 802.11 Devices .. .................
4.6.1.1 Example of small networks ................
4.6.1.2 Example of large networks ................
4.6.2 ACM-Capable Devices ................ .......
4.7 Conclusions ..................... .............

5 CROSS LAYER INTERACTIONS IN WIRELESS SENSOR NETWORKS .

5.1 Introduction . .
5.2 Related W ork . .

5.3 A Constrained Queueing Model for Wireless Sensor Netwo

5.3.1 Network Model ...........
5.3.2 Traffic Model ............
5.3.3 Queue Management ........
5.3.4 Session-Specific Requirements .
5.4 Stochastic Network Utility Maximization in
5.4.1 Problem Formulation ........
5.4.2 The ANRA Cross Layer Algorithm .
5.4.3 Performance of the ANRA Scheme
5.5 Performance Analysis .
5.6 Case Study .................
5.7 Conclusions and Future Work .

Wireless

Sensor

. 100
. 102

)rks .

SNetworks

6 THRESHOLD OPTIMIZATION FOR RATE ADAPTATION ALGORITHMS IN
IEEE 802.11 W LANS . .

6.1 Introduction . .
6.2 Related W ork ......................
6.3 Reverse Engineering for the Threshold-Based Rate /
6.4 Threshold Optimization Algorithm .
6.4.1 Learning Automata ...............
6.4.2 Achieving the Stochastic Optimal Thresholds
6.5 Performance Evaluation ................
6.6 Conclusions and Future Work .

............
............
. .
............
. .
............
. .

7 STOCHASTIC TRAFFIC ENGINEERING IN MULTI-HOP COGNITIVE WIRELESS
MESH NETW ORKS ................................. 157

Introduction . .
Related W ork . .
System M odel .....................
Stochastic Traffic Engineering with Convexity ..

157
160
163
167

100

. 104
S104
S105
S108
. 109
S110
. 111
S112
. 114
. 118
S125
. 128

S130

130
133
135
143
144
145
150
155

7.4.1 Form ulation . .
7.4.2 Distributed Algorithmic Solution with the Stochastic
A approach . .
7.5 Stochastic Traffic Engineering without Convexity .
7.6 Performance Evaluation ...................
7.7 Conclusions.. .......................

8 CONCLUSIONS ..........................

REFERENCES ................... ...........

BIOGRAPHICAL SKETCH .......................

Primal-Dual

167

169
173
178
183

185

186

198

LIST OF TABLES

Table page

2-1 Average admitted rates for multimedia flows ..... 44

2-2 Average admitted rates for multimedia flows ..... 47

4-1 Data rates v.s. SINR thresholds with maximum BER = 10 90

6-1 SNR v.s. BER for IEEE 802.11b data rates ... 152

7-1 Available paths for edge routers ..... .. 179

7-2 Convergence rates when Y is affected by all five primary users 182

7-3 Convergence rates when Y is not affected by any of the primary users 182

LIST OF FIGURES

Figure page

2-1 An example of multi-hop wireless networks ...... ...... 23

2-2 A conceptual example of the network capacity region with two multimedia flows.
The minimum data rate requirements reduce the feasible region of the optimum

solution

2-3

2-4

2-5

A single-hop wireless cellular network with three users .......

Impact of different values of J on the performance of QADP ...

Impact of different values of J on the average experienced delays .

. . . 2 7

2-6 Queue backlog dynamics for all users .........

2-7 Price dynamics in QADP for all users .. .......

2-9 Average queue backlogs in the network for J = 50000

3-1 Network topology with interconnected queues .....

3-2 Comparison of the energy consumption of the MaxW
M ES algorithm . .

3-3 The average queue backlogs in the network for J = 50

3-4 The average queue backlogs in the network for J = 35

eight algorithm

and 150 .

0 and 500 ..

3-5 Comparison of the lifetime of the MaxWeight algorithm and the MES algorithm

4-1 Hierarchical structure of wireless mesh access networks .............

4-2 An illustrative example of multiple Nash equilibria .................

4-3 Markovian chain of NETMA with two players ....................

4-4 Performance evaluation of the wireless mesh access network with N = 5 and
c = 3 . . .

4-5 The trajectory of frequency negotiations in NETMA when N = 5 and c = 3 .

4-6 The trajectory of power negotiations in NETMA when N = 5 and c = 3 .....

4-7 Performance evaluation of the wireless mesh access network with N = 20 and
c = 3 . . .

4-8 Performance evaluation of the wireless mesh access network with/without the
pricing scheme e . . .

and

4-9 The trajectories of power utilization prices of each ACM-capable AP 98

4-10 The trajectories of actual power utilized by each ACM-capable AP ...... ..99

5-1 Topology of wireless sensor networks ... 106

5-2 Example network topology ............................. 125

5-3 Average network utility achieved by ANRA for different values of J ...... ..126

5-4 Average network queue size by AN RA for different values of J ... 127

5-5 Sample trajectories of the admitted rates of two sessions for J = 5000 127

5-6 Sample trajectories of the virtual queues for J = 5000 ... 128

5-7 Sample trajectories of the hop selections for J = 5000 ... 129

6-1 Average throughput v.s. Doppler spread values ... 153

6-2 The trajectories of 0, and Od in achieving the stochastic optimum values .. 154

6-3 Evolution of the probability vector P . 154

6-4 Evolution of the probability vector Pd ... 155

7-1 Architecture of wireless mesh networks ... 157

7-2 Example of cognitive wireless mesh network . ... 179

7-3 Cognitive wireless mesh networks with convexity . 181

7-4 Trajectory of the probability vector of router A's first path .. 183

7-5 Trajectories of the probability vectors on other paths ... 184

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DYNAMIC RESOURCE ALLOCATION AND OPTIMIZATION IN WIRELESS
NETWORKS

By

Yang Song

August 2010

Chair: Yuguang "Michael" Fang
Major: Electrical and Computer Engineering

Due to the hostile wireless medium and the limited resources in wireless networks,

how well wireless networks can perform and how to make wireless networks provide

better service are critical and challenging problems. These motivate our research in

both theoretical analysis and protocol designs in time-varying wireless networks. In

this dissertation, we aim to address the dynamic resource allocation and optimization

problems in wireless networks, spanning wireless ad hoc networks, wireless mesh

networks, wireless sensor networks, wireless local area networks, and cognitive radio

networks.

Our contributions can be summarized as follows. First, we propose an online

dynamic pricing scheme which maximizes the network revenue subject to stability in

multi-hop wireless networks with multiple QoS-specific flows. Secondly, we design

a novel green scheduling algorithm in multi-hop wireless networks with stochastic

arrivals and time-varying channel conditions which minimizes the energy expenditure

subject to network stability. Thirdly, we develop a negotiation-based algorithm which

attains an c-optimal solution to the non-convex joint power and frequency allocation

problem in cooperative wireless mesh access networks. In addition, we analyze the

existence and the inefficiency of the Nash equilibrium in non-cooperative wireless

mesh access networks and proposed pricing schemes to improve the equilibrium

efficiency. Fourthly, we propose a queueing based model to capture the cross layer

interactions in multi-hop wireless sensor networks and designed a joint rate admission

control, traffic engineering, dynamic routing and scheduling scheme to maximize

the overall network utility. Fifthly, we analyze the thresholds-based rate adaptation

algorithms in IEEE 802.11 WLANs from a reverse engineering perspective and propose

a threshold optimization algorithm to enhance the performance of IEEE 802.11 WLANs.

Finally, we investigate the stochastic traffic engineering problem in multi-hop cognitive

radio networks and derive a distributed algorithm based on the stochastic primal-dual

approach for convex scenarios as well as a general solution based on the learning

automata techniques for non-convex scenarios.

CHAPTER 1
INTRODUCTION

WANETs are self-configuring and stand-alone networks of nodes connected by

wireless links. They have attracted extensive attention as ideal networking solutions

for scenarios where fixed network infrastructures are not available or reliable. In

addition, multimedia transmissions have become an indispensable component of

network traffic nowadays. However, the issue of QoS provisioning for multimedia

transmissions is remarkably challenging in multi-hop wireless ad hoc networks. Time

the QoS of multimedia transmissions. Therefore, while wired networks have mature

solutions and established protocols for providing QoS, novel solutions which incorporate

salient features of wireless transmissions need to be developed to support multimedia

transmissions in multi-hop wireless networks. In addition, from the network provider's

perspective, it is imperative to design a pricing mechanism which maximizes the network

revenue while addressing the QoS requirements of users. Such a scheme should also

ensure a network-wide stability under stochastic traffic arrivals and time varying channel

conditions. In Chapter 2, we proposed a QoS-aware dynamic pricing scheme which

provably ensures the network-wide stability while attaining a solution which is arbitrarily

close to the global maximum revenue, with a controllable tradeoff with the average delay

in the network. Moreover, a weight assignment mechanism is devised to address the

service differentiation issue for multiple flows in the network with different delay priorities.

In Chapter 3, we investigate the issue of Green Computing in wireless networks,

which is an important concern raised from computer scientists and engineers that

attempts to utilize the computing resources efficiently while introducing minimum

impact on the environment. In multi-hop wireless networks, we first identify that

the well-established MaxWeight, or back-pressure scheduling algorithm, is not

energy-optimal. The scheduling process of the MaxWeight algorithm neglects the

enormous energy consumption in retransmissions which are not negligible in wireless

networks with fading channels. To address this, we proposed a minimum energy

scheduling algorithm which significantly reduced the overall energy expenditure

compared to the original MaxWeight algorithm. The energy consumption induced by

the proposed algorithm can be pushed arbitrarily close to the global minimum solution.

Moreover, the improvement on the energy efficiency is achieved without losing the

throughput-optimality.

A wireless mesh network is characterized by a multi-hop wireless backbone

connecting wired Internet entry points, or gateways, and wireless access points (AP)

which provide network access to end users. In a WMN, how to assign multiple channels

and power levels to each AP is of great importance to maximize the overall network

utilization and the aggregated throughput. In Chapter 4, we investigate the non-convex

throughput maximization problem in WMNs for both cooperative and non-cooperative

scenarios. In the cooperative case, we model the interactions among all APs as an

identical interest game and present a decentralized negotiation-based throughput

maximizing algorithm for the joint frequency and power assignment problem. We prove

that this algorithm converges to the optimal frequency and power assignment solution,

which maximizes the overall throughput of the wireless mesh network, with arbitrarily

high probability. In the case of non-cooperative APs, we prove the existence of Nash

equilibria and show that the overall throughput performance is noticeably inferior to the

cooperative scenario. To bridge the performance gap, we develop a pricing scheme to

combat the selfish behaviors of non-cooperative APs. The overall network performance

in term of aggregated throughput is significantly improved.

Recent years have witnessed a surge in research and development of WSNs

for their broad applications in both military and civilian operations. Before the wide

deployment of wireless sensor networks, a systematic understanding about the

performance of multi-hop wireless sensor networks is desired. However, finding a

suitable and accurate analytical model for wireless sensor networks is particularly

challenging. An appropriate model should reflect the realistic network operations

with emphasis on the distinguishing features of wireless sensor networks such as

the challenge of automatic load balancing among multiple sink nodes, the task of

dynamic network scheduling and routing under time varying channel conditions, and

the instantaneous decision-making on the number of admitted packets in order to

ensure the network-wide stability. To capture the cross layer interactions of multi-hop

wireless sensor networks, in Chapter 5, we proposed a constrained queueing model to

and automatic load balancing in wireless sensor networks through a set of interconnected

queues. To demonstrate the effectiveness of the proposed constrained queueing

model, we investigate the stochastic network utility maximization problem in multi-hop

wireless sensor networks. Based on the proposed queueing model, we develop an

adaptive network resource allocation scheme which yields a near-optimal solution to

the stochastic network utility maximization problem. The proposed scheme consists of

multiple layer components such as joint rate admission control, traffic splitting, dynamic

routing, and an adaptive link scheduling algorithm. Our proposed scheme is essentially

an online algorithm which only requires the instantaneous information of the current time

slot and hence remarkably reduces the computational complexity.

IEEE 802.11 WLAN has become the dominating technology for indoor wireless

Internet access. In order to maximize the network throughput, IEEE 802.11 devices,

i.e., stations, need to adaptively change the data rate to combat the time varying

channel environments. However, the specification of rate adaptation algorithms is

not provided by the IEEE 802.11 standard. This intentional omission encourages the

studies on this active area where a variety of rate adaptation algorithms have been

proposed. Due to its simplicity, the thresholds-based rate adaptation algorithm is

predominantly adopted by vendors. The data rate increases if a certain number of

consecutive transmissions are successful. Although widely deployed, the obscure

objective function of this type of rate adaptation algorithms, commonly based on the

heuristic up/down mechanism, is less comprehended. In Chapter 6, we study the

thresholds-based rate adaptation algorithm from a reverse engineering perspective. The

implicit objective function, which the rate adaption algorithm is maximizing, is unveiled.

Our results provide an analytical model from which the heuristics-based rate adaptation

algorithm, such as ARF, can be better understood. Moreover, we propose a threshold

optimization algorithm which dynamically adapts the up/down thresholds. Our algorithm

provably converges to the set of stochastic optimum thresholds in arbitrary stationary yet

potentially fast-varying channel environments and the performance in term of throughput

is enhanced remarkably.

In Chapter 7, we investigate the stochastic traffic engineering (STE) problem in

multi-hop cognitive radio networks. More specifically, we are particularly interested in

how the traffic in the multi-hop cognitive radio networks should be steered, under the

influence of random returns of primary users. It is worth noting that given a routing

strategy, the corresponding networks performance, e.g., the average queueing

delay encountered, is a random variable. In multi-hop cognitive radio networks, this

distinguishing feature of randomness, induced by the unpredictable behaviors of primary

users, must be taken into account in protocol designs. We formulate the STE problem in

a stochastic network utility maximization framework. For the case where convexity holds,

we derive a distributed cross layer algorithm via the stochastic primal-dual approach,

which provably converges to the global optimum solution. For the scenarios where

convexity is not attainable, we propose an alternative decentralized algorithmic solution

based on the learning automata techniques. We show that the algorithm converges to

the global optimum solution asymptotically under mild conditions. Finally, Chapter 8

concludes this dissertation.

CHAPTER 2
REVENUE MAXIMIZATION IN MULTI-HOP WIRELESS NETWORKS

2.1 Introduction

transmissions become an indispensable component of network traffic nowadays. For

example, people can watch live games while listening to online musical stations at the

same time. Therefore, supporting multimedia services in multi-hop wireless networks

effectively and efficiently has received intensive attention from the community. However,

the issue of QoS provisioning for multimedia transmissions is remarkably challenging

in multi-hop wireless networks. Time varying channel conditions among wireless links

impose severe adverse impact on the QoS of multimedia transmissions. Therefore,

while wired networks have mature solutions and established protocols for providing QoS,

novel solutions which incorporate salient features of wireless transmissions, need to be

developed to support multimedia transmissions in multi-hop wireless networks.

Multimedia flows usually impose application-specific requirements on the minimum

average attainable data rates from the network. Furthermore, different multimedia flows

may have distinct rate requirements. For example, multimedia streams for high quality

video-on-demand (VoD) movie transmissions usually require larger minimum data rates

on average than those of online music transmissions. Consequently, a natural question

arises that how the network resource should be allocated such that all the minimum

rate constraints are satisfied simultaneously. In addition, due to the nature of wireless

transmissions, decentralized solution with low complexity is strongly desired. In this

chapter, we investigate the resource allocation problem in multi-hop wireless networks,

from a network administrator's perspective. For each flow, the network charges a certain

amount of admission fee in order to build up a system-wide revenue. The price imposed

on each flow is subject to adaptation in order to obtain the optimum revenue. Hence, a

dynamic pricing policy is desired by the network administrator to maximize the overall

network revenue subject to the stability of the network. To achieve this, we propose

a QoS-aware dynamic pricing algorithm, namely, QADP, which provably accumulates

a network revenue that is arbitrarily close to the optimum solution while maintaining

network stability under stochastic traffic arrivals and time varying channel conditions.

Meanwhile, the minimum date rate requirements from all multimedia flows are satisfied

simultaneously.

Besides minimum data rates, multimedia flows, especially wireless video transmissions,

usually impose additional requirements on maximum end-to-end delays. For example,

a multimedia stream for video surveillance may need a lower data transmission rate

compared to high quality video-on-demand movie transmissions, whereas a much more

stringent delay requirement is imposed. Therefore, the network inclines to allocate more

network resource to those delay-imperative multimedia transmissions provided that the

minimum data rate requirements of all flows are satisfied. Unfortunately, however, an ab-

solute guarantee for arbitrary delay requirements is extremely difficult, if not impossible,

due to the lack of accurate delay analysis in multi-hop wireless networks. For example,

[1] derives a lower bound on the delay performance of arbitrary scheduling policy in

multi-hop wireless networks. However, the upper bound of delay is unspecified. In fact, it

is shown that in wireless scenarios, even to decide whether a set of delay requirements

can be supported by the network is an AP-hard problem and thus is intrinsically difficult

to solve [2]. Even worse yet, time varying wireless channel conditions and stochastic

traffic arrivals significantly exacerbate the hardness of sheer delay guarantees. In light of

this, in this work, we alternatively aim to provide a service differentiation solution for the

delay requirements of all flows. More specifically, the network provides a set of service

levels, denoted by = 1, L where level one has the highest priority in the system

with respect to delay guarantees. Note that L can be arbitrarily large. Each multimedia

flow, according to the upper layer application, proposes a service level request to the

level ten whereas a VoD online movie transmission demands a service level of two1 By

meticulously assigning weights to multimedia flows, QADP algorithm provides a service

differentiation solution in a way that the guaranteed maximum average end-to-end

delay for service level one traffic isj times less than that of level j transmissions, where

j = 1, L. In other words, level one transmissions have the minimum upper bound

for end-to-end delay and thus represent the highest priority. Therefore, by following

QADP, a QoS-aware revenue maximization solution, subject to the stability of the

network, is provided. QADP algorithm is inherently an online dynamic control based

algorithm which is self-adaptive to the changes of statistical characteristics of traffic

arrivals. Moreover, our scheme enjoys a decoupled structure and hence is suitable for

decentralized implementation which is of great interest for protocol design in multi-hop

wireless networks. Note that our results are applicable to some special interesting cases

of network topology such as single-hop wireless cellular networks.

The rest of this chapter is organized as follows. Section 2.2 compares our work

with other existing solutions on network optimization with multiple flows. System model

and problem formulation are provided in Section 2.3. Our proposed solution, i.e., QADP

algorithm, is introduced in Section 2.4 followed by the performance analysis in Section

2.5. Simulation results are provided in Section 2.6 and Section 2.7 concludes this

chapter.

2.2 Related Work

There exists a rich literature on how the network resource should be allocated

efficiently among multiple competitive transmitting flows. According to the network

model, they can be roughly divided into two categories, i.e., fluid based approach and

1 We emphasize that this application-to-service-level mapping is arbitrary and can be
specified by the network before transmissions.

queue based approach. In fluid based algorithms, e.g., [3-7], the characteristics of

a particular flow are uniquely associated with fluid variables. For example, the flow

injection rate is a commonly used fluid variable which is determined by the source node.

However, whenever a change of the flow injection rate occurs, it is usually assumed

that all the nodes on the path will perceive this change instantaneously. In addition,

the knowledge of up-to-date local information on intermediate nodes, e.g., shadow

prices [4, 6, 7], is usually vital for the flow control algorithm implemented on the source

node. Simply put, state information of a particular flow is usually assumed to be shared

and known by all nodes on its path instantaneously and accurately, although some

exceptions are discussed, e.g., [8]. In general, fluid based approach does not consider

the practical queue dynamics in real networks. Moreover, fluid based algorithms usually

utilize a dual decomposition framework, such as in [3, 4, 7] which heavily relies on

convex optimization techniques [9]. Consequently, a fixed point solution is attained.

However, in time varying environments such as wireless networks, a fixed operating

point is hardly optimal2 On the contrary, queue based approach models the network

as a set of interconnected queues. Each source node injects packets into the network

which traverse through the network hop by hop until reach the destination. Every packet

needs to wait for service in the queues of intermediate nodes along the path, which

reflects the reality in practical networks. In addition, queue based approach usually

adopts a dynamic control based solution where responsive actions are adjusted on the

fly by which a long term average optimum is achieved. In this work, we utilize a queue

based network model where an optimal dynamic pricing policy is developed. For more

discussions on fluid based algorithms, refer to [10, 11] and the references therein.

2 In fact, in the work, we analytically show that the imposed price needs to be
adjusted dynamically in order to achieve the optimum.

The interconnected queue network model has attracted significant attention since

the seminal work of [12] where the well-known MaxWeight scheduling algorithm is

proposed. Neely extends the results into a general time varying setting in [13, 14],

based on which a pioneering stochastic network optimization technique is developed

[15, 16]. For a comprehensive treatment on this area, refer to [17] and the references

therein. Unfortunately, in the literature, few work has been devoted to addressing the

issue of QoS provisioning in practical wireless settings, which is of special interest in

supporting multimedia transmissions in multi-hop wireless networks. For example,

[18-20] investigate the delay constraints by assuming that the queue behaves as an

M/M/1 queue. However, this may not be realistic due to the complex interaction of

queues induced by the scheduling algorithm. In [2, 21, 22], delay guaranteed scheduling

algorithms are studied. However, the derived delay bound is up to a logarithmic factor

of the proposed delay requirement, given that the delay requirements satisfy certain

per-server and per-session conditions [2]. Finding a general policy that is able to

achieve arbitrary feasible delay requirements is still an open problem. In addition, while

existing solutions on prioritized transmissions are available, e.g., [23, 24], the question

of how to achieve a minimum data rate guarantee concurrently, as well as maximizing a

system-wide revenue, is unspecified. In [25, 26], lazy packet scheduling algorithms are

proposed. However, they require the knowledge on future stochastic arrivals as a priori,

while in our scheme, a.k.a., QADP algorithm, such information is not needed.

Revenue maximization problem has been studied extensively in the literature

for many different settings. Nevertheless, in general, either QoS provisioning is not

particularly addressed[27-30], or the network model is restricted to wired networks

where the channel conditions of the system are assumed to be time-invariant and

remain unchanged [31, 32]. Our work is inspired by [33]. However, our work differs

from [33] in the following crucial aspects. First, [33] studies a single hop network with

only one access point (AP) while our work investigates a general multi-hop wireless ad

hoc network where exogenous arrivals may enter the network via any node. Secondly,

[33] assumes a Markovian user traffic demand while our work is applicable to arbitrary

traffic demands. Thirdly and most importantly, [33] does not consider the issue of

QoS provisioning for multiple transmission flows, which is the main focus of our work.

To the best of our knowledge, this work is the first work to address the problem of

revenue maximization, subject to network stability while providing QoS differentiations to

quality-sensitive traffics such as multimedia video transmissions.

2.3 Revenue Maximization with QoS (Quality of Service) Requirements

2.3.1 System Model

We consider a static multi-hop wireless network represented by a directed graph

g = (AF, ), illustrated in Figure 2-1, where A is the set of nodes and S is the set

of links. The numbers of nodes and links in the network are denoted by N and E,

respectively. A link is denoted either by e e S or (a, b) e S where a and b are the

transmitter and the receiver of the link. Time is slotted, i.e., t = 0, 1, .... For link (a, b),

the instantaneous channel condition at time slot t is denoted by Sa,b(t). For example,

Sa,b(t) can represent the time varying fading factor on link (a, b) at time t. Denote S(t)
as the channel condition vector on all links. We assume that S(t) remains constant

during a time slot. However, S(t) may change on slot boundaries. We assume that

there are a finite but arbitrarily large number of possible channel condition vectors

and S(t) evolves following a finite state irreducible Markovian3 chain with well defined

steady state distribution. However, the steady state distribution itself and the transition

probabilities are unknown. At each time slot t, given the channel state vector S(t), the

network controller chooses a link schedule, denoted by I(t), from a feasible set Ts(t),

which is restricted by factors such as underlying interference model, duplex constraints

3 Note that the Markovian channel state assumption is not essential and can be
relaxed to a more general setting as in [14].

or peak power limitations. For a wireless link (a, b), the link data rate Pa,b(t) is a function

of I(t) and S(t). We denote y(t) as the vector of link rates of all links at time slot t.

S" Flow 1

Flow 2

Flow 3

Figure 2-1. An example of multi-hop wireless networks

There are C commodities, a.k.a., flows, in the network, where each commodity, say

c, c = 1, 2, .. C, is associated with a routing path Pc = {c(0), c(1), c(Vc)} where

c(0) and c(jc) are the source and the destination node of flow c while c(j) denotes the

j-th hop node on its path. Without loss of generality, we assume that every node in the

network initiates at most one flow4 However, multiple flows can intersect at any node in

the network. Each node maintains a separate queue for every flow that passes through

it. For each flow c, denote Ac(t) as the exogenous arrival to the transport layer of node

c(0) during time slot t. We assume that the stochastic arrival process, i.e., Ac(t), has an

expected average rate of Ac. More specifically, we have

St-1
lim 1 E(Ac(T))= Ac Vc. (2-1)
t-oo t
r=o

For a single queue, define the overflow function [13] as

1 t-1
g(B) = lim sup Pr(Q(r) > B) (2-2)
t-*oo t

4 If a node initiates more than one flow in the network, we can replace this node with
multiple duplicate nodes and the following analysis still holds.

where Q(r) is the queue backlog at time r. We say the queue is stable if [13, 14]

lim g(B) -> 0. (2-3)
B-oo

A network is stable if all the queues in this network are stable.

Denote A = {A,, Ac} as the arrival rate vector of the network. Note that all

the arrival rate vectors are defined in an average sense. A flow control mechanism

is implemented where during time slot t, an amount of Rc(t) traffic is admitted to the

network layer for flow c. Apparently, we have Rc(t) < Ac(t). For simplicity, we assume

that there are no reservoirs to hold the excessive traffic, i.e., incoming packets are

dropped if not admitted. However, we emphasize that our analysis can be applied to

general cases where transport layer reservoirs are deployed to buffer the un-admitted

traffic. The network capacity region, a.k.a., the network stability region, denoted by Q,

is defined as all the admission rate vectors that can be supported by the network, in the

sense that there exists a policy that stabilizes the network under this admission rate.

Moreover, we are particularly interested in multimedia transmissions where each flow

c has specific QoS requirements. To be specific, each flow c has a minimum data rate

requirement ac as well as an application-dependent prioritized service level request,

denoted by c, 1 < c < L. Denote a = {ac, ... ac} and = {, c} as the

minimum rate vector and the service level request vector of the network where has

the highest priority in terms of the delay guarantees provided by the network. In addition,

we assume that the minimum rate vector, i.e., a, is inside of the network capacity region

Q. Since that if a is inherently not feasible, we cannot expect to find any policy to meet

those demands and the only solution is to increase the network's information-theoretic

coding, or utilizing MIMO techniques with multiple antennas. Nevertheless, in this

work, we restrict ourselves to the specific question of how to find a simple yet optimal

policy for the network if such requirements are indeed theoretically attainable. We

emphasize that, however, the answer to this question is by no means straightforward
due to the intractability of quantifying the underlying network capacity region. Moreover,
time varying channel conditions and stochastic exogenous arrivals make the problem

even more challenging. It is our main objective to develop an optimum policy without
knowing the network capacity region and the statistical characteristics of random arrival

processes as well as time varying channels. We will formulate the QoS-aware revenue
maximization problem next.
2.3.2 Problem Formulation

Denote the queue backlog of node n for flow c as Qc(t). Note that Q (,) 0 since
whenever a packet reaches the destination, it is considered as leaving the network. The

queue updating dynamic of Qc(t) is given as follows. Forj = 1, Kc 1, we have

Qco)(t + 1) < [QC()(t) o(t)] c + ),(t) (2-4)

and forj = 0,

Qgo)(t + 1) = [QCQ)(t) It),c(t)] + R(t) (2-5)

where [x]+ denotes max(x, 0) and pnc((t), it(t) represent the allocated data rate of the

incoming link and the outgoing link of node n, by the scheduling algorithm, with respect

to flow c. Note that (2-4) is an inequality since the previous hop node may have less
packets to transmit than the allocated data rate p'c)(t).
During time slot t, pc(t) is charged for flow c as the per unit flow price. The
functionality of the price is not only to control the admitted flows, but also, more

importantly, to build up a system-wide revenue from the network's perspective. We

further assume that each flow is associated with a particular user and thus we will
use flow and user interchangeably. Every user c is assumed to have a concave,
differentiable utility function Oc(Rc(t)) which reflects the degree of satisfaction by
transmitting with data rate Rc(t). At time slot t, user c selects a data rate which

optimizes the net income, a.k.a., surplus, i.e.,

Rc(t) = argmax, (Oc(r)- r x p(t)) Vc = 1,- C. (2-6)

Without loss of generality, we assume that

Oc(r) = log(1 +r). (2-7)

However, we stress that the following analysis can be extended to other heterogeneous

forms of utility functions straightforwardly. Note that the fairness issue of multiple flows

can be solved by choosing utility functions properly. For example, a utility function

of log(r) represents the proportional fairness among competitive flows. For more

discussions, refer to [4] and [10].

From the network administrator's perspective, the overall network-wide revenue

is the target to be maximized. Meanwhile, the stability of the network as well as the

QoS requirements from multimedia flows need to be addressed. Formally speaking, the

objective of the network is to find an optimal policy to

QoS-Aware Revenue Maximization Problem:

t-1
maximize D = lim inf-1 Y () (2-8)
t-*oo t
r=0

s.t.

(a) the network is stable,

(b) the minimum data rate requirements, i.e., a, are satisfied,

(c) the guaranteed maximum end-to-end delays for multiple multimedia flows are
prioritized according to the service levels of t,

where

O(t) = E( Rc(t)pc(t)) (2-9)
C
is the expected overall network revenue during time slot t, with respected to the

randomness of arrival processes and channel variations.

Reduced Feasible Region

a2

Figure 2-2. A conceptual example of the network capacity region with two multimedia
flows. The minimum data rate requirements reduce the feasible region of the
optimum solution

However, the QoS-aware revenue maximization problem is inherently challenging

due to unawareness of future random arrivals and stochastic time varying channel

conditions. Even worse yet, the QoS requirements significantly complicate the problem,

as illustrated in Figure 2-2. In unconstrained cases, the feasible region of the optimum

solution A* is essentially the whole network capacity region. However, as shown in

Figure 2-2, the original optimum solution may not even be feasible under the minimum

data rate requirements. Besides, finding a policy which achieves the new constrained

optimum solution, denoted by X*, is a nontrivial task as well. In addition, due to the

multi-hop nature of wireless transmissions, decentralized solutions with low complexity

are remarkably favorable. To address the aforementioned concerns, in next section,

we will propose an optimal online policy, namely, QADP, which provably generates a

network revenue that is arbitrarily close to the optimum solution of (2-8). Meanwhile, the

imposed requirements of the problem, i.e., (a), (b) and (c), are achieved on the fly.

In the following, we first assume that the arrival rate vector A lies outside of the

capacity region Q for all time slots, i.e., a heave traffic scenario.

2.4 QoS-Aware Dynamic Pricing (QADP) Algorithm

In this section, we propose an online policy, i.e., QADP algorithm, which solves the

QoS-aware revenue maximization problem in (2-8).

We first introduce some system-wide parameters to facilitate our analysis. Define
Rmax as the upper bound of admitted traffic of flow c during one time slot, i.e., Rc(t) <

Rcax, Vc, t. For example, R cx can represent the hardware limitation on the maximum
volume of traffic that a node can admit during one time slot, or simply the peak arrival

rate within one time slot if such information is available. Let pmax be the maximum data

rate on any link of the network, which may be determined by factors such as the number

of antennas, modulation schemes and coding policies. In addition, for each flow c, we

introduce a virtual queue Yc(t) which is initially empty, and the queue updating dynamic

is defined as

Y,(t + 1) = [Y(t) R(t)]+ +a Vc. (2-10)

Note that virtual queues are easy to implement. For example, the source node of flow

c, i.e., c(0), can maintain a software based counter to measure the backlog updates of

virtual queue Yc(t). In addition, for each flow c, we define

1 2
S= N(pmax)2 + (Rcax)2 ( ac) Vc. (2-11)
2

Denote 01, Oc as the weights which will be calculated and assigned to all flows,

where C denotes the number of flows in the network. Let J be a tunable5 positive large

number determined by the network. In addition, we assume a maximum value of the

allocated weight, denoted by Omax, i.e., Oc < Oax, Vc. The proposed QADP algorithm is

given as follows.

Part I: Weight Assignment
For all multimedia transmissions, find the flow with the minimum value of ac x
fc, c = 1, C, say, flowj. For each flow c, assign an associated weight, denoted

5 The impact of J on the performance of QADP algorithm will be clarified shortly.

by Oc, which is calculated by

0max x ;x
OC = 0x Vc= 1, ... ,C. (2-12)
Oac X c

Part II: Dynamic Pricing
For every time slot t, the source node of flow c, i.e., c(O), measures the value of
Qc(0)(t) and Yc(t). If Q(,0)(t) > YC(t), the instantaneous admission price is set as

ec (cW)(t Y(t))
pc(t) = (2-13)

and pc(t) = 0 otherwise.

Part III: Scheduling
For every time slot t, find a link schedule I*(t), from the feasible set Ts(t), which
solves
max ab(t a,b (2-14)
I(t)ETs( (a,b)GE

where
a,b = max (Oc(Qf(t)- Qb(t))) (2-15)
c:(a,b)ePc
if 3c, such that (a, b) e Pc, and a,b = 0 otherwise.

END

It is worth noting that Part I of QADP can be precalculated before actual transmissions.

The weight assignment can be implemented either by the network controller which

knows the QoS requirements of all flows, or by mutual information exchanges among

multiple multimedia flows, in a decentralized fashion. The value of Oc represents the

QoS-wise "importance" of flow c and remains unchanged unless the QoS requirements

from flows, i.e., (a, t), are updated, by which a new weight calculation is triggered.

The dynamic pricing part is the crucial component of QADP By following (2-13),

not only the incoming admitted rates can be regulated effectively, but also the overall

average network revenue can be maximized, as will be shown shortly. Note that after

the weight assignment, in order to compute pc(t), the source node of flow c, i.e., c(0),

which is considered as the edge node of the network, requires only local information,

i.e., current backlogs of the source data queue and the virtual queue. It is interesting

to observe that if Q (o)(t) < Yc(t), the admission is free! Intuitively, a small value of
Qg(0)(t) indicates a deficient arrival rate of flow c. On the contrary, a large value of Yc(t)

means that the average "service rate" is less than the average "arrival rate" and thus

the virtual queue is building up. Note in (2-10) that this indicates that the average of

Rc(t) falls below the arrival rate, i.e., ac. Therefore, when Q(,0)(t) < Yc(t), the network
provides free admission to encourage more incoming packets in order to satisfy the QoS

constraints. We will make this intuition precise and rigorous in the following section.

The third part of QADP is a weighted extension to the well-known MaxWeight

scheduling algorithm [12, 13, 34]. Instead of the exact difference of queue backlogs,

we deliberately select the weighted difference of queue backlogs as the weight of a

particular link in the scheduling algorithm. Intuitively, if a flow is assigned with a larger

value of Oc, the links associated with it will have a higher possibility of being selected

for transmissions by QADP. Therefore, by assigning proper values of Oc to flows with

different QoS requirements, a service differentiation can be achieved which provides

more flexibility to previous schemes in the literature, e.g., [13, 15, 16]. In addition,

as indicated by (2-13), a higher priority needs to pay at a higher price. Note that to

calculate (2-14), QADP needs to solve a complex optimization problem which requires

a global information on channel states, i.e., S(t). However, availed of the prosperous

development of distributed scheduling schemes, such as [35-38], the difficulty of

centralized computation can be circumvented, which provides QADP algorithm the

amenability for decentralized implementations.

2.5 Performance Analysis

In this section, we provide the main result on the performance of QADP algorithm.

Theorem 2.1. Define D* as the optimum solution of (2-8). For QADP algorithm, we
have

(a) Revenue Maximization
1 t-1
lim inf O(T) > D* (2-16)
t-+oo t J
=0
where K is a constant and is given by

K = Oc6c (2-17)
C

and 6c is defined in (2-11).

(b) Network Stability
The network is stable under QADP algorithm, i.e., for every queue in the
network, (2-3) is satisfied.

(c) QoS Provisioning
By following QADP algorithm, any feasible minimum data rate requirements a
can be satisfied. In addition, the guaranteed maximum average end-to-end delays
for multimedia flows with service level j arej times larger than that of level one
transmissions.

It is of great importance to observe that in (2-16), the achieved performance of

QADP algorithm can be pushed arbitrarily close to the optimum solution D* by selecting

a sufficiently large value of J. The proof of Theorem 2.1 is provided in the rest of this

section.

2.5.1 Proof of Revenue Maximization

Recall that the minimum rate vector a is assumed to lie inside the capacity region

Q. Therefore, there exists a small positive number > 0 such that a + 1 e Q where 1 is

a unity vector with dimension C.

Lemma 1. For any feasible input rate vector, there exists a stationary6 randomized

policy, denoted by RAND, which generates

E (p / c Pn WO(t)) = 0 Vn, c, t (2-18)

6 Stationary means that the probabilistic structure of the randomized policy does not
change with different values of queue backlogs.

and

E(()(t))> ac+ Vt, c (2-19)

where O, (t) is the exogenous arrival on node n for flow c during time slot t.

Proof. (Sketch) The proof of Lemma 1 utilizes standard techniques as in [13, 15, 16].

The basic idea is to reduce the exponentially large dimension of extreme points to

a finite set with dimension E + 1 by Caratheodory's Theorem. Then a randomized link

schedule selection is implemented among all reduced E+1 extreme points. The detailed

proof is omitted.

We emphasize that Lemma 1 is only an existence proof in the sense that the

randomized algorithm RAND cannot be implemented in practice. This is because

that RAND requires a prior knowledge on the network capacity region, i.e., the

underlying steady state distribution of Markovian channels, and hence is computationally

prohibitive. However, the existence of RAND plays a crucial role for the performance

analysis of QADP algorithm, which, in contrast, is an online adaptive policy and does not

require the statistical characteristics of the stochastic arrivals and time varying channel

conditions.

Recall that a virtual queue Yc(t) is introduced for every flow c and the queue

updating dynamic is given by (2-10). As a result, the minimum data rate requirement

is converted to a queue stability problem since if the virtual queue Yc(t) is stable,

the average service rate, i.e., the time average of Rc(t), needs to be greater than the

average arrival rate, i.e., ac. Define Z(t) = [Q(t); Y(t)] as all the real data queues and

virtual queues at time slot t. If we can ensure that the network is stable with respect to

Z(t), the backlogs of real queues are bounded and the minimum rate requirements are

achieved at the same time.

Define a system-wide potential function (PF) as

(2-20)

PF(Z(t))= PFC(Z(t))
C

where

PFC(Z(t)) = ( c(Q()2
2 n

Oc(YCt))2)

Note that PF(Z(t)) is a scalar-valued nonnegative function. Define

A(Z(t)) = E(PF(Z(t

1)) PF(Z(t))Z(t))

as the drift of the potential function PF(Z(t)).
For flow c, we take the square of both sides of (2-4) and (2-5) and obtain7

(Qo) (t + 1))2 < (Q )()2

- (/c),c(t)) 2Qc2 (t) (/t

),c( Q

P ),())

forj = 1, ,K 1, and

(o)(t -+ 1))2 < (Q )(t))2

Rc(t))

forj = 0. Similarly, for (2-10), we have

(Ye(t + 1))2 < ( Y(t))2 + (Rc(t))2 + (ac)2

ac).

7 We use the fact that for nonnegative real numbers a, b, c, d, if a < [b
a2 < b2 + c2 + d2 2b(c d), as given in Lemma 4.3 on [17].

(2-21)

(2-22)

(2-23)

(2-24)

d, then

(p( ),c(t))

(Rc(t))2 2 Q(c)(t) (pL),c(t)

2 Yc(t)(R(t) -

By combining the inequalities above, we have

PF(Z(t + 1)) PF(Z(t))

< E- O+ 0 Q ((t)R(t) Oc Yc(t)( Rc(t) a)

SOQ(t) (f(t) p ,(t)) (2-25)
n
where

=c = 0 N(Ipmax)2 (Rmax)2 + (ac)2. (2-26)

Note that (2-25) is summed over the whole network. If node n is not on the path of

flow c, pc(t) = (t) = 0. Moreover, pn(t) = 0 for the source node of flow c and

p~ (t) = 0 for the destination node of flow c. Next, we sum over all flows to derive the
network-wide potential function difference as

PF(Z(t + 1)) PF(Z(t))

< K B Ot)_ n,c(t))
n,c
S Q Yc(t)R(t) (t)(R(t) a)
C C

where K = c Ec. Therefore, we have

A(Z(t)) < K- OcQ(t)E ( o(t) p-inc(t) Z(t))
n,c
+ YOCQW(o)(t)E(Rc(t)|Z(t))
C
OcYc(t)E(Rc(t) aZ(t)). (2-27)
C

Next, for a positive constant J, we subtract both sides of (2-27) by JE (Yc Rc(t)pc(t)lZ(t))

and have

A(Z(t))- JE( R(t)p,(t)|Z(t)

< K OC Q(t)E (po (t) C(t)l Z(t))
n,c
+YOCQco)(t)E(Rc(t) Z(t))
C
Oc Y(t)E(Rc(t) cZ(t))
C

-JE Rc(t)pc(t)|Z(t). (2-28)

Note that (2-28) is general and holds for any possible policy.

For an arbitrarily small positive constant 0 < e < emax define the e-reduced network

capacity region, Qe, as all possible input rate vectors such that

Q, = {Alc + e Vc} (2-29)

where Q is the original network capacity region. We will discuss about how to obtain emax

shortly.

Define D* as the optimum value of the reduced problem to (2-8) where Q is

replaced by Qe. It can be verified that [14]

lim D* -> D* (2-30)

where D* is the optimum value of the original revenue maximization problem in (2-8),

i.e., the target of QADP algorithm.

Specifically, we denote r, (0), r,(1), .. r*(t), and p*,c(0), p,c(1), p,c(t),..

as the optimum sequences of admitted rates and prices, for flow c, which achieve D*.

Define ?E as the time average of the optimum sequence of r,(0), r,(1), rc(t), .

Therefore, following the definition of (2-29), we have r- + c E Q. By Lemma 1, we claim

that there exists a randomized policy, denoted by RAND, which yields

E(,p '(t) -n(t) (t)) = Vc,n =c(0) (2-31)

and

E( (t) n,(t)) = Vc, n z c(O) (2-32)

and

E(rc(t) + ) > ac + Vc. (2-33)

Denote the RHS of (2-28) as Without loss of generality, we assume e < c. Therefore,
for randomized policy RAND, we have

VRAND K-e Oc O(t)+ OCY(t)
n,c C

-JE (rc(t) + )P(tlZ(t) (2-34)
( C

where p ,c(t) is the corresponding price which induces a rate of rc(t) + by (2-6).
Lemma 2. QADP algorithm minimizes the RHS of (2-28) over all possible policies.

Proof. The scheduling part of QADP algorithm in Section 2.4 is a weighted version of
(G not) -- (t))
MaxWeight algorithm [12, 13, 34] which maximizes En,c OcQ(t)( () c(t))

Z(a,b)E la,b(t)ba,b if 3c, such that (a, b) e Pc, for every time slot t. The dynamic pricing
part of QADP is essentially maximizing

S(Oc(Yc(t)- Q (o)(t))Rc(t) +JRc(t)pc(t)) (2-35)
C
for every time slot. By (2-6) and (2-7), we see that QADP finds an optimum price p*(t)
which maximizes

1
M = O(Yc(t) Q~0)(t))( 1) + J( p(t)). (2-36)
(D Wt

Define W = Oc(Qg(O)(t)- YC(t)).

Case 1: If W > 0, we have

M = J + W- (Jp(t)

1
W ( ).
Pc M

Take the first order derivative, we have

V V J
M' = J.
(Pc(t))2

The second order derivative brings us

W
(Pc(t))3

<0.

Therefore, M is a concave function and the optimum value is achieved by

(t) W

Case 2: If W < 0, M is a decreasing function with respect to pc(t). Therefore, p*(t)
0.

Following Lemma 2, we conclude that for QADP algorithm,

< WRAND < K-

-JE ( ,c(t) -
C c

Oc QC)(t) .OCYC(t)
\n,c C

0e)p (*)|Z(t) .

(2-41)

We take expectation with the distribution of Z(t), on both sides of (2-41). By the fact

that Ey(E(X Y))

E(X), we have

E(PF(Z(t + 1)))- E(PF(Z(t)))

SK+ JO(t) E c(-
\ nc

SOCYCt))

-JE ( (r'c(t)
C c

(2-37)

(2-38)

(2-39)

(2-40)

e)p. (t)

(2-42)

Note that (2-42) is satisfied for all time slots. Therefore, we take a sum on time slots

-r = 0,... T -1 and have

E(PF(Z(T)))

E(PF(Z(0))) < TK

T-1
S ~cE ( OCQ,(,T) +
r=0 n,c c

T-1O \
(243) by T and( ) rearrange terms to obtain
T=0 c
Divide (2-43) by T and rearrange terms to obtain

T-1

T=0

C Y (T))

4 )

(2-43)

(2-44)

where the nonnegativity of the potential function is utilized. We first take lim infT- on

both sides of (2-44) and have8

1
liminf E T- O CT)
T-oo T -' \ En
Tn )\ n r

ST-l
lim inf T- JE
T--oo T
TO

( ( ( (T)
,c "7

- 0C YC(T,))

- C)P T)

(2-45)

T-1
< K lim inf 1 JO(T).
S T-oo T
T-0

8 We assume that the initial queue sizes of real data queues and virtual queues are
bounded.

T-1

TT=0
1 T-1 E(PF(Z(O)))
_ K T JO(Y ) T
=-0

Note that the first term of (2-45) is nonnegative and thus we have

lim inf O(T)

ST-1
> nf E Y (rc (T) -+ 0p ) .
7 T T=0 ( C

Recall that the analysis above applies to arbitrarily small 0 < e < emax. We observe that

I T-1
lim lim infT E ( lT-) + 0)P c(( ) D*. (2-46)
c-+O T-+ T>
7- T=0 C

Therefore, by (2-30), taking the limit of e 0 yields the performance bound of (2-16) in

Theorem 2.1.

2.5.2 Proof of Network Stability

To prove the stability of the network, we take lim sup on (2-44) and obtain

1T-1
lim sup 1E I OQ(r) + OCYC())
T-+oo T
TO n,c c

< K J lim sup 1 O() (2-47)
( T-+ T-O 1

since the second term of (2-44) is also nonnegative. Note that if it satisfies that O(t) <

Omax for all time slot t, (2-47) becomes

T-1
lim sup 1 $$E O CQC() S OYC)Y(7) T-_ o T nc c < K JOmax (2-48) Recall that the above analysis applies to any 0 < e < max. In addition, by the definition of (2-29), we have max P max maxa where = 1, ... C (2-49) I where pmax is the maximum possible data rate on a link. Moreover, by (2-6) and (2-7), we have Omax = C where C is the number of flows in the network. Therefore, we have lim supI T-E C QC(\ Cy($$ K +JC
T-oo T-0 n,

Using the fact that

X(t) < Y(t) Vt = lim supX(t) < lim sup Y(t), (2-50)

we have, for every flow c, the average queue length on its routing path is bounded by

1 T K JC
lim sup E ( QC(J ()) < max (2-51)
T-+oo 7-0 max

Finally, by applying Markov Inequality, we conclude that all data queues in the network

are stable, based on the fact that the RHS of (2-51) is bounded.

2.5.3 Proof of QoS Provisioning

Similar to (2-51), for every virtual queue Yc(t), we can obtain

T-1
1 K + JC
lim supT E(Y,(T)) < KJCx (2-52)
T-oo T 0^ -
r-=0

Therefore, by similar analysis and the definition of virtual queues, we conclude that the

virtual queues are stable and thus the minimum data rate requirements imposed by

multimedia flows are achieved.

Next, we show that QADP indeed provides a service differentiation solution on the

guaranteed maximum end-to-end delays for all multimedia flows. Denote the actual

experienced average delay of flow c as w~. By Little's Law, wc is approximated9 by

9 Note that we consider a heavy loaded network. Propagation delays are assumed to
be negligible compared to queueing delays and thus are omitted.

average overall queue length on the path of flow c
C average incoming rate of flow c
lim supT, E ( Q ( o ))
MU (2-53)
lim supT, yT- 1 E(Rc(T))

In light of the stability of virtual queue Yc(t), we have lim supTX, 1 y- J E(Rc(T)) >

ac. Hence, we can obtain

S< maxJC (2-54)

Equivalently speaking, to differentiate the guaranteed maximum delay bound, we need
to find a set of weights such that

Ocajct = Od dd (2-55)

is satisfied for any pair of multimedia flows c and d. Therefore, it is straightforward
to verify that the weight assignment algorithm in QADP indeed provides a service
differentiation solution where the guaranteed maximum end-to-end delays of multimedia
flows are distinguished according to application-dependent service level requests,
which completes the proof of Theorem 2.1. The performance of QADP algorithm will be

evaluated numerically in Section 2.6.
2.6 Simulations

2.6.1 Single-Hop Wireless Cellular Networks

We first consider a single-hop wireless cellular network with downlink multimedia
transmissions, as shown in Figure 2-3. The base station (BS) is associated with three

users with infinite backlogged traffic. A separate queue is maintained by the base station
for every user. In addition, at each time slot, BS can only transmit to one particular user.
A wireless link is assumed to have three equally possible channel states, i.e., Good,

Medium, Bad. The corresponding transmission rates for three channel states are 20,

15 and 10 bits per slot, respectively. Therefore, the base station encounters a complex

opportunistic scheduling problem where the revenue maximization problem, network

stability as well as the QoS requirements need to be addressed simultaneously.

Base Station
1 2 3

UseUser 2r 3

Figure 2-3. A single-hop wireless cellular network with three users

Without loss of generality, we assume that the minimum average rate requirements

for user 1, 2, 3 are a = [1, 2, 3] bits per slot. In addition, to provide service differentiation,

the network offers three prioritized service levels, e.g., Platinum, Gold and Silver,

where level Platinum possesses the highest priority in terms of end-to-end delay upper

bound. We assume that user 1 has a service level request for Platinum while user

2 and 3 demand for level Gold and Silver, respectively, according to the upper layer

applications. Other system parameters are assumed to be Rlax = 20 bits per slot for

all flows and Omax = 100. We next implement QADP algorithm for different values of J

where J = [50, 100, 500, 1000, 5000, 10000, 20000, 50000, 100000]. Every experiment is

simulated for 500000 time slots.

Figure 2-4 depicts the system revenue, i.e., the solution of (2-8) by QADP

algorithm, with respect to different values of J. As shown in (2-16) and demonstrated

pictorially in Figure 2-4, the achieved system revenue by QADP converges gradually

to the optimum solution as J grows. Note that the values of system revenues are

almost indistinguishable when J > 50000. Figure 2-5 illustrates the actual experienced

average delays of all three flows with different values of J, where the delays of user 1

to user 3 are compared from left to right. It is worth noting that, not only the maximum

guaranteed end-to-end delays are distinguished analytically by QADP, but also the

Impact of Different Values of J

50 100 500 1000 5000 100002000050000100000
Values of J

Figure 2-4. Impact of different values of J on the performance of QADP
Impact of Different Values of J
60
User 1
User 2
50 User3
0 40

40
-u

50 100 500 1000 5000 10000200005000000000
Values of J

Figure 2-5. Impact of different values of J on the average experienced delays

actual experienced average delays are prioritized for all three flows with distinct service

level requests. More specifically, user 1, i.e., the Platinum user, enjoys a delay which

is less than half of that of user 2 and one third of that of user 3, for all values of J, as

demonstrated in Figure 2-5. However, as shown by Figure 2-4 and Figure 2-5 jointly,

while a larger value of J yields an improvement on the performance of QADP, the

end-to-end delays of all three flows are augmented concurrently. Therefore, by tuning

J, a tradeoff between optimality and average delays can be achieved. The time average

admitted rates of multimedia flows are shown in Table 2-1. We observe that in all cases,

the average rates of flows exceed the minimum data rate requirements specified by a.

The sample paths of price adaptations and queue backlog evolutions are illustrated

in Figure 2-7 and Figure 2-6, respectively, for the first 100 time slots with J = 50000.

Table 2-1. Average admitted rates for multimedia flows
Flow 1 Flow 2 Flow 3
J = 50 4.01 2.71 3.07
J = 100 4.04 2.72 3.09
J = 500 4.05 2.91 3.11
J = 1000 4.31 3.07 3.16
J = 5000 4.51 3.57 3.42
J= 10000 4.63 3.94 3.62
J = 20000 4.75 4.28 4.09
J = 50000 5.09 4.71 4.61
J = 100000 5.13 4.88 4.85

Note that, in Figure 2-7, the prices imposed for three users are dynamically adjusted

at every time slot. It is worth noting that whenever a data queue in Figure 2-6 has a

tendency to build up, the price imposed by QADP algorithm, as shown in Figure 2-7,

rises correspondingly, which in turn discourages the excessive admitted rate and thus

all queues in the network remain bounded. As a result, the stability of the network is

achieved.

2.6.2 Multi-Hop Wireless Networks

We next consider a multi-hop wireless network with a topology shown in Figure

2-1. There are three multimedia flows exist in the network, denoted by Flow 1, 2, 3.

The routing paths of flows are specified by P1 = {A, B, C, D}, P2 = {F, G, C, D} and

P3 = {E, F, G, H}. Without loss of generality, we assume a two-hop interference model

which represents the general IEEE 802.11 MAC protocols [36, 37]. Other configurations

are the same as the single-hop scenario described above except that the possible

link rates are assumed to be 40, 30, 20 bits per slot for three channel conditions. We

observe that in this network topology, link C -> D and link F -> G are shared by two

different flows. Therefore, in the scheduling part of QADP, the particular flow with a

larger weighted queue backlog difference should be selected.

In Figure 2-8, we specifically depict the dynamics of three virtual queues for the

first 400 time slots with J = 50000. Unlike the single-hop case in Figure 2-6, the virtual

Figure 2-6. Queue backlog dynamics for all users

0.7
-e- price for user 1
0.6 price for user
price for user 3

S0.3
0.2 ii

0 20 40 60 80 100
Time Slots

Figure 2-7. Price dynamics in QADP for all users

50
-4-Queue 1
-Virtual Queue 2
40 ...0o Queue 2
Virtual Queue 3
Queue3
-'-Virtual Queue 1
| 30 0
C O
I 20- 0: 00 0o

0 20 40 60 80 100
Time Slots

queues behave remarkably different in this multi-hop scenario. It is worth noting that

while the virtual queues of user 1 and 3 have relatively low occupancies, the virtual

queue associated with user 2 suffers a larger average backlog. Intuitively, due to the

underlying two-hop interference model, link G -> C needs to be scheduled exclusively

in the network for successful transmissions. In other words, link G -> C is the bottleneck

of the network. Therefore, to ensure network-wide stability, a much more stringent

regulation is enforced on the admitted rate of flow 2. As a consequence, although

remains bounded, the virtual queue of flow 2 accumulates more backlogs compared

to other competitive flows. In addition, we compare the time average queue backlogs

of all data queues in the network, from left to right, in Figure 2-9. We can observe that

the data queues on the path of flow 1 have fewer average backlogs due to the highest

20
>15
10

0 50 100 150 200 250 300 350 400
Time Slots

Figure 2-8. Virtual queue backlog updates in QADP for J = 50000

70

60 Q1
20 50 I
40 4F

IN'

1CII 0;!
Queues

Figure 2-9. Average queue backlogs in the network for J = 50000

priority with respect to the service level, i.e., Platinum. On the contrary, the queues on

the path of flow 3 have larger backlogs compared to other two flows. It is noticeable that

Q3 and Qj have considerably larger average queue sizes. This is because that QF has

to share the link rate of F -> G with Q2. Nevertheless, Q3 possesses a smaller share of

bandwidth than Q2 due to the lower prioritized service level associated with flow 3. Even

worse yet, both link F -> G and G -> H have less opportunity to be scheduled due to

their locations and the underlying interference model. Therefore, the average backlogs

on the path of user 3 has higher occupancies compared to other two competitive flows.

The average data rates are provided in Table 2-2. Note that the minimum average rate

requirements of all multimedia flows are satisfied simultaneously, as expected. The

tradeoff between optimality and average delay, which is controlled by different values of

J, as well as the network service differentiation in terms of delays are analogous to the

single-hop scenario discussed above. Duplicated simulation figures are omitted.

Table 2-2. Average admitted rates for multimedia flows
Flow 1 Flow 2 Flow 3
J = 50 3.04 2.08 3.88
J = 100 3.12 2.03 3.98
J = 500 3.17 2.04 4.60
J = 1000 3.30 2.03 5.04
J = 5000 4.13 2.01 6.45
J = 10000 4.36 2.02 7.32
J = 20000 5.08 2.03 8.12
J = 50000 6.43 2.01 9.18
J= 100000 7.63 2.01 9.89

2.7 Conclusions

We consider a multi-hop wireless network where multiple QoS-specific multimedia

flows share the network resource jointly. To maximize the overall network revenue,

we propose a dynamic pricing based algorithm, namely, QADP, which achieves a

solution that is arbitrarily close to the optimum, subject to network stability. Moreover,

a weight assignment mechanism is introduced to address the service differentiation

issue for multimedia flows with different delay priorities. In tandem with the virtual

queue technique which provides minimum average data rate guarantees, the QoS

requirements of multiple multimedia streams are addressed effectively.

In this work, the application-dependent service level requests and the users'

utility functions are assumed to be attained truthfully. The mechanism design for

strategy-proof user information acquisition seems interesting and needs further

investigation. Moreover, in this work, we assume that the information of channel states,

i.e., S(t), is available for QADP algorithm, which is either acquired by the central base

station or approximated by the distributed local scheduling algorithms. As a future

work, an online channel probing mechanism needs to be incorporated into QADP, as

suggested in [39].

CHAPTER 3
ENERGY-CONSERVING SCHEDULING IN WIRELESS NETWORKS

3.1 Introduction

There has been a lot of interest over the past few years in characterizing the

network capacity region as well as designing efficient scheduling algorithms in multi-hop

wireless networks. Due to the stochastic traffic arrivals and time-varying channel

conditions, supporting high throughput and high quality communications in multi-hop

wireless networks is inherently challenging. To utilize the scarce wireless bandwidth

resource effectively, scheduling algorithms which can dynamically allocate the network

resource, i.e., select active links, are investigated intensively in the community. For

example, MaxWeight algorithm, a.k.a., back-pressure algorithm, has been extensively

studied in the literature, e.g., [5, 14, 34, 37, 40], following the seminal work of [12]. The

MaxWeight algorithm enjoys the merit of self-adaptability due to its online nature. In

addition, MaxWeight algorithm is known to be throughput optimal [17]. That is to say,

the MaxWeight algorithm can stabilize the network under arbitrary traffic load that can

be stabilized by any other possible scheduling algorithms. Therefore, the MaxWeight

algorithm attracts significant attention and becomes an indispensable component for link

scheduling in network protocol designs, e.g., [41-43].

While the throughput-optimality of the MaxWeight algorithm is well understood,

the energy consumption induced by the MaxWeight algorithm is less studied in the

literature. However, due to the scarcity of energy supplies in wireless nodes, it is

imperative to study the energy consumption of the scheduling algorithm which is of

special interest in energy-constrained wireless networks such as wireless sensor

networks. Is the throughput optimal MaxWeight scheduling algorithm also energy

optimal? In this work, we show that the answer to this question is no. The reason is that

the vast energy consumption during packet retransmissions are completely neglected

by the MaxWeight algorithm. For example, in [16], an energy optimal control scheme

is proposed where a minimum power expenditure is achieved. However, as in other

related works, e.g., [17], the wireless channels in [16] are assumed to be error-free, i.e.,

all the transmissions are assumed to be successful. Nevertheless, in practice, wireless

channels are error-prone and data transmissions are subject to random failures due to

the hostile channel conditions. Therefore, before a packet can be successfully removed

from the transmitter's queue, several transmissions may have occurred, including the

original attempt and the posterior retransmissions, which deplete a significant amount

of energy for the transmitter. However, in the traditional MaxWeight algorithm, such

energy-consuming retransmissions induced by channel errors are overlooked. Intuitively,

from energy-saving perspective, the possibility that a particular link is selected for

transmissions should rely on not only the queue difference between the transmitter

and the receiver, which is the design rationale of the traditional MaxWeight algorithm,

but also the potential energy consumption of retransmissions induced by erroneous

channels. We will make this intuition precise and rigorous in the following sections.

In this work, we propose a minimum energy scheduling (MES) algorithm which

consumes an amount of energy that can be pushed arbitrarily close to the global

minimum solution. In addition, the energy efficiency attained by the MES algorithm

incurs no loss of throughput-optimality. The proposed MES algorithm significantly

reduces the overall energy consumption compared to the traditional MaxWeight

algorithm and remains throughput optimal. Therefore, our proposed MES algorithm is

more favorable for dynamic link scheduling and network protocol designs in energy-constrained

wireless networks such as wireless sensor networks.

The rest of this chapter is organized as follows. Section 3.2 describes the system

model used in this work. The proposed MES algorithm is introduced in Section 3.3

where the performance analysis is provided. Simulation results are shown in Section 3.4

and Section 3.5 concludes this chapter.

3.2 System Model

We consider a static multi-hop wireless network denoted by a directed graph (A/, C)
where A is the set of nodes and C denotes the set of links in the network. We use |X|
to represent the cardinality of set X. Time is slotted by t = 0, 1, 2, .. and in every time

slot, the instantaneous channel state of a link (a, b) C is denoted by Sa,b(t) where a
and b are the transmitter and the receiver of the link. In this work, we use S(t) to denote

the channel state vector of the whole network. Note that S(t) remains constant within

one time slot, however, it is subject to changes on time slot boundaries. We assume that
S(t) has a finite but potentially large number of possible values and evolves following an

irreducible Markovian chain with well defined steady state distributions. Nevertheless,

the steady state distribution and the transition probabilities are unknown to the network.
Given an instantaneous channel state Sa,b(t), the transmission of a packet on link (a, b)

is successful with a probability of pa,b(t), if link (a, b) is active and suffers no interference
from concurrent transmissions. From the network's perspective, at each time slot t, an

interference-free link schedule, denoted by I(t), is selected from a feasible set Q(t),

which is constrained by the underlying interference model, e.g., K-hop interference
model [44], as well as other limitations such as duplex constraints and peak power
limitations. Denote Ub(t) as the nominal link rate if link (a, b) is selected by the network
and the channel is error-free, i.e., pa,b(t) = 1. The actual data rate of link (a, b) is hence

represented by Ua,b(t) = b(t)Pa,b(t). We assume that Ua,b(t) is upper bounded by

a constant umax for all (a, b) e In practice, umax can be determined by the number
of antennas equipped in a single node as well as the coding/modulation schemes

available to the network. We denote u(t) as the link rate vector of the network at time t.

Apparently, u(t) is a function of both I(t) and S(t).
The network consists of |t| flows indexed by f = 1, 2, .. |T. Each flow f is

associated with a routing path Rf = [no, n, -. nf] where nf,j = 0, f denotes
the nodes on the path of flow f. At each time slot t, the stochastic exogenous arrivals

of flow f, i.e., the number of new packets that are initiated by node no, is denoted by

Af(t). For the ease of exposition, we assume that Af(t) is i.i.d. for every time slot with

an average rate of Af. In addition, the arrival processes of all flows are assumed to be

independent. We further assume that the maximum number of new packets generated

by a flow during one time slot is upper bounded, i.e., Af(t) < Amax, Vf, t. We emphasize

that the i.i.d. assumption incurs no loss of generality and our model can be extended to

cases where Af(t) is non-stationary in a straightforward fashion, as in [14].

Every node in the network maintains a separate queue for each flow that passes

through it. Denote Q (t) as the queue backlog at time t, for node n, where f is one of

the flows that traverses through n, i.e., ne cRf. We assume that Qf(t) = 0, Vt if n = n.

That is to say, if a packet reaches the destination, we consider the packet as leaving the

network immediately. Define the overflow function of Qf as
1 t-1
O(M) = l sup 1 Pr(Qj'() > M). (3-1)

The queue is stable if [14]

lim O(M) 0 (3-2)
M--oo

and the network is stable if all individual queues are stable concurrently.

For notation brevity, we define the network admission rate vector as A = {An,f, Vn, f}

where Anf is the average exogenous arrival rate of flow f on queue1 Q,. We have

Anf = Af if n = no and An,f = 0 otherwise. Denote A as the network capacity region

[12], namely, the set of all feasible admission rate vectors, i.e., A, that the network

can support, in the sense that there exists a scheduling algorithm which stabilizes the

network under traffic load A. It is shown in [12] and [14] that A is convex, closed and

bounded.

1 Note that with a slight abuse of notation, we use Qf to denote both the queue itself
and the number of packets in the queue.

Without loss of generality, we consider an energy consumption model as follows.
Recall that the successful transmission probability of link (a, b) is pa,b(t). Therefore,
for a particular packet to be transmitted from a to b, on average, a number of '
P,,b(t)
transmissions are needed in order to "erase" this packet from the queue of node a.
From the transmitter's perspective, however, every transmission, either the original
attempt or retransmissions, costs the same amount of energy. Denote aa,b as the energy
needed to transmit a packet on link (a, b). For example, aa,b can be proportional to the
distance between node a and b. Therefore, in order to transmit a packet successfully
on (a, b), node a needs to spend a total energy of on average. For the receiver b,
o r(t)e. For the receiver b,
we assume that the energy depletion on successful packets receptions are dominant,
i.e., the energy spent for overhearing and short ACK messages are neglected. Denote

3a,b as the energy consumed for a successful packet reception in demodulation and
decoding on node b. With a data rate2 of Ua,b(t) on link (a, b), the overall energy spent
during time slot t is given by

Ga,b(t) = Ua,b(t) Pa,bt a,b
\Pa,b(t) 0
= ,b(t) (a,,b + Pa,bPa,b(t)). (3-3)

Note that due to the stochastic nature of wireless channels, Ga,b is a random variable.
We stress that the simple energy consumption model above is not essential and our
analysis can be extended to other more complex forms of energy models straightforwardly,
as will be shown in the next section.

2 The unit of data rate in this work is defined as packets/slot. It is worth noting that
other units such as bits/slot are also applicable.

In addition, we assume that the energy consumption of the whole network during

one single time slot is upper bounded, i.e.,

SGa,b(t) < Gmax, Vt. (3-4)
(a,b)GC

Therefore, to minimize the energy consumption, the objective of the network is to

find a scheduling algorithm which solves

Minimum Energy Scheduling Problem:

T-1
minimize C = limsup G (t) (3-5)
T-*oo T 't
t=0
s.t.

the network remains stable, and

G(t)= E (a. hab(t) Pab) +ab (3-6)

is the expected overall network energy consumption during time slot t, with respect

to the randomness of arrival processes and channel variations. Note that ha,b(t) is the

actual number of successfully transmitted packets on link (a, b) during time slot t and3

ha,b(t) < Ua,b(t)

In the next section, we will propose a minimum energy scheduling (MES) algorithm

which minimizes the average network energy consumption asymptotically subject

to network stability. In addition, the proposed MES algorithm is throughput optimal,

in the sense that the MES algorithm can ensure the network stability for all feasible

network admission rate vectors in the network capacity region. Restated, the set of

feasible arrival rates supported by the MES algorithm is the superset of all other possible

3 The inequality holds when node a has less packets to transmit than the allocated
data rate Uab(t).

scheduling algorithms, including those with a priori knowledge on the futuristic arrivals

and channel conditions.

3.3 Minimum Energy Scheduling Algorithm

3.3.1 Algorithm Description

The minimum energy scheduling (MES) algorithm is given as follows.

MES ALGORITHM:

At every time slot t:

Every link (a, b) e finds the flow f* which maximizes

max (2Q;(t) Pab(t) 2Q(t)- J-ab (3-7)
f:(a,b)GRf ( Pa,b~t) b

where J is a positive constant which is tunable as a system parameter.

Hab(t) = 2Q(t) Ja,b 2Q(t) Jb (3-8)
Pabt)
where [x]+ denotes max(x, 0).

For the network, a link schedule I*(t) is selected which solves

max Uab(t)Hab(t) (3-9)
I(t)GQ(t)
(a,b) E

END

Note that the proposed MES algorithm is different from the MaxWeight algorithm

proposed in [12, 13]. In the original MaxWeight algorithm, the weight of a particular link

(a, b) is the queue difference between node a and b. However, in the MES algorithm, as

indicated by (3-8), the link weight Hab(t) is related to the potential energy consumption

on this link under the current channel condition as well. More specifically, the link

weight in the MES algorithm is the queue difference subtracted by a weighted energy

consumption factor, i.e., J .b(t + /,b where J represents the weight. Intuitively, if the

current channel is unfavorable, i.e., Pa,b(t) is small, the energy required for a successful

transmission is remarkably large due to the retransmissions and thus the link should

be selected less likely. Therefore, the link weight in the MES algorithm, i.e., Ha,b(t), can

be viewed as a balance between the queue difference and the energy consumption

under the current channel condition. More specifically, in the original MaxWeight

algorithm, if a link has a larger queue difference, the link is more likely to be selected

for transmissions. However, in the MES algorithm, both the queue difference and the

energy expenditure are taken into consideration. In addition, as indicated by (3-7), by

replacing J ((t + -3a,b with other metrics, our MES algorithm can incorporate other

forms of energy consumption models straightforwardly.

Observe that if J = 0, the MES algorithm reduces to the original MaxWeight

algorithm, i.e., the energy consumption during retransmissions are omitted. Therefore,

the MaxWeight algorithm is a special case of our MES algorithm. On the other hand, if

we let J -> oc, the performance of the MES algorithm can be pushed arbitrarily close to

the global minimum solution, as will be shown analytically in Section 3.3.3. However, as

illustrated in (3-7) and (3-8), when J increases, the network becomes more reluctant

to transmissions, for the sake of energy conservation, unless the accumulated queue

backlogs are significantly large. Intuitively, a larger average queue size induces a

longer experienced delay for transmissions. Therefore, the system parameter J is

essentially a control knob which provides a tradeoff between the energy-optimality and

the experienced delay in the network.

Note that similar to the traditional MaxWeight algorithm, the calculation in (3-9)

is centralized. However, following [12], much progress has been made in easing the

computational complexity and deriving decentralized solutions for the centralized

MaxWeight algorithm, e.g., [35-38, 40, 42-49]. It is worth noting that solving (3-9) is

equivalent to finding the maximum weight independent set in the conflict graph, which

is combinatorial in nature and thus is intrinsically difficult to solve. A natural heuristic,

denoted by GreedyMax, is described as follows. First, the link with the highest weight

in the network is selected. Next, the link with the second highest weight which does

not involve conflicts with any previously selected links, is selected and the iteration

continues until no link can be added. Based on GreedyMax, a novel pre-partition

based approach is introduced in [35]. The authors prove that if the topology of the

network satisfies certain conditions, a.k.a., local pooling factor conditions, GreedyMax

can achieve the same throughput as MaxWeight. Furthermore, tree based topology

are shown to satisfy the local pooling conditions. In light of this, [35] utilizes graph

algorithms to partition the whole network into trees where each tree is allocated an

orthogonal channel. As a result, the whole network can operate with simple GreedyMax

algorithm and achieve the same throughput as MaxWeight. This line of research is

further simplified by [36] where the author proves that a local greedy maximal algorithm

can obtain the same performance as the global GreedyMax algorithm. Specifically, in

each time slot, a link only needs to compare its weight with local neighboring links to

decide a feasible transmission schedule. Therefore, in tandem with the tree pre-partition

method in [35], the complex scheduling algorithm in (3-9) can be implemented in a

fully distributed fashion. Another feasible direction is to utilize the distributed random

access approximation schemes, e.g., [37, 38, 42]. For example, in [42], each node in

the network utilizes an IEEE 802.11 MAC protocol where the channel access probability

is dynamically adjusted in accordance to the link weight, i.e., Hb(t) in our scenario.

The effectiveness of such random access based distributed approximations is studied

extensively in [43] and [42] We note that, although distributed implementation is not

the focus of this work, our proposed MES algorithm can be approximated well by the

solutions suggested in the above papers.

If an admission control mechanism is implemented in the network to regulate the

overwhelming arrivals, in order to achieve an average minimum rate provision for each

flow, we can utilize the concept of virtual queues introduced in [17, 34], where for each

flow f, we define a virtual queue yf(t) which is initially empty, i.e.,

Y(0) = 0, Vf (3-10)

and the virtual queue updating dynamic is defined as

y,(t+ 1) = [y(t) R(t)]++ 6 Vf (3-11)

where Rf(t) is the admitted traffic for flow f at time slot t and 6f is a feasible minimum

rate requirement of flow f. Intuitively, if each of the virtual queues in the system is stable,

the average arrival rate should be less than the average departure rate of the virtual

queue and hence we have
T-1
lim Rf(t) > 6f, Vf (3-12)
T-oo T Z- t
t=0
which is exactly the desired minimum rate requirement for flow f. Moreover, the virtual

queues are easy to implement. For example, the source node of flow f can maintain

a software based counter to measure the backlog updates of virtual queue Yf(t).

Therefore, the minimum rate requirements can be incorporated into the proposed

scheme straightforwardly.

3.3.2 Throughput-Optimality

We first show that the proposed MES algorithm is throughput optimal, as given in

the following theorem.

Theorem 3.1. The proposed MES algorithm is throughput optimal, i.e., for an arbitrary

network admission rate vector A which is inside of the network capacity region A, MES

stabilizes the network under A.

Proof. We first provide the queue updating equation of Qf. Note that a packet is

removed from the transmitter's queue if and only if it is received by the receiver

successfully. Therefore, the queue updating dynamic is given by

Q(t + 1) < [Q( ut(t) + u ,(t) + An,(t) (3-13)

where uu(t) and uf(t) are the allocated data rate on the outgoing link and the
incoming link of node n, with respect to flow f. Note that uu(t) = uLt(t)p(tt)

and pout(t) is the current successful transmission probability on this link. Note that

Unt(t) = 0, Vt if node n is the destination node of flow f and un,(t) = 0, Vt if node n is
the source node of flow f. Furthermore, An,f(t) = Af(t) if n is the source node of flow f

and An,f(t) = 0 otherwise.
From (3-13), we have

(Qf(t + 1))2 (Qnf(t))2 < ((umax)2 + (Umax max)2)
-2Q(t) (t(t) (t)- A(t)) (3-14)

Next, we sum (3-14) over the whole network on all data queues and obtain

-(Qfn(t + 1))2 (Qn(t))2
n,f n,f
< B 2 Q(t) ( (t) u(t) An(t)) (3-15)
n,f
where

B = |FITI| ((umax)2 + (max + Amax)2) (3-16)

is a constant.
Denote Q(t) = {Qf(t), Vf, n} as the instantaneous queue backlogs in the network.

We take the conditional expectation with respect to Q(t) on (3-15) and have

E (Qf(t+ 1))2|Q(t) E Y(Qf(t)) Q(t)
n,f / \n,f
< B 2 2 Q/(t)E (ut(t) u,(t) ,n(t)| (t)) .
n,f
Define
t u' out
GmES(t) = E h ot (t) n,f in ) in
G MES {+ \ out{+\ P\nf outn,f t / n, Q\i\
Q = hnE (, otn + h"n.(t)Pn. Q
n~f v ^n~f) / /

where ae (I f,) denotes the energy needed for a packet transmission (reception) on the

outgoing (incoming) link of node n, with respect to flow f.

out
GMES(t) = E out -(t) nt+\ nf n, in ,inf
n,f Ot) + ht n t) nn
n,f Pn.f t)

as the expected network-wide energy consumption during time slot t, by following the

proposed MES algorithm. Apparently, we have

E (GES(t)) = GMES(t).

Next, we add both sides by JGES(t) where J is a positive constant, and have

E 2(Q Q(t 1)) |Q(t) -E n(Q,'(t)) Q(t)
n,f /n,f
+JGMESt)

< B 2 Q (t)E (uft(t) i(t) Af(t)Q(t))
n,f

< B 2 (t)E (ut(t) u ,(t) An,, f(t)Q(t))
n, f
out
+JE un, ft( +- un,(t) IQ(t) (3-17)
n,f Pn, f ) + )

Denote the R.H.S. of (3-17) as 0. It is of great importance to observe that, from

the algorithm description of the MES algorithm above, at every time slot t, the MES

algorithm essentially minimizes the R.H.S. of (3-17) over all possible scheduling

algorithms.

Since A lies in the interior of the network capacity region A, it immediately follows

that there exists a small positive constant c > 0 such that

A + = {(A1,1 +c), --, (An,f + ), } E A. (3-18)

By invoking Corollary 3.9 in [17], we claim that there exists a randomized scheduling
policy, denoted by RA, which stabilizes the network while providing a data rate of

E uyf(t) U (t)Q(t) =Anf+C, Vt (3-19)

where uO,(t), uin(t) are the link data rates induced by the randomized policy RA.
Therefore, we have

ERA = B 2c f Qn5(t)

JE (E n (uot) + U ,(t),) IQ(t) (3-20)

Note that the last term in (3-20) is the actual energy consumption by RA algorithm
during time slot t. Following (3-4), we have

0RA < B- 2c Qf(t) JGmax. (3-21)
n,f
In light of (3-17), we have

E (E,,(Q(t + 1))21Q(t)) E (Cn, (Qf(t)) 2o (t))
JGMES(t) < MES < RA
< B- 2e n, Q (t) + JGmax. (3-22)

Next, we take the expectation with respect to Q(t) on (3-22) and have

E (,(Q (t 1))2) E (: ,(Q (t))2)
+JGMES(t) < B 2eE (En,r Q(t)) + JGmax. (3-23)

Note that the above inequality holds for any time slot t. Hence, we sum over from time

slot 0 to T 1 and obtain

E (E,r(Q (T))2) E (E,.r(Qo(0))2)

+J GMES(t)

< TB 2c Eo1 E E (Q(t)) + TJGmax. (3-24)

Next, we rearrange terms and divide (3-24) by T and have
T-1
2c4 E (Q(t))
t=0 n,f
max E (Ynr(Q((O))2)
< B JGmx T

1 T- E (,(Qf(T))2 (3-25)
-JT T GMES(t) T (3-25)
t=0
Note that the last two terms of (3-25) are both non-positive. By taking lim sup,,, on

both sides of (3-25), we attain

1 1 B + JGmax
lim sup E ((t)) < 2 < T-*oo T 2c
t=0 n,f

Therefore, for every individual queue Q,, we have

S 1 B + JGmax
lim sup E (Q((t)) < < (3-27)
to
Finally, by invoking Markov inequality, we have

1 -B +1 JGmax
lim sup T Pr (Q(t) > M) < 2M (3-28)
T-+oo T n 2zM
t-0
Therefore, by taking limMv,, we obtain the stability result of the MES algorithm and thus

completes the proof.

3.3.3 Asymptotic Energy-Optimality

In this section, we show that the MES algorithm yields an asymptotic optimal

solution to (3-5), i.e., the average energy consumption induced by the MES algorithm

can be arbitrarily close to the global minimum solution of (3-5), by selecting a sufficiently

large value of J.

Theorem 3.2. Define CMES and C* as the average energy consumption induced by

the MES algorithm and the optimal (minimum) solution of (3-5), respectively. The

performance of the MES algorithm is given by

B
CMES < C* + B (3-29)

where B is defined in (3-16). Therefore, by choosing J oo, the performance of the

MES algorithm can be pushed arbitrarily close to the optimum solution C*.

Proof. First, denote the optimal sequence of link rates, which generates the optimum

solution C*, as u*(O), u*(1), -. u*(t), ... Next, let us consider a deterministic policy,

denoted by DE, which allocates exactly the optimum link data rates on every time slot t.

Similar to (3-22), we have

E (,,(Qf(t + 1))2 Q(t)) E (Yn,(Qf(t)) 2|IQ(t))

+JGoES(t) < EMES < DE (3-30)

where

SDE

SB 2 Qt)E (ot(t) u, un(t)- An, f(t)Q(t))
n,f
out
+JE u o -p)- u + t(t) |Q(t) (3-31)
n, f Pn, f

It is worth noting in (3-31), only the data rates are replaced by the ones generated by

the DE algorithm whereas all other values remain the same. Note that in this case, the

optimal data rates u*pt(t) and u.in(t) are known as a priori by the DE algorithm and

thus are constants with respect to Q (t).

We take the expectation on both sides of (3-30) and have

E (Q(t 1+)) )-
\ n,f
+JGMES(t) < B
-2E Qf(t)E (u*"t
JE u tn n,f
\ n,ff

+JE un, f(t) t
n, f Pnu

E (Q (t))2
\n,f

u(t) U (t) A)

f n in

We emphasize that, however, in this scenario, at arbitrary time slot t, the value of

nf -( uf(t- ,f (3-33)

could be either non-positive or nonnegative. In other words, the relationship of (3-19)

does not hold. To circumvent this, we first sum (3-32) over time slots t = 0, T T- 1

and divide it by T. Thus, we attain
1 T-1
J GMES(t) < B
t=0

1 -0
J E u ,,ft (t) )

T nT
t=o
^(EQf())2) gT-1
t-0

where

(3-34)

(3-32)

uin t in
Un, f n, )

E(t) = E Qf(t)E (u t(t) (t) An, (t))) .
\ nf ,)

By taking lim supT ,, we have

T-1
Jlimsup GES(t)< B JC*
T t=
T-l
-2 im sup(). (3-35)
T-oo
t=0

Note that

lim(AB)= lim(A)lim(B) (3-36)

if lim(A) and lim(B) exist and are bounded. Recall that in the previous section, we have

shown that
T-1
iimsup E(Q(t)) (3-37)
T t=0 n,f

exists and is finite. Moreover, since u*"t(t) and u. n(t) are the optimum solution to

(3-5), the network stability is achieved under A. Therefore, we have

T-1 T-1
1 1
lim sup ufut(t) > lim sup un '7(t) + An
T-oo Y T-oo T n
t=o0 t=o

since otherwise, the network stability cannot be ensured by u*. Finally, we conclude that

I T-1
lim sup E(t) (3-38)
T-oo /
T- t=0

is nonnegative and thus the last term of (3-35) can be omitted. Consequently, by

dividing J on both sides of (3-35), we have

T-1
1 B
lim sup GMES(t) <- C*. (3-39)
T--oo J J
t=0

Therefore, Theorem 3.2 holds.

3.4 Simulations

We consider a multi-hop wireless network illustrated in Figure 3-1. There are three

flows in the network, denoted by Flow 1, 2, 3. The routing paths of flows are specified

by R, = {A, B, C, D}, R2 = {F, G, C, D} and R3 = {E, F, G, H}. The exogenous

arrival processes are Bernoulli processes with an average rate of 5 packets per slot

for all three flows. Without loss of generality, we assume a two-hop interference model

which represents the general IEEE 802.11 MAC protocols [36, 37]. The nominal link rate

of a wireless link is assumed to be 20 packets per slot. For a particular link, there are

three equally possible channel states, i.e., Good, Medium, Bad where the corresponding

successful transmission probabilities are 0.8, 0.6, 0.3, respectively.

------------------.......

c .Flow 1

D
(r/"--- Flo--- Flow 2

Flow 3

Figure 3-1. Network topology with interconnected queues

For the ease of exposition, we assume that aa,b = a,b = 50-J for all links in

the network [50]. We investigate the average energy consumption induced by the

MaxWeight algorithm and the proposed MES algorithm with different values of J, i.e.,

50, 100,200,500,800, 1000, 2000, 5000, 8000, 10000, 12000, 14000, 16000, 18000,

20000, 25000. Every simulation is executed for 10000 time slots.

Figure 3-2 depicts the average energy consumption per time slot for the MaxWeight

algorithm and the MES algorithm with different values of J. As illustrated in Figure

3-2, when J = 0, the MES algorithm reduces to the original MaxWeight algorithm

and yields the same amount of energy expenditure. However, as J increases, the

energy consumption induced by the MES algorithm decreases remarkably. In addition,

as shown analytically in Theorem 3.2, the MES algorithm approaches to the global

minimum energy expenditure gradually as J increases. To achieve a better understanding

on the impact of larger values of J, we compare the average queue backlogs of all data

260 ******** HEEHEEHEEUE 700000000000000
-, Energy Consumption by the MaxWeight Algorithm
j 2500
2400
0- 2300
02200 Energy Consumption by
r_ the MES Algorithm
S2100
S2000
1900
1800
0 0.5 1 1.5 2 2.5
Different Values of J x 10o

Figure 3-2. Comparison of the energy consumption of the MaxWeight algorithm and
the MES algorithm

queues, with J = 50, 150, 350 and 500, in Figure 3-3 and Figure 3-4, where the

queues in the network are indexed by 1 to 9 in the order of QA, QA, Q~, Q2, Q2, Q2,

Qj, Qj and Qj. It is worth noting that, as J grows, the average queue backlogs in

the network increases correspondingly. Following Little's Law, larger queue backlogs

yield longer network delays. Therefore, while the network-wide energy expenditure is

noticeably reduced, a larger value of J yields a longer average delay in the network. As

a consequence, a tradeoff between the energy-optimality and the experienced delay can

be attained by tuning J properly.

So far, throughout this work, we have assumed that the energy in each node is

constrained yet sufficiently large. Next, we investigate the performance of the MES

algorithm in multi-hop wireless networks where each node has a limited and finite

amount of energy. More specifically, the same network in Figure 3-1 is considered,

however, we assume that each node in the network has a battery with an initial energy

of 1 Joule. We compare the performance of the MaxWeight algorithm with the MES

algorithm for different values of J, in terms of the network lifetime, which is defined as

the time instance when the first node in the network depletes the battery completely.

The values of J are 50, 200, 500, 800, 1000, 2000, 5000, 8000, 10000, 15000 and 20000.

J=50
120

0)
0 100
U

S40
0)
20

1 2 3 4 5 6 7 8 9
Indices of Queues

J= 150
120

o 100
U

40
0)

1 2 3 4 5 6 7 8 9
Indices of Queues

Figure 3-3. The average queue backlogs in the network for J 50 and 150

Figure 3-5 pictorially compares the network lifetime induced by the MaxWeight

algorithm and that of the MES algorithm for different values of J. Similar to the previous

scenario, when J is small, the MES algorithm yields similar performance, in terms of the

network lifetime, compared to the MaxWeight algorithm. However, when J increases, the

MES algorithm significantly outperforms the traditional MaxWeight algorithm. It is worth

noting that when J = 20000, the MES algorithm prolongs the network lifetime by more

than twice as much as that of the MaxWeight algorithm! By the same token, a tradeoff

between the network lifetime and the experienced delay of the network can be controlled

effectively by tuning the value of J.

3.5 Conclusions

In this chapter, we investigate the energy consumption issue of the scheduling

algorithms in multi-hop wireless networks. We show that the traditional MaxWeight

algorithm, which is well known to be throughput optimal, is not energy optimal due to

J = 350
120

M 80
o 60

1 2 3 4 5 6 7 8 9
Indices of Queues

J = 500
120

60

S2000
250
1 0 2 3 4 5 6 08 1 8

Indices of QueuesJ x

Figure 3-. CompThe average queue backlogs in the network falgor J ithm350 and the MES

algorithm

the overlook of the energy consumption induced by inevitable packet retransmissions.
In light of this, we propose a minimum energy scheduling (MES) algorithm which
s3500 < ...--.. ,-

significantly reduces the energy consumption compared to the original MaxWeight
algorithm. In addition, we analytically show that the MES algorithm is essentially energy

Diffoptimal in the sense that the average energy expenditure of the MES algorithm can beValues of J
Figure 3-5. Comparison of the lifetime of the MaxWeight algorithm and the MES
algorithm

the overlook of the energy consumption induced by inevitable packet retransmissions.

In light of this, we propose a minimum energy scheduling (MES) algorithm which

significantly reduces the energy consumption compared to the original MaxWeight

algorithm. In addition, we analytically show that the MES algorithm is essentially energy

optimal in the sense that the average energy expenditure of the MES algorithm can be

pushed arbitrarily close to the global minimum solution. Moreover, the improvement on

the energy efficiency is achieved without losing the throughput-optimality. Therefore,

the proposed MES algorithm is of great importance for network protocol designs in

energy-constrained multi-hop wireless networks such as wireless sensor networks.

CHAPTER 4
CHANNEL AND POWER ALLOCATION IN WIRELESS MESH NETWORKS

4.1 Introduction

Metropolitan wireless mesh networks gain enormous popularity recently [51]. The

deployment of wireless mesh networks not only facilitates the data communication by

removing cumbrous wires and cables, but also provides a means of Internet access

scheme, which is a further step towards the goal of "communicating anywhere anytime".

No matter where the location is or the purpose that the wireless mesh network is

deployed, the same conceptual layered architecture is utilized. Figure 4-1 illustrates the

hierarchical structure of wireless mesh networks. The peripheral nodes are the access

points (AP) which provide wireless access for the end users, or clients. Each AP is

associated with a mesh router. They can be manufactured in a single device with two

separate functional radios [52] [53], or simply connected with Ethernet cables [54]. The

mesh routers are capable of communicating with each other via the wireless backbone.

The central node is a gateway mesh router which functions as an information exchange

between the wireless mesh access network and other networks such as the Internet.

Both the routing algorithmic design and channel assignment for backbone mesh routers

are interesting issues and attract tremendous attention [55-57].

AP Access Point
0 G Gateway Node
D R Wireless Mesh Router

Figure 4-1. Hierarchical structure of wireless mesh access networks

In this work, we investigate an important issue which needs to be solved in

wireless mesh networks. As in Figure 4-1, the AP and its associated clients form

a regular WLAN cell, which operates with the de facto IEEE 802.11 standards.

The throughput of one cell depends on the signal-to-interference-plus-noise ratio

(SINR) experienced at the receiver where the interference mainly comes from the

other operating cells. For example, if each of the cells operates with IEEE 802.11 b

standard, we can utilize a different frequency band such as IEEE 802.11 a or WiMAX

[58], for the inter-cell communication among mesh routers and hence causes no

interference to intra-cell transmissions. However, the co-channel interference from other

operating cells is inevitable due to the limitation of available transmission channels,

e.g., 3 non-overlapping channels in our example. Most current off-the-shelf APs are

capable of adjusting the transmission rate according to the measured channel condition

which is indicated by transmission bit error rate (BER). Given a particular modulation

scheme, BER is uniquely determined by the SINR experienced by the receiver of the

link. Generally speaking, higher SINR value yields lower BER and higher data rate.

Therefore, the mutual interference dramatically degrades the transmission rate of each

cell and the aggregated throughput of the whole network [59]. Each AP attempts to

tune the physical parameters such as operating frequency1 and transmission power in

order to maximize the SINR and hence the throughput. In our work, we investigate the

issue of maximizing the overall throughput of the network, defined as the summation

of throughput of all cells, by finding the optimal frequency and transmission power

allocation strategy. Also, due to the concern of scalability and computational complexity,

we prefer a decentralized solution to the throughput maximization problem.

Unfortunately, the throughput maximization problem is challenging. For example,

the frequency and power selected by one AP affects the SINR of other APs, and vice

versa. Worse yet, if the APs belong to different regulation entities, the non-cooperative

APs may only want to maximize their own cell's throughput rather than the overall one.

Therefore, the throughput maximization problem is coupled and finding the optimum

1 We will use frequency and channel interchangeably.

solution is not straightforward. Moreover, traditional site-planning methods in cellular

networks are not feasible either. For example, the network administrator may want to

add more APs when more users are joining the network or disable some APs where

the associated users fail to pay the bill. The network topology is not static, although the

changes take place slowly. Therefore, the demand for adaptability and light computation

burden requires a decentralized solution for the throughput maximization problem.

In this work, we analyze the throughput maximization problem for both cooperative

and non-cooperative scenarios. In the cooperative case, we model the interaction

among all APs as an identical interest game and present a decentralized negotiation-based

throughput maximizing algorithm for the joint frequency and power assignment. We

show that this algorithm converges to the optimal frequency and power assignment

strategy, which maximizes the overall throughput of the wireless mesh access network,

with arbitrarily high probability. In the cases of non-cooperative APs, we prove the

existence of Nash equilibria and show that the overall throughput performance is usually

inferior to the cooperative cases. To bridge the performance gap, we propose a linear

pricing scheme to combat with the selfish behaviors of non-cooperative APs.

The rest of this chapter is organized as follows. Section 4.2 outlines the system

model we considered in the work. The cooperative wireless mesh access networks

and the non-cooperative counterpart are investigated in Section 4.3 and Section 4.4,

respectively. An extension of our model is discussed in Section 4.5 and the performance

evaluation is provided in Section 4.6. Finally, Section 4.7 concludes this chapter.

4.2 System Model

In this work, we consider a wireless mesh access network illustrated in Figure 4-1.

Each AP and corresponding clients form a cell. Without loss of generality, we assume

that all the cells operate with IEEE 802.11b standard and the interference exclusively

comes from the cells with same frequency. Furthermore, the distance between cells

are sufficiently large in the sense that the accumulated interference experienced at the

receiver only affects the SINR value and not block the whole transmission. We assume

that the channels are slow-varying additive white Gaussian noise (AWGN) channels.

The channel gains of each pair of nodes are assumed to be constant over the time

period of interest. As we are interested in the maximum achievable throughput, we

consider the worst case where all APs are saturated. In other words, the APs always

have packets to transmit and they can communicate with each other via the backbone

mesh routers with negligible delay. Also, we assume that the APs are transmitters and

clients are receivers due to the dominance of downlink traffic, as assumed2 in [61]

[62] and [63]. We only focus on the joint frequency and power allocation where the

contention behavior is less relevant and thus omitted. Therefore, we can simplify our

model as that all the APs are transmitting data to the associated clients consistently. We

assume that each AP is capable of adjusting the operating frequency and power as well

as acquiring the SINR values measured at the client by short ACK messages.

Let us first consider the simplest case where there is only one cell in the wireless

mesh access network, i.e., a single WLAN. In the following two sections, we assume

that the APs have pre-determined and fixed modulation and coding schemes. In other

words, upon receiving the SINR value3 measured by the client, denoted by 7, the AP

tunes the physical parameters in order to maximize the throughput, which is defined as

R*(7) = maxRi x (1 Pe(y, Ri)) (4-1)
Rf

where Ri is the raw data rate specified by the IEEE 802.11 standard and R*, i.e., the

throughput of this cell, is a non-decreasing function of received SINR 7. Pe is the error

2 The dominance of the downlink traffic is verified by the experimental measurements
in [60] as well.

3 Although there is no interference in this case, we adopt SINR instead of SNR for
notation consistency.

probability of the transmission channel, which is a function of SINR value providing the

modulation and coding scheme [64]. Apparently, if there is only one cell in the mesh

access network, the AP will boost the power as much as possible to increase the value

of 7 and thus the throughput is maximized.

We now consider the cases where N cells coexist in the wireless mesh access

network. Let pi and f denote the power and frequency for the i-th AP, respectively. We

use p = [pi, p2, PN] and f = [fl, f2, fN] to represent the power and frequency

assignment vector for all N APs. Therefore, for each cell i, the value of SINR4 i.e., 7i, is

a function of (p, f). The throughput of one cell depends not only on the power level and

frequency of itself, but also those of other APs in the network. Therefore, the throughput

maximization problem is coupled and by no means straightforward.

In the following two sections, we will discuss the scenarios where the APs are

cooperative and non-cooperative, respectively, under the assumption that the modulation

and coding schemes of APs are pre-loaded and fixed. In Section 4.5, we will extend our

analysis by considering the scenario where the APs are capable of adaptive coding and

modulation. The performance evaluation of all scenarios are provided by simulations in

Section 4.6.

4.3 Cooperative Access Networks

In this section, we consider the scenarios where all APs in the wireless mesh

access network are cooperative. The transmission power of APs are quantized into

discrete power levels for simplicity. From the system point of view, we want to find a joint

frequency and power level assignment such that the overall throughput in the whole

4 Throughout the chapter, the term SINR of the cell represents the average SINR
among all the clients in the cell, which can be obtained by a moving average of the
reported SINR value.

network is maximized. Our objective function can be written as

N N
Unetwork(p, f)= R*(7) = R (p, f) (4-2)
i= l i= l

where R, is defined in (4-1).

However, finding the optimal frequency and power assignment which maximizes

(4-2) is non-trivial. The interdependency makes the problem coupled and difficult to

solve by traditional optimization methods [65]. A combination of (p, f) is named a profile

and a naive approach to solve the problem is to investigate all profiles exhaustively.

However, this is impossible in practice. For example, in a medium-size wireless mesh

access network with 20 APs where each has 3 frequency channels and 10 power levels,

the search space is (3 x 10)20 profiles! Obviously, the centralized algorithms are not

favorable in the wireless mesh access network due to the scalability concern. Next, we

will introduce a decentralized negotiation-based throughput maximization algorithm,

from a game-theoretical perspective.

4.3.1 Cooperative Throughput Maximization Game

The APs in the wireless mesh access networks are considered as players, i.e.,

decision makers of the game. We model the interaction among APs as a Cooperative

Throughput Maximization Game (CTMG), where each player has an identical objective

function Ui, as
N
Ui(p, f) Unetwork(P, f) R (p, f) Vi. (4-3)
j= 1
For each player i, all possible frequency and power level pairs form a strategy space

;, which has a size of c x /, where c is the number of frequency channels available and

I is the number of feasible power levels. Define

Q = 0 x 02 X X VN. (4-4)

Then, the N players autonomously negotiate about the joint frequency-power profile

in Q in order to find the optimal profile which maximizes (4-3). However, due to the

interdependency among N players caused by mutual interference, one question of

interest is that whether this negotiation will eventually meet an agreement, a.k.a., a

Nash equilibrium. The importance of Nash equilibria lies in that a possible steady state

of the system is guaranteed. If the game has no Nash equilibrium, the negotiation

process never stops and oscillates in an everlasting fashion. In addition, we are

concerning about what the performance of the steady states would be, if exist, in terms

of overall throughput of the whole network. We provide answers to these questions in

the following.

Lemma 3. The CTMG is a potential game.

A potential game is defined as a game where there exists a potential function P

such that

P(a', ai) P(a", ai) = U,(a', ai) Ui(a", ai) Vi, a', a" (4-5)

where U, is the utility function for player i and a', a" are two arbitrary strategies in 0i.

More specifically, we have a' = [pl, f,'] and a" = [p", f,"]. The notation of a_i denotes the

vector of choices made by all players other than i. Potential games have been broadly

applied in modeling the interactions in communication networks [66]. The popularity is

on account of the nice properties of potential games, such as

- Potential games have at least one Nash equilibrium.

- All Nash equilibria are the maximizers of the potential function, either locally or
globally.

There are several learning schemes available which are guaranteed to converge to
a Nash equilibrium, such as better response and best response [67] [68].

For detailed description about potential games, readers are referred to [67] and [69],

which investigate the potential game theory in engineering context.

We observe that in the cooperative case, each player has the same utility function

as in (4-3), which is the overall throughput of the network. Apparently, one potential

function of the game is the common utility function itself, i.e.,

P = U = U2 =... = UN. (4-6)

In fact, the games where all players share the same utility function are called identical

interest games [70], which is a special case of potential games and hence all the

properties of potential games can be applied directly.

In the literature, both best response and better response are popular learning

mechanisms that have been utilized in potential games [71-73]. At each step of the best

response approach, one of the players investigates its strategy space and chooses the

one with maximum utility value. This updating procedure is carried out sequentially. The

primary drawback of the best response is the computational complexity, which grows

linearly with the cardinality of the strategy space. An improvement of the best response

is the so-called better response, where at each step, the player updates as long as

the randomly selected strategy yields a better performance. The dramatically reduced

computation is the tradeoff with the convergence speed. Both the best response and the

better response dynamics are guaranteed to converge to a Nash equilibrium in potential

games [66]. However, there may be multiple Nash equilibria in a potential game and the

performance of different equilibria may vary dramatically. Therefore, although the best

response and the better response could guarantee the convergence, they may reach an

undesirable Nash equilibrium with inferior performance.

A B D C

1 2 2 1
1 2 1 2

Figure 4-2. An illustrative example of multiple Nash equilibria

Let us consider an illustrative example in Figure 4-2. There are four labeled APs

in the network. A and B are close to each other, and so are C and D. Without loss of

generality, we assume that the APs have the same power and only adjust the operating

frequencies in an order of A -> B -> C -> D to avoid the interference. The adaptation

continues with the best response mechanism until a Nash equilibrium is reached.

Suppose that there are two frequency channels available, say 1 and 2. First, A randomly

selects one channel, say 1. B will pick 2. Next, C has the chance to update. Since C

is closer to B than A, channel 1 will be selected. Finally, D will choose channel 2. By

inspection, we claim that profile 1 2 2 1 is a Nash equilibrium since no player is

willing to update its strategy unilaterally. Meanwhile, we observe that another profile

1 2 1 2 is also a Nash equilibrium. Obviously, the second Nash equilibrium

generates much less interference than the first Nash equilibrium and hence yields

superior performance in terms of overall throughput. However, the best response only

leads to the less desirable Nash equilibrium.

In fact, the existence of multiple Nash equilibria is observed in [71] by simulations.

However, the authors fail to specify which one would be the steady state of their game

due to the limitation of the best response, even in a statistical fashion. Recall that

the Nash equilibria are the maximizers of the potential function in potential games,

converging to an inferior Nash equilibrium analogously indicates being trapped at a local

optimum of the potential function. However, it is the global optimum, i.e., the optimal

Nash equilibrium, that is the desirable steady state which we are yearning for.

Next, we introduce a negotiation-based throughput maximization algorithm

(NETMA) which can converge to the optimal Nash equilibrium with arbitrarily high

probability.

4.3.2 NETMA- Negotiation-Based Throughput Maximization Algorithm

We assume that the APs are homogeneous and each has a unique ID for routing

purpose. Each AP maintains two variables Dpre and Dcr. The AP has the knowledge of

its current throughput and records it in Dcu,. Whenever there is a change of throughput

caused by exterior interference5 the AP sets Dpre = Dcur and resets Dcur with the newly

measured throughput. When the wireless mesh access network enters the negotiation

phase6 NETMA is executed. The detailed procedure of NETMA is provided as follows.

NETMA:

- Initialization: For each AP, a pair of frequency and power level is randomly
selected. Set Dpre = D,,r, the current throughput.

Repeat:

1. Randomly choose one of the AP, say k, as the updating one, i.e., each AP
2. For the updating AP k:

(a) Randomly chooses a pair of frequency and power level, say f' and
p', from the strategy space Ok. Then the AP computes the current
throughput with f' and p' and records it into D,,r.
(b) Broadcasts a short notifying message which contains its unique IDk to all
the other APs in the mesh access network.
3. For each AP other than k, say j:

(a) If the 7j value changes, records the previous throughput into Dpre and the
current throughput into D,,r. Remains unchanged otherwise.
(b) Upon receiving the notifying message, a three-value vector of [Dpre, D,,r, IDj]
is sent back to the k-th AP.
4. After receiving all the three-value vectors by counting the identifiers IDj, the
k-th AP computes the sum throughput before and after f' and p' are selected,
which are denoted by Ppre and Pcur.

5 We assume the channel is slow-varying and the change of throughput for a single
cell is due to the mutual interference only.
6 To reduce the negotiation overhead, a negotiation phase can be initiated by the
network administrator after a new contracted user joins or a current user terminates the
service, or on a daily basis.

5. For a smoothing factor 7 > 0, the k-th AP keeps f' and p' with probability

ePa/T 1
1= e 1(4-7)
ePcur/T + ePpre/7 1 + e(Ppre-Pcu)/17
6. The k-th AP broadcasts another short notifying message, which indicates the
end of updating process and a specific number 6, to all the other APs.

Until: The stopping criteria F is met.

Note that in step 6, the specific format of 6 depends on the predefined stopping

criterion F. For example,

* If the stopping criterion is the maximum number of negotiation steps, 6 is a counter
which adds one after each updating process.

If the stopping criterion is that no AP has updated for a certain number of steps, 6
is a binary number where 1 means updating.

If the stopping criterion is that the difference between sum throughput obtained in
consecutive steps are less than a predefined threshold c, 6 is the calculated sum
throughput after each updating process.

We can have other stopping criteria Fs and corresponding formats of 6 as well.

The NETMA algorithm is inspired by the work in [74], where a similar algorithm

was first introduced in the context of stream control in MIMO interference networks. The

distinguishing feature of this type of negotiation algorithms, from the better response and

the best response, is the randomness deliberately introduced on the decision making in

step 5. The rationale can be illustrated in Figure 4-2 intuitively. If there is no randomness

in decision making i.e., 7 = 0, the four APs may get trapped at a low efficiency Nash

equilibrium 1 2 2 1. However, with the randomness caused by nonzero 7, they may

reach an intermediate state 1 2 2 2 and arrive at the optimum Nash equilibrium

1 2 1 2 eventually. Moreover, the updating rule in step 5 also implies that if f' and

p' yield a better performance, i.e., Ppre Pcur < 0, the k-th AP will keep them with high

probability. Otherwise, it will change with high probability.

The steady state behavior of NETMA is characterized in the following theorem.

Theorem 4.1. NETMA converges to the optimal Nash equilibrium in CTMG with

arbitrarily high probability.

Proof. The proof of Theorem 4.1 follows similar lines of the proof in [74] and [75].

Figure 4-3. Markovian chain of NETMA with two players

First, we observe that the joint frequency-power negotiation generates an

N-dimensional Markovian chain. Figure 4-3 illustrates the Markovian chain introduced

by NETMA with two players, say A and B. Let x and y be the choices for each player,

where x e CA and y e OB. In other words, player A can choose a frequency-power pair

from [xI, xcx] and player B can choose from [yi, ycx/]. Note that at an arbitrary

time instant, only one of the players can update. In Figure 4-3, for example, state (xi, yi)

can only transit to a state either in the same row or the same column, not anywhere

else. This is true for every state in the Markovian chain. Let Si, denote the state of

(xi, yi). We have

Pr(Sm,,Si,) =

eP(Sm.n)/7
2xcx x(eP(sm.n)l/+eP(Sij) '
if m = i or n = j

0,

otherwise.

(4-8)

where 7 is the smoothing factor in step 5 of NETMA and P(Si,) is the value of the

potential function, i.e., (4-21), at the state of Sj.

Let us derive the stationary distribution Pr* for each state. We examine the

balanced equations. Writing the balance equations [76] at the dashed line, we obtain
cxl cxl
Pr*(Si,l) x Pr(Sl,kS,) = Pr*(Sl,k) x Pr(S1i,S1,k). (4-9)
k=2 k=2

By substituting (4-9) with (4-8), we have

cx/ Pr*(Si,) x e P(Sl,k)/
Pr*(eSl.) x ePsl)l /-_+ep~sl,1/ -
k 2k=2 1" l,k eP(Sx ,)/, eP(5 ,k)/

Observing the symmetry of equation (4-10) as well as the Markovian chain, we

note that the set of equations in (4-10) are all balanced if for arbitrary state S in the

strategy space Q, the stationary distribution is

Pr*(S) = Cep()/' (4-11)

where /K is a constant. By applying the probability conservation law [77] [76], we obtain

the stationary distribution for the Markovian chain as

eP(3) /
Pr*(S) =e (4-12)
Cs5so eP(S;)/-

for arbitrary state S e 2.

In addition, we observe that the Markovian chain is irreducible and periodic.

Therefore, the stationary distribution given in (4-12) is valid and unique.

Let S* be the optimal state which yields the maximum value of potential function P,

i.e.,

S* = argmaxs,,-P(S,). (4-13)

From (4-12), we have

lim Pr*(S*)=1 (4-14)
T--0

which substantiates that NETMA converges to the optimal state in probability.

Finally, the analogous analysis can be straightforwardly extended to an N-dimensional

Markovian chain and thus completes the proof.

In NETMA, there is no central computational unit required. The joint frequency-power

assignment is achieved by negotiations among cooperative APs and the maximum

overall throughput is achieved with arbitrarily high probability. The autonomous behavior

and decentralized implementation make NETMA suitable for large scale wireless mesh

access networks. Moreover, NETMA has fast adaptability for the topology change of

the wireless mesh access networks. NETMA does not depend on any rate adaption

algorithms, nor on any underlying MAC protocols. In our simulation in Section 4.6, we

use IEEE 802.11 b as the MAC layer protocol. However, it can be easily extended to

arbitrary MAC protocol with multi-rate multi-channel capability, such as IEEE 802.11 a.

In addition, even with the existence of exterior interference source, such as coexisting

WLANs, NETMA works properly as well since the objective of NETMA is to maximize

the overall throughput of the network in the current wireless environments. The tradeoff

between algorithmic performance and convergence speed is controlled by parameter

7 in step 5, where large 7 represents extensive space search with slow convergence.

On the contrary, small represents limited space search with fast convergence. Note

that the smoothing factor 7 here is analogous to the concept of temperature in simulated

annealing [78]. Therefore, it is advisable that at the beginning period of the negotiation,

the value of 7 is set with a large number and keeps deceasing as the negotiation

iterates. We choose r = 10/k2 in our simulations, where k denotes the negotiation step.

In step 1, we require that each AP updates with a probability of 1/N. For example,

we can utilize a random token mechanism where each updating AP randomly selects

an AP as the next updating AP, i.e., passing the token. Note that in NETMA, even an

erroneous operation happens, for example, two APs update at the same time in our

case, it only prolongs the convergence time for NETMA yet does not affect the final

output of NETMA. This is because that such an error, as verified in [72] via extensive

simulations, has no influence on the statistically monotonic-increasing tendency of the

potential function.

4.4 Non-Cooperative Access Networks

In the previous section, we discuss the scenarios where all APs in the wireless

mesh access network are cooperative, and the overall throughput is maximized

by negotiations among autonomous APs using the NETMA mechanism. However,

cooperation is not always attainable. Although the functionality of relaying packets

for each other can be achieved by incentive mechanisms such as [79], the adjustable

parameters inside each cell cannot be enforced and effectively controlled. The N APs

may belong to distinct self-interested users and they care about exclusively their own

throughput rather than the overall aggregated throughput. In other words, the utility

function of each selfish user is

Ui = R*(-i) (4-15)

where Rf is the throughput of the i-th cell, defined in (4-1). Analogous to CTMG, we

can formulate the interaction among N selfish APs as a Non-cooperative Throughput

Maximizing Game (NTMG) where each AP is attempting to find the frequency-power

pair which maximizes its own SINR value as well as the corresponding throughput. As in

the cooperative case, each player's utility function depends on the frequency and power

of itself as well as those of others. However, NTMG is no longer an identical interest

game.

Lemma 4. In NTMG, all the APs will transmit with the maximal power at the Nash

equilibrium, if exists.

Proof. The proof of Lemma 4 is straightforward. For a single player, we have

'i =pigi (4-16)
YkET,(f) Pkgki + NiV,

where gd is the channel gain from cell i's transmitter to j's receiver and N, is the

Gaussian noise at the i's receiver. T,(f,) denotes the set of cells which operate at

the same frequency f, other than cell i. Note that given other players' strategies, yi

is a monotonic increasing function of p, and so is U,. Assume at a Nash equilibrium

of NTMG, the k-th AP has a power level of Pk satisfying 0 < pk < Pmax, where pmax

denotes the maximum power defined by MAC layer. The k-th AP is inclined to increase

its power Pk in order to yield a higher value of U,, which contradicts the definition of Nash

equilibrium. Thus, at the Nash equilibrium of NTMG, if exists, all the APs will operate at

the same power level, i.e., pmax.

Based on Lemma 4, the NTMG can be viewed as a simplified game where each

player has the same power and only adjusts the frequency to minimize the interference.

Moreover, according to (4-15) (4-16) and the assumption of uniform environment, the

NTMG is equivalent to the following simplified game where each player has the utility

function7 as

U, = ( Pmax9ki + N) (4-17)
kEG,(f,)
and U, is a function of frequency assignment vector f exclusively.

As in the cooperative case, the frequency selection among N players is mutually

dependent. The question arises that whether this frequency adjusting dynamic

converges, or equivalently, whether NTMG has a Nash equilibrium. We provide the

answer of this question in the following theorem.

7 The negative sign comes from the convention that utility functions are the ones to be
maximized.

Theorem 4.2. There exists at least one Nash equilibrium in NTMG.

Proof. Let us first consider the simplified game. For each player, the utility function is

given as

U, = -( pgki+ N,) (4-18)
k.F, (f,)
= -( Pk9gk,i x6( fk) Ni) (4-19)
k#i,kcN

where
J 1, ifk =0
6(k) 1, if k (4-20)
0, otherwise.

We conjecture that one of the feasible potential functions is

P=- xE pkgki. (4-21)
iEN kGE (f)

The verification is as follows.

2P = Pkk,
ieN kc(F(fi)

kGT,( f) j i jEN kG j( J)

+ Pk9kj}
kGe(f),k#i

= Y.Pkgki6(fk fi)
{k#ikGN
+ pig,(fi )

+ E Pk9kji (4-22)
j ijCN kcG (fj),k i

Note that

Pkgi(fk fi) = pgik6(f fk) (4-23)

for any pair of i, k. We have

P= Pkgk,6(fk- f) + XQ(i) (4-24)
k i,kGN )
where

Q(-i)= Pkj (4-25)
j ijeCN kc.Fe (),k/ i
and Q(-i) is independent of f;. Therefore, for arbitrary two frequencies a' and a" of
player i, we have

QOa(-i) = a,,(-i) (4-26)

and

P(a', a_i) P(a", a_i)

{kCE (al) Pkgki x QX ( i)}

{ ZEC (al') pkgk, x Qa X i()}
U,(a', a_,)- U,(a",a_i). (4-27)

Therefore, according to the definition in (4-5), the simplified game is a potential
game and has at least one Nash equilibrium. Thus, the existence of Nash equilibrium in
NTMG is obtained from the equivalence derived from Lemma 4.

As shown in Lemma 4, at the equilibrium, the non-cooperative APs will always
transmit at the maximum power level. This seem to be the best choice for each one of
the APs. However, it is usually not a favorable strategy from a social-welfare point of
view. To bridge the performance gap, we propose a linear pricing scheme to combat
with the selfish behaviors, i.e., the players are forced to pay a tax proportional to the
utilized resources. For example, we could impose a price to all selfish APs for the power

they utilize. Hence for each AP, the utility function becomes

U, = R (-i) Appi (4-28)

where A' represents the power utilization price specific for the i-th AP and pi is its

transmission power. Therefore, the more power AP uses, the more tax it has to pay. By

imposing power prices properly, a more desirable equilibrium may be induced, from a

social-welfare point of view. We define the corresponding game as a Non-cooperative

Throughput Maximization Game with Pricing (NTMGP).

Let us first investigate the impact of prices on the behaviors of players. If Ai = 0,

where no price is imposed, the i-th AP will transmit at the maximum power and causes

extra interference to other APs. However, if we impose an unbearably high price, say

AP = oo, the AP would rather not to transmit at all. Based on these observations,

we propose a heuristic linear pricing scheme to improve the overall throughput in

non-cooperative wireless mesh access networks.

To enforce the scheme, we introduce a pricing dictator unit (PDU) into the network

which determines the prices for all APs and informs them timely. In addition, we assume

that the PDU has the monitoring capability and is aware of the operating frequencies

of each cell. There are two prices charged by the PDU for each non-cooperative AP

Besides the power utilizing price Ai, a frequency switching price A' is imposed on the

i-th AP whenever it changes the operating frequency. The price setting process is

described as follows.

Price setting process:

Phase I:

The PDU sets A' = .. AN = 0 and A = .. AN' = 0 and all APs play NTMG
until converges, i.e., a Nash equilibrium is reached.
The PDU collects the current throughput information from each cell, denoted
by Mi, where / is the index of the cell.

Phase II:

The PDU sets = A" = oo.
For each AP indexed by i = 1, .. N :

1. The PDU sets Ai = oo for the i-th AP and let the APs play the NTMGP.
Upon convergence, the PDU collects the overall throughput, say V,, in the
current price setting.
2. Calculate the power utilizing price for the i-th AP as

Vi YE l"i Mi
A = P1 M (4-29)
Pmax

3. Reset A, = 0.

Output:

Power utilizing price vector A~ = [', .. AN]

Frequency switching price vector Af = [oo, o]

In the price setting process above, the PDU imposes zero prices for all APs initially. As

a consequence, all APs will transmit with pmax at the equilibrium, as shown in Lemma

4. Upon convergence, the PDU fixes the frequency switching price to infinity which

discourages the non-cooperative APs from switching channels thereafter. In (4-29),

=1,ji Mj is the sum throughput for all cells other than i, when the i-th AP transmits
with the maximal power due to the zero power price. Similarly, V, is the sum throughput

of other cells when the i-th AP is silent due to the unaffordable power price. Therefore,

in (4-29), the power utilization price charged for the i-th AP, a.k.a., A', can be viewed

as a compensation to the impact it causes on the overall throughput of other cells.

The more power it utilizes, the more severe it affects the other players and thus the

more it pays, as illustrated in (4-28). Hence, by imposing taxes deliberately, the selfish

behaviors of non-cooperative APs are effectively discouraged and a more desirable

equilibrium can be induced, in term of overall throughput of the whole network. The

proof of existence of Nash equilibrium in NTMGP is straightforward. Note that the utility

function of (4-28) is quasi-concave with respect to power. The existence of pure strategy

Nash equilibrium follows directly from the results of [80]. We will present the detailed

performance evaluation of CTMG, NTMG and NTMGP in Section 4.6.

4.5 An Extension to Adaptive Coding and Modulation Capable Devices

So far, we have assumed that the throughput of a cell is given in the form of (4-1),

which is not a continuous function. To be precise, in both Section 4.3 and Section 4.4,

we are confined to the traditional IEEE 802.11 family devices where the modulation

and coding schemes are pre-determined and fixed. For example, Table 4-1 provides

a mapping between SINR values and corresponding rates [81], the feasible data

transmission rate is a discrete set and is usually much less than the theoretical channel

capacity. We name such devices as legacy IEEE 802.11 devices.

Table 4-1. Data rates v.s. SINR thresholds with maximum BER = 10-5
Rate(Mbps) Minimum SINR (dB)
1 -2.92
2 1.59
5.5 5.98
11 6.99

However, thanks to the advance of coding techniques, the maximum data transmission

rate can be largely closed to the theoretical Shannon capacity in AWGN channels

[82]. Note that achieving this requires variable-rate transmissions by matching to the

instantaneous SINR, which can be implemented in practice through adaptive coding

and modulation (ACM) techniques [83]. This motivates us to extend our results to

ACM-capable devices. More specifically, we are considering more advanced and

powerful APs which attempt to tune the frequency and power in order to maximize

C,(7i) = W, log,2(1 +i) (4-30)

where C, is the Shannon capacity of the i-th cell, and W, is the bandwidth. Note that in
this scenario, the maximum achievable transmission rate, as denoted by the Shannon

capacity, is a continuous variable with respect to y,, rather than discrete, as exemplified
in Table 4-1. It is worth noting that the only difference in this scenario is the alternative
objective function of each AP. Therefore, all the results we have obtained so far are

extendable8 to this special scenario with merely a change of the objective function.
Furthermore, due to the continuity of the objective function in (4-30), the aforementioned

heuristic pricing scheme in Section 4.4 can be improved. Without loss of generality, we

assume that all the cells have unity bandwidth. Restated, we assume that the PDU
is a centralized device which knows the channel environment sufficiently by greedy

acquiring. In addition, the PDU is assumed to have monitoring capability and is aware of

the operating frequency of each cell. The tailored pricing scheme for this ACM-capable
scenario is described as follows.

* First, the PDU sets A = .. A = 0 and A' = A^ = 0 and all APs play selfishly,
until a Nash equilibrium is achieved.

The PDU sets \} = .. A = oo.

For each AP indexed by i = 1, N, the PDU sets

8 More specifically, in the cooperative case, we replace (4-2) with Unetwork(P, f)
Ni_1 C,('yi) whereas in the non-cooperative case, (4-15) and (4-28) are replaced by
U, = C,(i) and U, C,(Qy) Appi, respectively.

kac
Pk X gkk X gik
S-, In 2 x (1 +Yk) x (YE, P, X 9gk + Nk)2
kET,
5 Pk X gkk X gik
2 In2 x (1 + ,) x (P kk)2
kEFT, 7k
5ik X 72
=gik k (4-31)
In 2 x (1 + k) Pk X gkk'

Then the PDU informs each AP the corresponding prices, i.e., A- = oo and A, calculated

in (4-31).

Note that the major difference in this pricing scheme lies in (4-31), where the A,

charged for the i-th AP is the Pigouvian price levied to discourage its reckless power

increase.

After obtaining the prices imposed by the PDU, each AP fixes the operating

frequency and calculates the optimum power p,, which maximizes U, = C,- pi,A

according to the following steps.

SU X (4-32)
9pi pi p

where
ac, gii
.--(4-33)
8pi In 2 x (Ek e, Pk X gki Ni) X (1 yi)
By setting (4-32) equal to zero, we have

1 k= PkE X gki + N, Pmax-34)
i In 2 x A, gIi pmin

where [x]b denotes max{min{b, x}, a} and pmx, Pmin represent the maximum and

minimum power of the device, respectively. Note that Ck ,, Pk x gki + Ni can be easily

measured when the i-th AP sets its transmission power to zero. Therefore, finding the

optimum power of each AP requires only local information and can be implemented in a

distributed fashion.

In (4-31), we observe that the prices imposed on APs depend on the power vector

p, and vice versa. Therefore, after each AP optimizes its power according to (4-34),

the PDU measures the new value of power, adjusts the price following (4-31), and then

announces the new price again. The pricing setting process in (4-31) and the utility

maximization process in (4-34) are executed in an iterative manner, until convergence.

The whole pricing scheme is summarized as follows.

Pricing Scheme:

1. The PDU initially sets zero prices. As a consequence, all APs will converge at a
Nash equilibrium with maximum power.

2. The PDU sets infinity for the frequency switching price which prevents the whole
network from unstable frequency oscillations.

3. The PDU sets and announces the power prices for each AP according to (4-31).

4. Informed by the PDU, each AP optimizes its transmission power following (4-34).

5. Go back to step 3 until the iteration converges.

The performance evaluation of this tailored pricing scheme is illustrated in the next

section.

4.6 Performance Evaluation

4.6.1 Legacy IEEE 802.11 Devices

In this subsection, we first investigate the performance of NETMA, NTMG and

NTMGP with legacy devices, i.e., the scenarios we considered in Section 4.3 and

Section 4.4. We assume the wireless mesh access network has N homogeneous APs.

The other simulation parameters are summarized as follows.

* Each AP has a maximum power pmax = 100mW and a minimum power pmi =
10mW and 10 different power levels as [10mW, 20mW, 100mW].

The noise experienced at each receiver is assumed identical and has a power of
2mW.

* All APs use IEEE 802.11 b standard as the MAC protocol. In other words, each AP
has four feasible data rate, 1, 2, 5.5, 11 Mbps and 3 non-overlapping channels, i.e.,
c = 3.

Without loss of generality, we assume that the received power is inversely
proportional to the square of the Euclidian distance.

The smoothing factor 7 decreases as r = 10/k2, where k is the negotiation step.

The stopping criteria for NETMA and NTMG are the maximum number of
iterations, denoted by w.

For the sake of simplicity, we utilize a table-driven rate adaption algorithm provided

in Table 4-1. Note that our results can also be applied to arbitrary propagation models,

rate adaptation algorithms and underlying multi-channel multi-rate MAC protocols.

4.6.1.1 Example of small networks

We first consider a small wireless mesh access network with 5 APs, i.e., N =

5. All APs are randomly located in a square of 10-by-10 area. The global optimum

solution is obtained by enumerating all feasible strategies, i.e., (3 x 10)5 profiles, as the

performance benchmark. We first investigate the cooperative scenario where NETMA

mechanism is applied. Next, the non-cooperative scenario is considered and each AP

operates at the maximum power and adjusts the frequency only. The stopping criteria

for both NETMA and NTMG are the maximum number of iterations where W = 200. The

performance comparison is shown in Figure 4-4.

N=5, c=3

34
S32
30
S28 NETMA
S...NTMG
26
H--
S24
22
20 1
0 20 40 60 80 100 120 140 160 180 200
Iteration

Figure 4-4. Performance evaluation of the wireless mesh access network with N = 5 and
c=3

N=5, c 3
-o- AP 1
28- AP2
-I --AP4
S26 I -AP5
N 24 I
22 I

S16 -
o i I

I 14-
12 1

0 20 40 60 80 100 120 140 160 180 200
Iteration

Figure 4-5. The trajectory of frequency negotiations in NETMA when N = 5 and c = 3

N=5, c 3
-*--AP1
90 -1 AP2
H si *.,-AP4
80 I -*-AP5

50* I I I

S

0 20 40 60 80 100 120 140 160 180 200
Iteration

Figure 4-6. The trajectory of power negotiations in NETMA when N = 5 and c = 3

As indicated by the OP curve, the global optimum obtained by enumeration

approach functions as the upper bound of the overall throughput. In Figure 4-4, we

observe that NETMA gradually catches up with the global optimum as negotiations

go. As expected, the non-cooperative APs yield remarkably inferior performance in

terms of overall throughput, depicted by the NTMG curve. The inefficiency is due to

the selfish behavior that APs transmit at the maximum power and are regardless of

the interference. The existence of Nash equilibrium in both CTMG and NTMG are

substantiated by the convergence of curves in Figure 4-4. Figure 4-5 and Figure 4-6

depict the trajectories of frequency negotiations and power level negotiations in NETMA,

respectively. At the initialization, each AP randomly picks a frequency and a power level

and negotiates with each other following NETMA mechanism, until the optimum Nash

equilibrium is achieved. Note that when the frequency vector and power vector converge

in Figure 4-5 and Figure 4-6, the corresponding overall throughput obtained by NETMA

catches the global optimum in Figure 4-4 simultaneously.

4.6.1.2 Example of large networks

We now consider a large wireless mesh access network with 20 APs. The

enumeration approach is no longer feasible in this scenario due to the enormous

strategy space. The 20 APs are randomly scattered in a d-by-d square, where the

side length d is a tunable parameter in simulations. We investigate both cooperative

and non-cooperative cases represented by NETMA and NTMG curves, where the

maximum number of iterations is set to w = 1000. Figure 4-7 pictorially depicts the

performance inefficiency of NTMG caused by the non-cooperative APs which transmit

at the maximum power. The average throughput per AP is calculated by averaging the

results of 50 simulations, for each value of the side length d. In Figure 4-7, it is worth

noting that as the side length d gets bigger, the performance gap between NETMA

and NTMG reduces. The reason is that when the area is large, the impact of mutual

interference is less severe and so is the performance deterioration. However, when the

network is crowded, i.e., d is small, the selfish behaviors are remarkably devastating.

To alleviate the throughput degradation by the non-cooperative APs, we implement

the linear pricing scheme introduced in Section 4.4. The throughput improvement is

illustrated as NTMGP in Figure 4-7. It is noticeable that by utilizing the proposed pricing

scheme, the efficiency of Nash equilibrium is dramatically enhanced, especially for

crowded networks. Therefore, the selfish incentives of the non-cooperative APs have

been effectively suppressed.

4.6.2 ACM-Capable Devices

In this subsection, we investigate the performance of our model with ACM-capable

devices. Since the major difference lies in the improved pricing scheme tailored for this

N =20, c= 3

10
95

7
65
B 9 ........ .. N**
'-p..' NETMA

100 110 120 130 140 150 160 170 180 190 200
Side Length

Figure 4-7. Performance evaluation of the wireless mesh access network with N = 20
and c = 3

specific scenario, we will only provide the performance evaluation of the tailored pricing

scheme, i.e., NTMGP, in order to avoid duplicate results. We consider a populated

network where 30 APs are randomly scattered in an 100m-by-100m square, i.e., N = 30.

All other simulation parameters are the same as in the previous subsection except that

the power is a continuous variable in this scenario. We first investigate the performance

in terms of overall achievable rate of the network when no pricing scheme is applied, as

a performance benchmark. Afterwards, the tailored pricing scheme is implemented to

improve the equilibrium efficiency, a.k.a., the overall performance at the equilibrium. As

shown in Figure 4-8, the overall achievable rate of the network is dramatically improved

by the pricing scheme. The PDU adapts the announced price for each AP according to

the optimum power, which is calculated by the previous announced price, in an iterative

fashion. The aggregated achievable rate converges as the price setting iteration goes,

as depicted by Figure 4-8.

Figure 4-9 shows the trajectories of the power utilization prices for each AP The

counterpart of actual utilized power is illustrated in Figure 4-10, where the curves start at

,,ax and evolve with the announced price in Figure 4-9.

As observed in Figure 4-8, to achieve a better performance, the price setting

process needs to be executed iteratively, comparing with the "one shot" heuristic

55

54

S53

149
0 10 20 30 40 50 60 70 80 90 100
Iteration

Figure 4-8. Performance evaluation of the wireless mesh access network with/without
the pricing scheme

Figure 4-9. The trajectories of power utilization prices of each ACM-capable AP

pricing scheme proposed for legacy devices. Therefore, the further improvement of the

it is worth noting that even at the first iteration, where the prices are determined by pmax,

the induced equilibrium yields remarkable superior performance than the case where no

pricing scheme is applied.

4.7 Conclusions

In this chapter, we investigate the throughput maximization problem in wireless

mesh networks. The problem is coupled due to the mutual interference and hence

challenging. We first consider a cooperative case where all APs collaborate with each

other in order to maximize the overall throughput of the network. A negotiation-based

throughput maximization algorithm, a.k.a., NETMA, is introduced. We prove that

54 f =
5 f-

Figure 4-10. The trajectories of actual power utilized by each ACM-capable AP

NETMA converges to the optimum solution with arbitrarily high probability. For the

non-cooperative scenarios, we show the existence and the inefficiency of Nash equilibria

due to the selfish behaviors. To bridge the performance gap, we propose a pricing

scheme which tremendously improves the performance in terms of overall throughput.

The analytical results are verified by simulations. In addition, we extend our model

and analytical results to the scenarios where more advanced APs are utilized, i.e., the

devices with the adaptive coding and modulation capability. In this scenario, we propose

a tailored pricing scheme which remarkably improves the overall performance in an

iterative fashion.

CHAPTER 5
CROSS LAYER INTERACTIONS IN WIRELESS SENSOR NETWORKS

5.1 Introduction

Wireless sensor networks have attracted significant attention in both industrial

and academic communities in the past few years, especially with the advances in

low-power circuit design and small size energy supplies which significantly reduce

the cost of deploying large scale wireless sensor networks. The sensor networks

can sense and measure the physical environment, e.g., temperature, speed, sound,

radiation and the movement of the object etc. In addition, wireless sensor networks have

become an important solution for military applications such as information gathering

and intrusion detections. Other implementations of the wireless sensor networks include

the healthcare body sensor networks, vehicular-to-roadside communication networks,

multimedia sensor networks and underwater communication networks. For more

discussions on the wireless sensor networks, refer to the survey papers such as [84]

and [85].

Since the sensor nodes in the network are usually placed in hostile environments,

wireless transmissions among sensor nodes are strongly preferred. In addition, due

to the restrained size of wireless sensor nodes, the computational capability of a

single node is limited. Therefore, the measured information is usually transmitted to a

remote data processing center (DPC) for further data analysis. Furthermore, due to the

unreliable wireless links, multiple data sinks may exist in the network which collect the

measured data and transmit the packets to the DPC node securely and reliably, possibly

through the Internet.

Before the wide deployment of wireless sensor networks, a systematic understanding

on the performance of the multi-hop wireless sensor networks is desired. However,

finding a suitable and accurate analytical model for wireless sensor networks is

particularly challenging. First, the time varying channel conditions among wireless links

100

significantly complicate the analysis for the network performance in terms of throughput

and experienced delay, even in an average sense [1, 86, 87]. Secondly, due to the

unpredictability of the behavior of the monitored object, the exogenous traffic arrival

to the network, i.e., the number of newly generated packets, is a stochastic process.

Therefore, to ensure the stability of the network, i.e., to keep the queues in the network

constantly finite, the analytical model of wireless sensor networks should comprise a

packets into the network. Thirdly, due to the hostile wireless communication links, a

dynamic routing scheme should be included in the analytical model. Moreover, the

model should capture the complex issue of wireless link scheduling which is significantly

challenging due to the mutual interference of wireless transmissions. Lastly, in order

to fully explore the network resource and to mitigate the network congestion, an

appropriate analytical network model should be able to dynamically deliver packets

through multiple data sinks and thus an automatic load balancing solution can be

achieved.

In the existing literature, most of the proposed models for wireless sensor networks

rely on the fluid model [6], where a flow is characterized by a source node and a

specific destination node, e.g., [3, 4, 6, 7]. However, this model is not applicable to the

cases where the generated packets can be delivered to any of the sink nodes, i.e., the

destination node is one of the sinks and is selected dynamically. Moreover, this fluid

model neglects the actual queue interactions within the wireless sensor network. In this

work, to study the cross layer interactions of the multi-hop wireless sensor networks, we

propose a constrained queueing model where a packet needs to wait for service in a

data queue. More specifically, we investigate the joint rate admission control, dynamic

sensor network through a set of interconnected queues. Due to the wireless interference

and the underlying scheduling constraints, at a particular time slot, only a subset

of queues can be scheduled for transmissions simultaneously. To demonstrate the

effectiveness of the proposed constrained queueing model, we investigate the stochastic

network utility maximization (SNUM) problem in multi-hop wireless sensor networks.

Based on the proposed queueing model, we develop an adaptive network resource

allocation (ANRA) scheme which is a cross layer solution to the SNUM problem and

yields a (1 c) near-optimal solution to the global optimum network utility where

e > 0 can be arbitrarily small. The proposed ANRA scheme consists of multiple layer

components such as joint rate admission control, traffic splitting, dynamic routing as

algorithm which only requires the instantaneous information of the current time slot and

hence significantly reduces the computational complexity.

The rest of the chapter is organized as follows. Section 5.2 briefly summarizes

the related work in the literature. The constrained queueing model for the cross layer

interactions of wireless sensor networks is proposed in Section 5.3. The stochastic

network utility maximization problem of the wireless sensor network is investigated in

Section 5.4, where a cross layer solution, a.k.a., the ANRA scheme is developed. The

performance analysis of the ANRA scheme is provided in Section 5.5. An example

which demonstrates the effectiveness of the ANRA scheme is given in Section 5.6 and

Section 5.7 concludes this chapter.

5.2 Related Work

To capture the cross layer interactions of multi-hop wireless sensor networks,

several analytical models have been proposed in the literature. For example, in [3-7,

88], the multi-hop network resource allocation problem has been studied through a fluid

model. Each flow, or session, is characterized by a source and a destination node where

single path routing or multi-path routing schemes are implemented. Most of the work

rely on the dual optimization framework which decomposes the complex cross layer

interactions into separate sub-layer problems by introducing dual variables. For example,

102

the flow injection rate, controlled by the source node of the flow, is calculated by solving

an optimization problem with the knowledge of the dual variables, a.k.a., shadow prices

[6, 7], of all the links that are utilized. However, there are several drawbacks for the fluid

based model. First, to calculate the optimum flow injection rate, the information along all

paths should be collected in order to implement the rate admission control mechanism.

In a dynamic environment such as wireless sensor networks, this process of information

collection may take a significant amount of time which inevitably prolongs the network

delay. Secondly, the optimization based solutions usually pursue fixed operating points

which are hardly optimal in dynamic wireless settings with stochastic traffic arrivals

and time varying channel conditions. Thirdly, the fluid model usually assumes that

the changes of the flow injection rates are "perceived" by all the nodes along its paths

instantaneously. The actual queue dynamics and interactions are neglected.

In contrast, following the seminal paper of [12], many solutions have been focused

on the queueing model for studying the complex interactions of communication

networks. Neely et.al. extend the results of [12] into wireless networks with time

varying channel conditions. For a more complete survey of this area, refer to [17].

The key component of the queue based solutions in these papers is the MaxWeight

scheduling algorithm [12, 14]. Intuitively, at a time slot, the network picks the set of

queues which (1) can be active simultaneously and (2) have the maximum overall

weight. It is well-known that the MaxWeight algorithm is throughput-optimal in the sense

that any arrival rate vector that can be supported by the network can be stabilized

under the MaxWeight scheduling algorithm. In addition, the MaxWeight algorithm is

an online policy which requires only the information about current queue sizes and

channel conditions. However, one notorious drawback of the MaxWeight algorithm is

the delay performance. The reason is that in order to achieve the throughput-optimality,

the MaxWeight algorithm explores a dynamic routing solution where long paths are

utilized even under a light traffic load. This phenomenon is substantiated via simulations

103

by [89]. In [89], the authors propose a variant of the MaxWeight algorithm where

the average number of hops of transmissions is minimized. Therefore, when the

traffic is light, the proposed solution provides a much lower delay than the traditional

MaxWeight algorithm. However, as a tradeoff, the induced network capacity region in

[89] is noticeably smaller than that of the original MaxWeight algorithm. Consequently,

it is difficult to provide a minimum rate guarantee on all the sessions in the network.

Our work is inspired by [89]. With respect to [89], however, our work innovates in the

following ways. First, different from [89], we focus on a heavy-loaded wireless sensor

network. Therefore, our solution incorporates a rate admission control mechanism

which is not considered in [89]. Secondly, rather than minimizing the overall number

of hops, we maximize the overall network utility which can also ensure the fairness

among competitive traffic sessions. Thirdly, we specifically provide a minimum average

rate guarantee for every session to ensure the QoS requirement. Fourthly, instead of a

single destination scenario as considered in [89], we extend the model to cases where

multiple data sink nodes are available. Each source node can deliver the packets to any

of the sinks. Moreover, the dynamic routing and the issue of automatic load balancing

is realized by the network on the fly. Finally, while [89] treats different sessions equally

when minimizing the overall number of hops, our model priorities all the sessions

with different QoS requirements. Therefore, a more flexible solution with service

differentiations can be achieved. We will present the constrained queueing model in

the next section.

5.3 A Constrained Queueing Model for Wireless Sensor Networks

5.3.1 Network Model

We consider a multi-hop wireless sensor network represented by a directed graph

G = {N, L} where N and L denote the set of vertices and the set of links, respectively.

We will use the notation of IAI to represent the cardinality of set A, e.g., the number

of nodes in the network is |N| and IL| is the number of links. The time is slotted as

t = 0, 1, and at a particular time slot t, the instantaneous channel data rate of link

(m, n) e L is denoted by pm.n(t). In other words, link (m, n) can transmit a number of

m .n(t) packets during time slot t. Note that the value of pm,n(t) is a random variable.
Denote p(t) as the network link rate vector at time slot t. In this work, we assume

that p(t) remains constant within one time slot but is subject to changes at time slot

boundaries. The value of p(t) is assumed to be evolving following an irreducible and

periodic Markovian chain with arbitrarily large yet finite number of states. However, the

steady state distributions are unknown to the network.

At time slot t, the network selects a feasible link schedule, denoted by I(t) =

{/(t), /(t), ... IIL(t)} where /1(t) = 1 if link / is selected to be active and /1(t) = 0
otherwise. The set of all feasible link schedules is denoted by Q(t) which is determined

by the underlying scheduling constraints such as interference models and duplex

constraints. Therefore, selecting an interference-free link schedule in the network graph

g is equivalent to the process of attaining an independent set in the associated conflict

graph g, where the vertices are the links in g and a link exists in 0 if the two original

links in g cannot transmit simultaneously.

5.3.2 Traffic Model

There are a number of |S1 source nodes in the wireless sensor network which

consistently monitor the surroundings and inject exogenous traffic to the network. The

generated packets need to be delivered to the remote data processing center (DPC) in a

multi-hop fashion. To simplify analysis, we assume that each source node is associated

uniquely with a session. The set of source nodes is denoted by S = {n0, no, nos}

where ns, s = 1, .. |S is the source node of session s. It is worth noting that the

following analysis can be extended straightforwardly to the scenarios where each source

node may generate multiple sessions.

There are |DI number of sinks in the network which are connected to the remote

data processing center via the Internet. In other words, the sink nodes can be

105

viewed as the gateways of the wireless sensor network. Denote the set of sinks as
D = {da, d2, dlDo}. In this work, we consider a general scenario where the data
packets from a source node can be delivered to the DPC via any of the sink node in
D. Therefore, different from the existing literature such as [3, 88, 89], the source nodes
do not specify the particular destination node for the generated packets. The selection
of the destination node is achieved by the network via dynamic routing schemes. The
network topology considered in this work is illustrated in Figure 5-1.
Data Processing Center (DPC)

O
Internet
Internet

..-.... : --- --,,.

0 -,

F Sink Node 'I Source Node

Figure 5-1. Topology of wireless sensor networks

For a particular node in the network, say node n, we denote Od as the number of
minimum hops from node n to the d-th data sink in set D. Define

n = d = ,... |D| (5-1)
d

as the minimum value of O for node n, i.e., the minimum number of hops from node
n to a sink node in set D. We assume that node n is aware of the value of n, as well
as those values of the neighboring nodes, which are attainable via precalculations by
traditional routing mechanisms such as Dijkstra's algorithm.

106

At time slot t, the exogenous arrival of session s, i.e., the number of new packets1

generated by the source node of session s, is denoted by As(t). We assume that there

is an upper bound for the number of new packets within one time slot, i.e., As(t) <

Amax, Vs, t. For ease of exposition, we assume that As(t) is i.i.d. over time slots with an

average rate of As. However, the data rates from multiple source nodes can be arbitrarily

correlated. For example, if the wireless sensor network is deployed for monitoring

purpose, it is very likely that a movement of the object will trigger several concurrent

Denote the vector A = {A1, f l Alsl} as the network arrival rate vector. The network

capacity region A is thus defined as all the feasible2 network arrival vectors that can

be supported by the network via certain policies, including those with the knowledge

of futuristic traffic arrivals and channel rate conditions. In this work, we consider a

heavy-loaded traffic scenario where the network arrival vector A is outside of the network

capacity region. Therefore, in order to achieve the network stability, a rate admission

control mechanism is implemented at the source nodes. More specifically, at time slot

t, we only admit a number of Xs(t) packets into the network from the source node of

session s, i.e., nO. Apparently, we have

Xs(t) < As(t), Vs, t. (5-2)

In addition, we assume that each session has a continuous, concave and differentiable

utility function, denoted by Us(Xs(t)), which reflects the degree of satisfaction by

1 We assume that the packets have a fixed length. For scenarios with variable packet
lengths, the unit of data transmissions can be changed to bits per slot and the following
analysis still holds.
2 Note that additional constraints may be imposed. For example, the constraints
on the minimum average rate and the maximum average power expenditure can be
enforced. For more discussions, please refer to [17].

107

transmitting Xs(t) number of packets. It is worth noting that by selecting proper utility

functions, the fairness among competitive sessions can be achieved. For example, if

Us(X5(t)) = Iog(Xs(t)), a proportional fairness among multiple sessions can be enforced

[4, 10, 11].

5.3.3 Queue Management

For each node n in the network, there are I N| number of queues that are

maintained and updated. The queues are denoted by Qn,h, where h = ,, .- |IN 1.

Note that I N| 1 is the maximum number of hops for a loop-free routing path in the

network. The packets in the queue of Qn,h are guaranteed to reach one of the sink

nodes in set D within h hops, as will be shown in Section 5.4. It is interesting to observe

that for a newly generated packet by session s, the source node, i.e., n,, can place it in

any of the queues of Q, ,h, where h = ,o, I INI 1, for further transmission. That is to

say, consecutive packets from the source node n, may traverse through different number

of hops before reaching a destination sink node in set D. Therefore, when a new packet

is generated, the source node needs to make a decision on which queue the packet

should be placed, namely, traffic splitting decision. In addition, the decision should be

made promptly on an online basis with low computational complexity.

With a slight abuse of notation, we use Qn,h to denote the queue itself and Qn,h(t) to

represent the number of queue backlogs3 in time slot t. For a single queue, say Qn,h, it

is stable if [13, 14]

lim g(B) -> 0 (5-3)
B-oo

where
T-l
g(B) = lim sup T-Pr(Qnh(t)> ).
too
The network is stable if all the individual queues in the network are stable.

3 In the unit of packets.

108

For a link (n,j) e L, we require that the packets from Qn,h can be only transmitted to

Q,h-1, if exists. Therefore, the queue updating dynamic for Qn,h is given by

OUnh ) ,(t) U (t) -+ m ('h +lt) Xh(t)nnO (5-4)
(nj)GL (m,n)EL s

where [A]+ denotes max(A 0) and un (t) represents the allocated data ate for the

transmissions of Qn,h Qj,h-1 on link (n,j), at time slot t, and
N-1
n,j (t) = Unj(t)
h=pn

where unj(t) = pnj(t) if Inj(t) = 1, i.e., link (n,j) is scheduled to be active during time

slot t, and un(t) = 0 otherwise. The notation of xh(t) denotes the number of packets
that are admitted to the network for session s and are stored in queue Qn,h for future

transmissions. The indicator function 6A = 1 if event A is true and 6A = 0 otherwise.
Note that the inequality in (5-4) incorporates the scenarios where the transmitter of a

particular link has less packets in the queue than the allocated data rate. We assume

that during one time slot, the numbers of packets that a single queue can transmit and
receive are upper bounded. Mathematically speaking, we have

SUmn '(t) < u., Vn, h, t, (5-5)
(m,n)EL
and

Sj (t) < uot, Vn, h, t. (5-6)
(nj)CL

5.3.4 Session-Specific Requirements

In this work, we consider a scenario where each session has a specific rate
requirement as. Therefore, to ensure the minimum average rate, we need to find a policy

109

that

lim Xs(t) > a, Vs. (5-7)
T T
t=0
In addition, we assume that each session in the network has an average hop requirement

/s. More specifically, define

INI-I
Ms(t) = hXh(t) (5-8)
h= no

where
INI-I
SXh(t) =X(t),Vs, t
h= no

We require that for each session s,

T-l
lim 5 Ms(t) < 3, Vs. (5-9)
t=0

Note that the average hop for a particular session s is related to the average delay

experienced and the average energy consumed for the packet transmissions of session

s. Therefore, by assigning different values of as and /s, a prioritized solution among

multiple competitive sessions can be achieved for the network resource allocation

problem.

5.4 Stochastic Network Utility Maximization in Wireless Sensor Networks

In the previous section, we propose a constrained queueing model to investigate

the performance of multi-hop wireless sensor networks. The model consists of several

important issues from different layers, including the rate admission control problem, the

dynamic routing problem as well as the challenge of adaptive link scheduling. To better

understand the proposed constrained queueing model, in this section, we will examine

the stochastic network utility maximization problem (SNUM) in multi-hop wireless sensor

networks. As a cross layer solution, an Adaptive Network Resource Allocation (ANRA)

scheme is proposed to solve the SNUM problem jointly. The proposed ANRA scheme is

an online algorithm in nature which provably achieves an asymptotically optimal average

110

overall network utility. In other words, the average network utility induced by the ANRA

scheme is (1 c) of the optimum solution where c > 0 is a positive number that can be

arbitrarily small, with a tradeoff with the average delay experienced in the network.

5.4.1 Problem Formulation

Recall that every session s possesses a utility function Us(Xs(t)) which is

continuous, concave and differentiable. Without loss of generality, in the rest of this

chapter, we will assume that Us(X,(t)) = Iog(X,(t)). Therefore, in light of the stochastic

traffic arrival as well as the time varying channel conditions, our objective is to develop a

policy which maximizes

Stochastic Network Utility Maximization (SNUM) Problem

SE(Us(X,(t))) (5-10)
S
s.t.

* The network remains stable.

* The average rate requirements of all |S1 sessions, denoted by a = {ac, .. Qsl},
are satisfied.

The average hop requirements of all |S1 sessions, denoted by = {/1, .. s\},
are satisfied.

Note that if the underlying statistical characteristics of the stochastic traffic arrivals

and the time varying channel conditions are known, the SNUM problem is inherently

a standard optimization problem and thus is easy to solve. However, due to the

unawareness of the steady state distributions, the SNUM problem is remarkably

challenging. In addition, in wireless sensor networks, dynamic algorithmic solutions

with low computational complexity are strongly desired. In the following, we propose

an ANRA scheme to solve the SNUM problem asymptotically. The ANRA scheme is

a cross layer solution which consists of joint rate admission control, traffic splitting,

dynamic routing as well as adaptive link scheduling components. Moreover, the ANRA

algorithm can achieve an automatic load balancing solution by utilizing different sink

nodes corresponding to the variations of the network conditions. The ANRA algorithm is

an online algorithm in nature which requires only the state information of the current time

slot. We show that the ANRA algorithm achieves a (1 c) optimal solution where c can

be arbitrarily small. Therefore, the proposed ANRA algorithm is of particular interest for

dynamic wireless sensor networks with time varying environments.

5.4.2 The ANRA Cross Layer Algorithm

Before presenting the proposed ANRA scheme, we introduce the concept of

virtual queues [16, 17, 34] to facilitate our analysis. Specifically, for each session s, we

maintain a virtual queue Ys, which is initially empty, and the queue updating dynamic is

defined as

Ys(t + 1) = [Ys(t) Xs(t)]+ as, Vs, t. (5-11)

Similarly, we define another virtual queue for every session s, denoted by Zs, and the

queue dynamic is given by

Zs(t + 1) = [Z(t) ]+ + Ms(t), Vs, t, (5-12)

where Ms(t) is defined in (5-8). Note that the virtual queues are software based

counters which are easy to maintain. For example, the source node of each session can

calculate the values of virtual queues Ys(t) and Zs(t) and update the values accordingly

following (5-11) and (5-12). In addition, we introduce a positive parameter J which is

tunable as a system parameter. The impact of J on the algorithm performance will be

discussed shortly. The proposed ANRA cross layer algorithm is given as follows.

ADAPTIVE NETWORK RESOURCE ALLOCATION (ANRA) SCHEME:

Joint Rate Admission Control and Traffic Splitting (at time t):

112

For each source node, say n, there are a number of queues, i.e., Qn,oh, h =
,no, I INI 1. Find the value of h which minimizes

Zs(t)h + Qn,h(t) (5-13)

where ties are broken arbitrarily. Denote the optimum value of h as h*. The source
node n0 admits a number of new packets as

Xs(t) = min ( ), A(t)) (5-14)

where

X5() 2 (Z(t)h* + Qn,h(t)) 2Ys(t) (5-15)

For traffic splitting, the source node no will deposit all Xs(t) packets in Qno,h*.

Joint Dynamic Routing and Link Scheduling (at time t):
For each link (m, n) e L, define a link weight denoted by Wm,n(t), which is
calculated as

Wm,n(t) max (Qm,h(t)- Qn,h-(t)) (5-16)
h=
Note that if Qn,h-1 does not exist, the transmissions from queue Qm,h to Qn,h-1 are
prohibited. At time slot t, the network selects an interference-free link schedule
I(t) which solves
max pmn(t)Wm,n(t). (5-17)
I(t)en (t)
If link (m, n) is active, i.e., lmn(t) = 1, the queue of Qm. is selected for transmissions
where
h = argmaxh,, ~~N-1 (Qm,h(t) Qn,h-(t)). (5-18)

END

Note that (5-17) is similar to the original MaxWeight algorithm, introduced in [12]

and generalized in [13, 14, 34]. The dynamic routing and link scheduling are addressed

jointly by solving (5-17), which requires centralized computation. However, following

[12], many work have been focused on the distributed solutions of (5-17). Although
the distributed computation issue is not the focus of this work, we emphasize that our

proposed ANRA scheme can be approximated well by existing distributed solutions such

as [36-38, 41, 42, 45, 46, 48].

113

For the packets placed at queue Qno,h, at most h hops of transmissions are needed

in order to reach one of the sink nodes in set D. This can be verified straightforwardly

due to the requirement that a transmission from Qm,h to Qn,h-1 can occur if and only if

h 1 > ,. Moreover, the joint rate admission control and the optimum traffic splitting

components of ANRA can be implemented by the source node in a distributed fashion.

Note that in order to calculate the instantaneous admitted rate, the source node of

session s needs only to know the local queue backlog information. Moreover, the

decision of traffic splitting requires only local queue information as well. Therefore,

at every time slot, the joint rate admission control and traffic splitting decision can be

made on an online basis in accordance to the time varying conditions of local queues.

Furthermore, we will show that this simple adaptive strategy does not incur any loss of

optimality. The achieved network utility induced by the ANRA scheme can be pushed

arbitrarily close to the optimum solution. Next, we will characterize the global optimum

utility in the network and provide the main performance results of the proposed ANRA

scheme.

5.4.3 Performance of the ANRA Scheme

In this section, we first characterize the global optimum solution of the SNUM

problem in (5-10). Define U* as the global maximum network utility that any scheme

can achieve, i.e., the optimum solution of (5-10). In order to achieve U*, it is naturally

to consider more complicated policies such as those with the knowledge of futuristic

arrivals and channel conditions. However, in the following theorem, we show that,

somewhat surprisingly, the global optimum solution of the SNUM problem can be

achieved by certain stationary policies, i.e., the responsive action is chosen regardless

of the current queue sizes in the network and the time slot that the decision is made.

Recall that we have assumed that for each session, As(t) is i.i.d. over time slots.

Denote A(t) as the vector of instantaneous arrival rates of all sessions, at time slot

t. Let A be the set of all possible value of A(t). Note that for every element in A, i.e.,

Aa, a = 1, A, we have 0 Aa Amax where -_ denotes the element-wise

comparison. We use r, to represent the steady state distribution of Aa.

Theorem 5.1. If the constraints in the SNUM problem are satisfied, the maximum

network utility, denoted by U*, can be achieved by a class of stationary randomized

policies. Mathematically, the value of U* is the solution of the following optimization

problem, with the auxiliary variables pk and Rk, as

ISI+1
max Y e Us(Z paRk) (5-19)
a s k=l

s.t.

The constraints in (5-10) are satisfied.

-4 R>k Aa.

pk > 0, Va, k.

lSIl Pak a- 1, Va.

Proof. We prove Theorem 5.1 by showing that for arbitrary policy which satisfies the

constraints in the SNUM problem, the overall network utility is at most U*, which is the

optimum utility attained by a class of stationary randomized policies. In other words, we

need to show that
ST-1
Up = lim sup y Y Us(X,(t)) T-oo t t=0 s

where UP is the average network utility under a policy P.

For each state in A, say Aa, define Ra as the set of nonnegative rate vectors that

are element-wise smaller than A,. Define CR, as the convex hull of set Ra. Therefore,

any point in CR, can be considered as a feasible network admitted rate vector given

that the current arrival rate vector is Aa. Note that every point in CR, is a vector with a

dimension of IS|-by-1. Therefore, it can be represented by a convex combination of at

most S| + 1 points, denoted by Rk, k = 1, IS| + 1, according to Caratheodory's

theorem. In light of this, we first consider a time interval from 0 to T 1. Denote N,(T)

115

as the set of time slots that A(t) = Aa. Therefore, we can rewrite (5-20) as

U= limsup INa(T) Us pk R .
T--oo a T s \k

Due to the stationary assumption, we have

1 +
UP = ~~~a Z lim sup Us p( RR .
a s T-koo 1

Note that we also assume that the utility function is continuous and bounded. Therefore,

the compactness of the utility functions is assured. Next, we focus on a subsequence of

time durations, denoted by T,, i = 1, .. ,oo. Denote

Unet(T,) a U
a s k=1

It is straightforward to verify that

S= lim sup UPet(-T).
i-0oo

Due to the compactness of the utility functions, following Bolzano-Weierstrass theorem

[90], we claim that there exists a subsequence of T,, i = 1, ., oo, such that

lim U, (sl p ( ) T- ) R 5.
k=l

Denote p as the values which generate Ua, i.e.,

U = Us p k R .
\k= 1

We have

U = lim sup Unet(-T) = Ta Us pR .
ia so k=1

According to the definition of U* in (5-19), we conclude that UP < U*.

116

Intuitively, Theorem 5.1 indicates that the global maximum network utility can

be achieved by certain randomized stationary policies. However, to calculate U*, the

stationary policy needs to know the steady state distributions which are difficult to

obtain in practice. In light of this, we propose an adaptive network resource allocation

scheme, namely, ANRA, which is an online solution and does not require such statistical

information as a priori. For notation succinctness, denote

UA(t)= E (Us(X(t)))
S
as the expected network utility induced by the ANRA scheme. The performance of the

ANRA algorithm, with a parameter J, is given as the following theorem.

Theorem 5.2. For a given system parameter J, we have
1 T-1
liminf- UA(t) > U* (5-21)
t=0

where B is a constant and is given by

B = IN|(INI 1) ((out0)2 (ui, Amax )2)

+ ((as)2 (/)) (Amax) (( )4 +
S
In addition, the constraints in the SNUM problem, i.e., (5-10), are satisfied simultane-

ously.

Proof. The proof of Theorem 5.2 is deferred to Section 5.5.

It is worth noting that if we let J -> oc, the performance induced by the ANRA

algorithm can be arbitrarily close to the global optimum solution U*. However, as a

tradeoff, a larger value of J also yields a longer average queue size in the network.

According to Little's Law, a larger queue size corresponds to a longer average delay

experienced in the network. Therefore, by selecting the value of J properly, a tradeoff

117

between the network optimality and the average delay in the network can be achieved.

5.5 Performance Analysis

In this section, we provide a proof to Theorem 5.2 in the previous section. Recall

that in (5-11) and (5-12), we introduce two virtual queues, i.e., Ys(t) and Zs(t) for

each session s. Therefore, the average rate and the average hop requirements from

all sessions are converted into the stability requirements for the virtual queues. For

example, the virtual queue update of Y,(t) is given by (5-11). If the virtual queue Y, is

stable, the average arrival rate should be less than the average departure rate of the

queue, i.e.,
T-1
a, < lim X,(t,
T ooT (t)
t=0
which is exactly the minimum average rate requirement imposed by session s. By the

same token, the average hop requirement of session s is converted to the stability

problem of the virtual queue Zs. Define Q(t) = (Q(t), Y(t), Z(t)), namely, all the data

queues and the virtual queues in the network. Our objective is to find a policy which

stabilizes the network with respect to Q while maximizing the overall network utility.

We first take the square of (5-4) and have

(Qn,h(t 1))2 < ( n,h(t))2

n,h 1 I ,m h+l( \t) + h(t) 6 )
+ \ ,J (t) --+ Um n n L -- xht)(n nO
(nnj)L ( mn L s

-2Qn,h(t) U( (t)mh(t) Xh(t)6n no
(nj)EL (m,n)EL s

Since we assume that each node generates at most one session, we have

Xh(t)6,no < Amax, Vt, n.
S

118

Note that if we allow that a node can initiate multiple sessions, we have

Xh(t)6n=no < ISlAmax, Vt, n
S
where |S1 is the number of sessions in the network.
In light of (5-5) and (5-6), we have

(Qn,h(t + 1))2 ( (tf))2 < (Utt)2 +
2Qn,h(t) U( n ,h (- (t)
\( nj) L ( m,n) m,
(nj)GL (m,n)EL

2
Uin Amax )

- Xsh(t)n
S

We next sum (5-22) over all the data queues in the network, i.e., Qn,h, and have

1))2

2Qn,h(t) (n h )
(n,j)eL

( nnh ( )2
n,h

E u, 'mh+l(t
( m,m nL
(m,n)EL

(5-23)

Xh S
s

where

Bi = INI(INI

1) ((ou) 2

(uin Amax)2).

Note that Ms(t), defined in (5-8), satisfies

Ms(t)< (INI

1)2Amax.

Next, we take the square of (5-11) and (5-12) and thus have

Y(t + 1))2

(Xs(t))2

(a)2- 2Ys(t)(Xs(t)

(/s)2- 2Zs(t) (s

(5-22)

and

(Z (t

1))2 < Z(t)2
1) ) < W

(MA(t))2

Ms(t)).

119

no)

S(Qn,h(t
n,h

< Y(t))2

Similarly, we sum over all the sessions and have

(Ys(t)) < B2

2Z
S

Y )( t) (X(t)

B2 = IS|(Amax)2

Also, we obtain

(zs(t
S

1))2

Zs(t)) 2 B

2 Zs(t)(3s
S

where

B3 = (s) IS (INI 1)4 (A ax)2.
Define the system-wide Lyapuno function as
Define the system-wide Lyapunov function as

Y(YSt)2
s

Next, we define the Lyapunov drift [14] of the system as

A = E (L(Q(t 1)) L(Q(t)) IQ(t)).

Define

B = B1 + B2 B3,

we have

A < B 2 Q,h(t)E u (t)
n,h (nj)CL

2E( Y(t)(Xs(t)-0 s) Q(t)
Ss

E ^m,h, _h
umnh(t)
(m,n)eL s

.2E Zs(t)(s

- Ms(t)) Q(t)) .

120

(YS(t
S

1) 2

where

as)

Ms(t))

S(Z(t))2.
S

(5-24)

L(Q(t))= (Qn,h(t))
n,h

Next, we subtract both sides by JE( Y Us(X,(t)) Q(t)) and have

A -JE ( U(X(t)) Q(t)) S )

2 Q,h(t)E ( nE -
n,h (nj) L (m

2E(z Qnoh(t)Xh(t) Q(t)
ns h )

m ,h+
Um, n
,n) L

- 2E

'(t) Q(t))

s Y(t)Xs(t) Q(t)

2E (Z(t)Ms(t) Q(t))
S

2 Zs(t)/3
S

JE (
S

Us(X1(t)) Q(t))

We rewrite the R.H.S. of (5-25) as

R.H.S. = B + 2 Y(t)as
s ?

2 Qn,h(t)E
n,h

E ( 2Ys(t)

)5(t)

S2Z,(t)Ms(t)
S

2 Z,(t) ,
S

5 2Qn, h(t)Xh(t)
s h

JY Us(X,(t)) Q(t) .

We observe that the dynamic routing and scheduling component of the ANRA scheme is
actually maximizing

SQn,h(t)E
n,h

u(jn (t)
(nj)CL

5 uh '(t) Q(
(m,n)CL

In addition, the joint rate admission control and traffic splitting component of the ANRA
scheme is essentially maximizing

E ( 2Ys(t)Xs(t)-

with the constraints of

S2Z,(t)M (t)
s ?

5 2Qo h(t)Xh(t)
s h

Ji Us(Xs(t)) Q(t)
(5-27)

xh(t)
h

X,(t), Vs, t.

2 Ys(t)as

(5-25)

n,h I(t) Um (t)
( nj)L (m n)ChL
(n j)GL (m,n)GL

t)).

(5-26)

(5-28)

To see this, we can decompose (5-27) to show that each session s only maximizes

JU, (X(t)) + 2Y(t)Xs(t)- 2Z,(t) hXh(t) 2Qnh(t)Xh(t). (5-29)
h h

Therefore, the proposed ANRA algorithm indeed minimizes the R.H.S. of (5-25) over all

policies.

Consider a reduced network capacity region, denoted by Ae, parameterized by

> 0, as

{A|An,h + A} (5-30)

where A is the original network capacity region and
T-1
Anh = Tm X (n no. (5-31)
t=0 s
Define U* as the global optimum network utility achieved in the reduced capacity

region. Apparently, we have lim,,o U* -> U*. In addition, denote X,(0), Xh(1),

S, X(t), ... as the optimum rate sequence which yields U,. Define X, as the

average of the optimum rate sequence of session s, in the reduced capacity region. It is

straightforward to verify that X, + is in the original network capacity region A. Therefore,

following a similar analysis as in [13-15, 91], we claim that there exists a randomized

policy, denoted by R, which generates

n,t m,'h+(t) -Xh* --32)
E ( un(t)- t) (t)) > + C (5-32)
(nj)GL (m,n)EL

if n is one of the source nodes and

E n (t) u Mhl(t) > (5-33)
(nj)GL (m,n)s L

for other nodes. Furthermore, policy R ensures

Et) >h +c
h

122

and

E (M,(t) +e < ,)

where M,,(t) is generated by Xh (t). Due to the fact that the proposed ANRA scheme

minimizes the R.H.S. of (5-25) overall all policies, including R, we have

A -JE( Us(X,(t)) Q(t)

< B 2c ( Qnh(t) + ) Ys(t)
\ nh S

SZS(t))
s /

We next take the expectation with respect to Q(t) and obtain

L(Q(t + 1)) L(Q(t)) JE ( U(X(t))

\ n,h s s /

JEF Us Xh (t) ).
\s h/

We sum over time slots 0, T 1 and have

L(Q(T)) L(Q(O)) -1) T
L(Q(T)) L(Q(0))- JE Us(X,(t)) < TB
t=0 s

JE Us( Xsh:(t)
t=0 h

(5-34)

(5-35)

since

E (ZQn,h(t) + Y (t) ZZ(t)")
\n,h s s /
is always nonnegative. Next, we divide the both sides of (5-35) by T and rearrange

terms to have

T JE Us(Xs(t))> -JE Us Xs(t) + L(Q(0))
t=- S t-0 S h
where the nonnegativity of the Lyapunov function is utilized. Since we assume that the

initial queue backlogs in the system are finite and the virtual queues are initially empty,

123

JE Us( Xsh(t)+)Q
\ s h

taking c -> 0 and lim inf-r,, yields the performance result of the ANRA algorithm stated

in Theorem 5.2.

We next show that the constraints of the SNUM problem are also satisfied. To

illustrate this, we show that the queues in the network, including real data queues and

virtual queues, are stable. Based on (5-34), we sum over time slots 0, T 1 and

have
T-1 T-1
S2eE Qnh(t + Y(t) + Zs(t) < L(Q(0))+ J(E Us(Xs(t)) TB.
t=0 n,h s s t=0 O
(5-36)

Due to Xs(t) < Amax and the assumptions on the utility function, we claim that Us(t) is

upper bounded and denote the maximum utility within one time slot as Umax, i.e.,

Us(t) < Umax, s, t.

(5-37)

Divide the both sides of (5-36) by T and we have

T-1
Y 2cE Qn,h(t) sY,(t) SZs(t) < L(Q(O)) J|S Umax B.
t=0 n,h s s(

By taking lim supT ,o, we have

T-1
lim sup' E (t) Y(t) Zs(t)U) < x B (5-38)
T- t=0 n,h s s

Note that the above analysis holds for any feasible value of c. Denote (p as the maximum

value of c such that A, is not empty. Finally, we conclude that

T-1
lim sup T E Qnh(t) Y E (t) Zs(t)) < J < C. (5-39)
t= s n,Sh s s

The stability of the network follows immediately from Markov's Inequality and thus

completes the proof.

It is worth noting that as shown in (5-39), a large value of J induces a longer

average queue size in the network. Therefore, a tradeoff between the algorithm

performance of the ANRA scheme and the average delay experienced in the network
can be controlled effectively by tuning the value of J.
5.6 Case Study
In this section, we demonstrate the effectiveness of the ANRA algorithm numerically
through a simple network shown in Figure 5-2. We stress that, however, this exemplifying
study case reproduces all the challenging problems involved in the complex cross layer
interactions in time varying environments, such as stochastic traffic arrivals, random
channel conditions and dynamic routing and scheduling etc. As shown in Figure 5-2, the
source nodes in the network are node A and B whereas the destination sink nodes are
denoted by E and F.

Sl dl

S I

B --"-----"- D ---
82 d2

Figure 5-2. Example network topology

There are six nodes and twelve links in the network. Therefore, node A and B each
maintains four queues, from hop 2 to hop 5, and node C and D each maintains five
queues, from hop 1 to hop 5, in the buffer. At each time slot, a wireless link is assumed
to have three equally possible data rates4 2, 8 and 16. The traffic arrivals are i.i.d. with
three equally possible states, i.e., 0, 10 and 20. The minimum rate requirements of the

4 Note that the unit of data transmissions is packet per slot.
4 Note that the unit of data transmissions is packet per slot.

125

two sessions are 5 and 8 and the average hop requirements of the sessions are 30

and 10. Without loss of generality, we assume that at a given time slot, two links with a

common node cannot be active simultaneously. For example, if link A -> B is active, link

B A, A C, C A, B D and D -> B cannot be selected.

Figure 5-3 depicts the average network utility achieved by the ANRA scheme for

different values of J where each experiment is executed for 50000 time slots. We can

observe from Figure 5-3 that the overall network utility rises as the value of J increases.

However, the speed of utility improvement decreases and the achieved network utility

converges to the global optimum utility U* gradually. To demonstrate the tradeoff of

different values of J, in Figure 5-4, we show the average queue size in the network for

J = 20, 50, 200, 500, 1000, 2000, 5000, 10000 and 20000. We can see that, as expected,

the average queue size increases as the value of J gets larger. Note that the average

queue size is related to the average delay in the network. Therefore, a tradeoff between

the network optimality and the average experienced delay can be achieved by tuning the

value of J.

24
22 .... ..... .... I' '. .
I20 i
|18
Z16
14 i
12
10~- -*----------------------------------------

0 20 50 200 500 1000 2000 5000 10000 20000 50000
Values of J

Figure 5-3. Average network utility achieved by ANRA for different values of J

In Figure 5-5, we illustrate the sample trajectories of the admitted rates of two

sessions with J = 5000, for the first 50 time slots. We can observe that each session

admits different amount of packets into the network adaptively following the time varying

conditions of the network. In addition, we depict the trajectories of the four virtual

queues with the same settings, in Figure 5-6, for the first 100 time slots. By comparing

126

20 so 2 5o00 1000 2000 5000 1ooO 200o
Values ofJ

Figure 5-4. Average network queue size by ANRA for different values of J

Figure 5-5 and Figure 5-6 jointly, we can observe that for the minimum rate virtual

queue, say Y1, whenever there is the tendency that the virtual queue is accumulating,

as depicted in Figure 5-6, the corresponding admitted rate by session 1 increases

in Figure 5-5. By the definition of the virtual queue, a larger backlog of Y1 indicates

that the average departure rate of the virtual queue, i.e., the average admitted rate,

is insufficient. Therefore, the source node of session 1 will attempt to increase the

admitted rate and thus the backlog of the virtual queue will decrease accordingly where

the stability of the virtual queue can be assured.

20
S --Session 1
-A- Session 2

15

: l
I

0i

0 A
0 10 20 30 40 50
Time Slots

Figure 5-5. Sample trajectories of the admitted rates of two sessions for J 5000

127

0)

m 60
in
06
O 40

20

Figure 5-6. Sample trajectories of the virtual queues for J = 5000

In Figure 5-7, the traffic splitting decisions of the two source nodes, i.e., the hop

selections of the source nodes, are illustrated. We can observe that both source nodes

incline to utilize the queues with the smaller number of hops. The queues with longer

hops, e.g., h = 3 or 4, are used only when the queue backlogs in the queues with

smaller hops are overwhelmed. In addition, we can see that on average, session 2

utilizes a smaller number of average hops than session 1. Recall that session 2 has

a much more stringent constraint on the average number of hops than session 1, i.e.,

10 v.s. 30. Therefore, the source node of session 2, i.e., n, inclines to deposit more

packets on the queues with smaller hop counts. As a consequence, by assigning

different values of rate and hop requirements, a service differentiation solution can be

achieved by the ANRA scheme among multiple competitive sessions in the network. In

addition, a near-optimal network utility can be attained simultaneously.

5.7 Conclusions and Future Work

In this chapter, we propose a constrained queueing model to capture the cross layer

interactions in multi-hop wireless sensor networks. Our model consists of components

from multiple layers such as rate admission control, dynamic routing and wireless

link scheduling. Based on the proposed model, we investigate the stochastic network

128

4 ? -- i
S--Session 1 I *
3.5 '-- Session 2 i
SII I II
II I I I
3 i : II I luII
31 in ; I u I;
SI I I I I

0.5 II
II

IU III
1. I ', I i Ii Ij I

0.5
nS

0 10 20 30
Time Slots

40 50

Figure 5-7. Sample trajectories of the hop selections for J = 5000

utility maximization problem in wireless sensor networks. As a cross layer solution, an

adaptive network resource allocation scheme, a.k.a., ANRA algorithm, is proposed. The

ANRA algorithm is an online mechanism which yields an overall network utility that can

be pushed arbitrarily close to the global optimum solution.

As a future work, energy-aware distributed scheduling algorithms are to be studied

and evaluated. In addition, the extension of our model to wireless sensor networks with

network coding seems interesting and needs further investigation.

129

CHAPTER 6
THRESHOLD OPTIMIZATION FOR RATE ADAPTATION ALGORITHMS IN IEEE 802.11
WLANS

6.1 Introduction

IEEE 802.11 WLAN has become the dominating technology for indoor wireless

Internet access. While the original IEEE 802.11 DSSS only provides two physical data

rates (1 Mbps and 2 Mbps), the current IEEE standard provides several available data

rates based on different modulation and coding schemes. For example, IEEE 802.11 b

supports 1 Mbps, 2 Mbps, 5.5 Mbps and 11 Mbps and IEEE 802.11 g provides 12

physical data rates up to 54 Mbps. In order to maximize the network throughput, IEEE

802.11 devices, i.e., stations, need to adaptively change the data rate to combat with

the time-varying channel environments. For instance, when the channel is good, a high

data rate which usually requires higher SNR can be utilized. On the contrary, a low

data rate which is error-resilient might be favorable for a bad channel. The operation

of dynamically selecting data rates in IEEE 802.11 WLANs is called rate adaptation in

general.

The implementation of rate adaptation algorithms is not specified by the IEEE

802.11 standard. This intentional omission flourishes the studies on this active area

where a variety of rate adaptation algorithms have been proposed [92-108]. The

common challenge of rate adaptation algorithms is how to match the unknown channel

condition optimally such that the network throughput is maximized. According to the

methods of estimating channel conditions, rate adaptation algorithms can be divided into

two major categories. The first one is called closed-loop rate adaptation. Most schemes

in this approach enable the receiver to measure the channel quality and sends back

this information explicitly to the transmitter for rate adaptation. For example, the receiver

records the SNR or RSSI value of the received packet and sends this information back

to the transmitter via CTS or ACK packet. Consequently, the transmitter estimates the

channel condition based on the feedback signal and adjusts the data rate accordingly.

130

can achieve a better performance than the open-loop counterpart. However, in practice,

the close-loop rate adaptation algorithms are rarely used in commercial IEEE 802.11

devices. This is because that the extra feedback information needs to be conveyed

reliably and hence an inevitable modification on the current IEEE 802.11 standard is

needed. This non-compatibility hinders the close-loop rate adaptation algorithms from

practical implementations in current off-the-shelf IEEE 802.11 products.

by the vendors, is labeled as open-loop algorithms. The widely utilized Auto Rate

Fallback algorithm, a.k.a., ARF, falls into this category. As many other open-loop rate

adaptation algorithms, ARF adjusts the date rate solely based on the IEEE 802.11

ACK packets. For example, in Enterasys RoamAbout IEEE 802.11 card [109], two

consecutive frame transmission failures, indicated by not receiving ACKs promptly,

induces a rate downshift, while ten consecutive successful frame transmissions triggers

a rate upshift [110]. Most commercialized IEEE products follow this up/down scheme

[95, 96]. In this chapter, we focus on the open-loop rate adaptation algorithms due to

the practical merits. More specifically, we consider a threshold-based rate adaptation

algorithm which works as follows. If there are 0, consecutive successful transmissions,

the data rate is upgraded to the next level. On the other hand, if Od consecutive

transmissions failed, a rate downshift is triggered. Since ARF is merely a special

case of this threshold-based rate adaptation algorithm, our analysis can be applied to

ARF and its variants as well.

As a tradeoff with simplicity, there are several challenges existing for the threshold-based

rate adaptation algorithm. First, due to the trial-and-error based up/down mechanism, it

inherently lacks the capability of capturing short dynamics of the channel variations

[96]. To tackle this issue, Qiao et.al. propose a fast responsive rate adaptation

solution in [92]. By introducing a measure of "delay factor", the responsiveness of the

threshold-based rate adaptation algorithm can be guaranteed. The second challenge

of the threshold-based rate adaptation algorithm is the indifference to collision-induced

failures and noise-induced failures. It is worth noting that in a multiuser WLAN, an ACK

timeout can be ascribed to either an MAC layer collision, or an erroneous channel.

However, the threshold-based rate adaptation algorithm is unable to distinguish them

effectively. Therefore, excessive collisions may introduce unnecessary rate degradations

which significantly deteriorate the system performance. Attributing the actual reason

of a transmission failure, or a packet loss, is named loss diagnosis [111] and has

attracted tremendous attention from the community. For example, Choi et.al. [112]

propose an algorithmic solution, which is specific to the threshold-based rate adaptation

algorithm, to mitigate the collision effect in multiuser IEEE 802.11 WLANs. Therefore,

the performance deterioration by the "indifference to collisions" can be compensated

effectively.

While the first two challenges of the threshold-based rate adaptation algorithm have

been tackled effectively, the third major obstacle, namely, the optimal selection of the

up/down thresholds, remains as an open problem. A systematic treatment on how to

select the values of 0, and Od in the threshold-based rate adaptation algorithm is lacking

in the literature, although several heuristic solutions are proposed [93, 94].

The contribution of this work is twofold. First, we analytically investigate the

behavior of the threshold-based rate adaptation algorithm from a reverse engineering

perspective. In other words, we answer the essential yet unresolved question, i.e.,

"What is the threshold-based rate adaptation algorithm actually optimizing?", by

unveiling the implicit objective function. As a result, several intuitive observations of

the threshold-based rate adaptation algorithm can be explained straightforwardly by

inspecting this objective function. Therefore, our reverse engineering model provides

an alternative means to understand the threshold-based rate adaptation algorithm. Our

work is a complement to the recent trend of reverse engineering studies on existing

132

heuristics-based networking protocols, such as TCP (transport layer) [4, 113, 114], BGP

(network layer) [115] and random access MAC protocol (data link layer) [116]. To the

best of our knowledge, this is the first work of studying threshold-based rate adaptation

algorithms from a reverse engineering perspective. Our results explicitly show that the

values of 0, and Od play an important role in the objective function and thus the network

performance hinges largely on the selection of the up/down thresholds. In light of this,

we propose a threshold optimization algorithm which dynamically tunes the up/down

thresholds of the threshold-based rate adaptation algorithm and provably converges

to the stochastic optimum solution in arbitrary stationary random environments. We

show that the optimal selection of the thresholds significantly enhances the system's

performance.

The rest of chapter is organized as follows. Section 6.2 briefly overviews the

state-of-the-art rate adaptation algorithms in the literature. The reverse engineering

model of the threshold-based rate adaptation algorithm is derived in Section 6.3.

In Section 6.4, the threshold optimization algorithm is proposed. The performance

evaluations are provided in Section 6.5 and Section 6.6 concludes this chapter.

6.2 Related Work

RBAR [103] proposes an SNR-based close-loop rate adaptation algorithm where

the rate decision relies on the feedback signal from the receiver. Specifically, the

receiver estimates the channel condition and determines a proper rate via the RTS/CTS

exchange. While consistently outperforms the open-loop rate adaptation algorithms

such as ARF, RBAR is incompatible with the current IEEE 802.11 standard by altering

the CTS frames [96]. In [105], a hybrid rate adaptation algorithm with SNR-based

measurements is proposed, where the measured SNR is utilized to bound the range

of feasible settings and thus shortens the response time to channel variations. [97] is

another example of the close-loop rate adaptation algorithms which attempts to improve

the throughput by predicting the channel coherence time, which is nevertheless difficult

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in practice. Chen et.al. introduce a probabilistic-based rate adaptation for IEEE 802.11

introduced. The basic idea is that by varying the ACK transmission rate, the appropriate

rate information is conveyed to the transmitter. Wang and Helmy [108] propose a

traffic-aware rate adaptation algorithm which explicitly relates the background traffic to

the rate selection problem. However, aforementioned solutions either rely on altering the

frame structures, or do not conform to the de facto IEEE 802.11 standard in commercial

usage1 CHARM [101] avoids the overhead of RTS/CTS by leveraging the channel

reciprocity. However, modifications on the IEEE 802.11 standardized frame structures,

e.g., beacons and probe signals, are still needed.

In light of the complexity and the incompatibility of the close-loop rate adaptation

algorithms, alternative open-loop rate adaptation algorithms are proposed. Although

usually providing inferior performance than the close-loop solutions, open-loop

algorithms soon become the predominant technique in commercial IEEE 802.11

devices due to the simplicity and the compatibility. The most popular open-loop rate

adaptation algorithm is the ARF protocol proposed by Ad and Leo [104] where a rate

upshift is triggered by ten consecutive successful transmission while two consecutive

failures induce a rate downshift. SampleRate [102] is another widely adopted open-loop

rate adaptation algorithm which performs arguably the best in static settings [96].

However, as pointed out by [96], SampleRate suffers from significant packet losses in

fast changing channels. ONOE [106] is a credit-based mechanism included in MadWiFi

drivers. The credit is determined by the number of successful transmissions, erroneous

transmissions and retransmissions jointly. However, it is pointed in [98] that ONOE is

less sensitive to individual packet failure and behaves over-conservatively.

1 For example, they largely rely on the RTS/CTS signaling mechanism which is hardly
used in practice.

As mentioned previously, one of the drawbacks of the simple open-loop rate

adaptation algorithms is the inability to discriminate the collision-induced losses and the

noise-induced losses. While several collision-aware rate adaptations are available in

the literature [95, 96, 99, 107], they base on the close-loop solutions which utilize either

RTS/CTS signaling or modifications on the standard. In [110], [112], Choi et.al. propose

a collision mitigation algorithm for open-loop ARF algorithm based on a Markovian

modeling. However, the channel conditions are assumed to be constant in their work. In

this work, on the contrary, we particularly focus on the threshold optimization to combat

with fast channel fluctuations rather than collisions. Therefore, in tandem with [110, 112],

our work provide a solution to jointly mitigate the channel fluctuations and the multiuser

collisions. For example, the up/down thresholds, i.e., 0, and Od can be first calculated by

our threshold optimization algorithm in Section 6.4. Next, they are subjected to further

adjustments following [112] to mitigate the collision effects.

6.3 Reverse Engineering for the Threshold-Based Rate Adaptation Algorithm

We consider a station in a multi-rate IEEE 802.11 WLAN. There are N stations in

the WLAN where each station, say i, has a transmission probability of pi. Note that the

equivalence of the p-persistent model and the IEEE 802.11 binary exponential backoff

CSMA/CA model has been extensively studied in [116] and [117]. Throughout this work,

we assume that the transmission probability of each station is fixed. The interaction of

the data rate and the transmission probability remains as future research. Without loss

of generality, we assume that the stations have a same transmission probability of p.

In this work, we focus on the widely deployed open loop threshold-based rate

adaptation algorithm. Not surprisingly, the speed of channel variations has a great

impact on the performance of the rate adaptation algorithm. While the original intention

of the threshold-based rate adaptation algorithm is to maximize the average throughput,

a natural question arises that whether this is indeed the case. In this section, to better

understand the impact of 0, and Od on the performance of the threshold-based rate

135

a reverse engineering approach. We assume that the RTS/CTS signals are turned off.

In a time slot, say t, we denote the channel state2 as s(t) and denote the successful

transmission probability, given the current transmission rate r(t) and the channel
condition s(t), as

Ps (s(t), r(t)) = p(1 p)"- (1 e(s(t), r(t))) (6-1)

where e denotes the frame error rate (FER) and is given by

e(s(t), r(t))= 1 (1 P(s(t), r(t)))L (6-2)

and Pe(s(t), r(t)) is the bit error rate (BER) which is determined by the current data

rate, i.e., modulation scheme, and the current channel condition. L is the frame length of
the packet. Similarly, we define PF(S(t), r(t)) = 1 Ps(s(t), r(t)) as the transmission

failure probability at time t. It is worth noting that both Ps and PF are functions of the
current data rate r(t) as well as the instantaneous channel condition s(t), which is

random. Particularly, we assume that the threshold-based rate adaptation algorithm

will increase the data rate by an amount of 6 if there are 0, consecutive successful
transmissions and decrease it by 6 if there are Od consecutive failures. Denote u = 0, 1

and d = Od 1 for notation succinctness. We define a binary indicator function ((t)
where ((t) = 1 means that the transmission at time slot t is successful and ((t) = 0

otherwise. Mathematically, the updating rule of the threshold-based rate adaptation

algorithm can be written as

2 Note that the number of feasible channel states can be potentially infinite.

136

r(t + 1) = (r(t) + o r(tr jFC(t-r- r...r._C(t-)i

+(r(t)- 6)Fr(t) oFc(t-1)-o F (t-d)-o

+r(t)Fo. (6-3)

where

1, if event x is true
Fx = (6-4)
0, if event x is false.

For example, Fc(t)1 = 1 if the transmission at time t is successful and Fc(t)1 = 0

otherwise. The symbol of o.w. denotes the event that neither 9, consecutive successful

nor Od consecutive failed transmissions happened. For simplicity, we assume that the

maximum allowable data rate is sufficiently large and the minimum data rate is zero, i.e.,

not transmitting. Define

h(t) = [r(t), r(t 1),... r(l), e(s(t), r(t)),... ,e(s(l), r(l))] (6-5)

as the history vector. In addition, we define

Z(t + 1) = {r(t + 1)h(t)} (6-6)

where S is the expectation operator.

Condition:

(C.1) The channel states between two consecutive successful transmissions or two

consecutive failed transmissions are independent random variables.

We emphasize that the restrictive condition (C.1) is not our general assumption in

this work. If (C.1) is satisfied, however, the derivation of the reverse engineering analysis

can be presented in a more concise form, as will be shown shortly.

137

First, we obtain

S{Fr(c )-= 1 (t-1)= 1 ... c(.t-u)=1h(t)}

= Pr {[F(t)-= = 1, F(t-1) = = 1, ... Fr(t-u)1 = lh(t)} (6-7)

If condition (C.1) is satisfied, (6-7) can be further decomposed as

S {FtC(t)=1C(t-1)=1 ... c(t-u)-1 h(t)}

= Pr {c(t) = 1lh(t)} x Pr {r(t-1)=1= 1lh(t)}

x Pr {Fr(t-_u)= lh(t)}

= ]Ps(s(t- k), r(t k)) (6-8)
k=-
Similarly, we have

S (r(t)=o0 ((t-1)=O ... (t-d)O I h(t)}

= Pr {r(t)o = 1, Fr(t-1)=o = 1, Fr(t-u)o = l|h(t)} (6-9)

If (C.1) is assumed to be valid, we can obtain

S ({F(t)=O (c(t- 1)0 ... F (t-d)= I h(t)}
d
= (1 Ps(s(t- k),r(t- k)) (6-10)
k=0
For notation succinctness, we will temporarily assume that (C.1) is satisfied. The

condition will be relaxed after the implicit objective function is revealed. Therefore, we

138

can write (6-6) as

Z(t + 1) = S {r(t + 1)lh(t)}

(r(t) 6) Ps(s(O
k=0
d
+(r(t) ) (1-
k=0

+r(t) 1 P P(

d
- (1 Ps(s(t
k=0

r(t) +6 [Ps(s(t
(k=0

fl(1
k=0

- Ps(s(t

-k),r(t- k))

Ps(s(t- k), r(t- k))

(t -k),r(t -k))

k),r(t k))

k), r(t- k))

Let us revisit (6-3), which can be rewritten as

+(r(t) 6)F(t)=oFc(t-1)=o F((t-d)=

+r(t)F.. r(t))

r(t) + 6(t)

(6-12)

x (r(t)

(r(t) 6)Fc(t) oF(t-1) F r(t-d) o

r(t)Fro.w. r(t) I.

(6-13)

139

k),r(t k))

(6-11)

r(t + 1) = r(t) + x ((r(t)

where

((t)

6)F (t)=1- FC(t-_u)=

6)Fc(t)-1Fc(t-1)-i F(t- )-

It should be noted that

S{((t)h(t)} = Ps(s(t k),r(t- k))
k=0
d
(i Ps(s(t k),r(t- k)). (6-14)
k=0

Therefore, we observe that (6-12) is a form of stochastic approximation with respect to

(6-11). Next, we present the reverse engineering theorem for the threshold-based rate

Theorem 6.1. The threshold-based rate adaptation algorithm of (6-3) is a stochastic

approximation which solves an implicit objective function U(t), with a constant stepsize

of 6, where U(t) is in the form of

U(t) = r(t) Ps(s(t k), r(t k))
k=0
d
(1i Ps(s(t- k),r(t- k)) }+C (6-15)
k=0

if condition (C. 1) is satisfied and /C is a constant with respect to rate r(t).

Proof. Theorem 1 follows directly from the previous analysis. Note that the threshold-based

rate adaptation algorithm can be written as

r(t + 1) = r(t) + (t) (6-16)

where ((t) is the stochastic gradient and satisfies

aU
S{~ (t)lh(t)} = r(t)h(t) (6-17)
ar(t)

Hence Theorem 1 holds.

If condition (C.1) does not hold, we can replace
U
f Ps(s(t k),r(t- k))
k=0

140

and
d
(1 Ps(s(t- k),r(t- k))
k=0
in (6-15) with

Pr {FC(t)1, F(t-,) h(t)} (6-18)

and

Pr {Fc(t)= 0,.. (t-d)= h(t)} (6-19)

respectively and the theorem remains valid.

It is worth noting that the objective function U(t) is a time-varying function which is

determined by the data rate as well as the channel conditions of the past 7 time slots

where 7 = max(O,, Od). Note that the data rate within the last 7 time slots always remains

unchanged. However, the channel fluctuations affect the successful transmission

probability Ps and thus alter the objective function U(t).

The partial derivative of the objective of U(t) given h(t), i.e.,
u d
(t = Ps(s(t k), r(t- k)) (1 Ps(s(t k), r(t k)) (6-20)
k=0 k=0

is also time-varying. If the probability that the last O, transmissions are all successful is

greater than the probability that the last Od transmissions are all failures, the station

tends to increase the data rate and vice versa. The speed of rate increasing or

deceasing is determined by the difference of these two probabilities. In other words,

the partial derivative in (6-20) could be either positive or negative which corresponds

to a rate upshift or a rate downshift. The absolute value of the instantaneous derivative

determines the speed of rate changing.

A direct computation of the partial derivative in (6-20) is challenging, if not

impossible, due to the uncertainty induced by the unpredictable stochastic channel.

Therefore, the threshold-based rate adaptation algorithm, described in (6-3), utilizes

an alternative stochastic approximation approach with the unbiased estimation of (t),

i.e., ((t) in (6-13), which significantly reduces the computational complexity since only

local information based on ACKs is required. This nature of simplicity and practicability

facilitates the popularity of the threshold-based rate adaptation algorithm and its variants

such as ARF

To achieve a better understanding of (6-15), let us consider the following extreme

cases from a reverse engineering standpoint.

* (u = 0, d = 0) = (0, = Od = 1): In this case, the threshold-based rate adaptation
algorithm increases the data rate if the current transmission is successful and
decreases otherwise. From (6-15), we can observe that this aggressive algorithm
merely compares the successful probability with the failure probability of the
current time slot and expects that the next time slot will remain the current channel
condition.

(u = +oc, d = 0) = (0, = +oC, Od = 1): From (6-15), we can see that the
derivative is always negative since the first part of (6-20) is zero. Therefore, the
threshold-based rate adaptation will keep decreasing data rate until the minimum
data rate is reached, i.e., zero, which is consistent with the intuition.

(u = 0, d = +oc) = (0, = 1, Od = +00): In this scenario, the second part of
(6-20) is always zero and thus the algorithm will keep upgrading data rate until the
maximum data rate is achieved.

(u = +oc, d = +oc) = (0, = oo, Od = +00): The derivative of (6-20) is always zero
and hence the data rate never changes.

Hence, by inspecting (6-15) and (6-20) directly, we provide an alternative means

to understand the behavior of the threshold-based rate adaptation algorithm. In

(6-3), we have assumed a constant stepsize 6 while in current off-the-shelf IEEE

802.11 devices, a discrete set of data rates are provided. However, continuous rate

adaptation can be achieved by controlling the transmission power jointly or deploying

Adaptive-Coding-and-Modulation (ACM) capable devices. Therefore, our reverse

engineering model, while fits in continuous rate scenarios, provides an approximate

model for discrete rate adaptation scenarios. We believe that the reverse engineering

result in this work provides a first step towards a comprehensive understanding on the

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implicit objective function of (6-15), the interactions of rate adaptations among multiple

IEEE 802.11 stations can be investigated in a game theoretical framework.

It is immediate to observe from (6-15) and (6-20) that the selection of 0, and Od

has significant impact on the performance of the rate adaptation algorithm. Ideally, the

rate adaptation algorithm attempts to find the data rate which maximizes the expected

throughput in the next time slot, i.e.,

r' = argmaxr x Ps(s(t + 1), r). (6-21)

Define

V= r x Ps(s(t+ 1), r). (6-22)

Therefore, if we can estimate the value of av and relate it by

OV
(t) (6-23)
Or

then the threshold-based rate adaptation algorithm is indeed optimizing the expected

throughput via the stochastic approximation approach. However, to achieve a derivative

estimation of a general non-Markovian system is non-trivial [118] [119]. Moreover, if the

channel appears totally random, e.g., non-stationary and fast fading, there exists no

effective optimization-based solution unless certain level of knowledge on the channel

randomness is available. Therefore, in the next section, we will consider a stationary

random channel environment. Nevertheless, the stochastic channel can be slow-varying

or fast-varying following arbitrary probabilistic distributions. We propose a threshold

optimization algorithm which provably converges to the stochastic optimum values of 0,

and Od and the overall performance of the network is remarkably enhanced.

6.4 Threshold Optimization Algorithm

In this section, we model the stochastic channel as a stationary random process

denoted by s(t). It is worth noting that if the channel is quasi-static, i.e., s(t) is a

143

piecewise constant function, the probing-based rate adaptation algorithms, e.g.,

SampleRate [102], can achieve good performance. It is interesting to observe that

even in a quasi-static environment, for different channel conditions, the optimal values of

0, and Od may be different. One feasible way to find the optimum values of 0, and Od with

different channel conditions, in a quasi-static environment, is the sample-path based

policy iteration approach introduced in [120] and [121], based on the Markovian model of

the threshold-based rate adaptation algorithm proposed in [110].

However, it is arguable that the channel environment is stochastic and time-varying

in nature. Therefore, it is not unusual that the channel condition has already changed

before the optimization algorithm has reached a steady state solution. The algorithm

will be thereby consistently chasing after a moving object and thus the thresholds are

always chosen suboptimally. This is more severe in a fast fading stochastic environment.

In light of this, we alternatively pursue the stochastic optimum values of the thresholds

in a time-varying and potentially fast changing channel environment. That is to say,

we attempt to find the set of values for 0, and Od which maximize the expected system

performance with respect to the random channel. Before elaborating further, we briefly

outline the learning automata techniques based on which our threshold optimization

algorithm is proposed.

6.4.1 Learning Automata

Learning automata techniques are first introduced in the control community where

in many scenarios, the system is time-varying and stochastic in nature. Therefore,

stochastic learning approaches are desired to address the stochastic control problem in

random systems. The basic idea of learning automata techniques can be described as

follows. We consider a stochastic environment and a set of finite actions available for the

decision maker. Each selected action induces an output from the random environment.

However, due to the stochastic nature, the outputs for a given input may be different

at different time instances. Based on observations, a learning automata algorithm is

expected to find an action which is the stochastic optimum solution in the sense that the

expected system's objective is maximized.

At each decision instance, the decision maker selects one of the actions according

to a probability vector. The selected action is fed to the stochastic environment as

the input and a random output is attained. The gist of learning automata techniques

lies in the provable convergence to the c-optimal solution, as will be defined later, in

arbitrary stationary random environment. Thanks to the practicality and applicability,

learning automata techniques have been broadly studied in various aspects of the

communication and networking literature such as [122][123] [124] [125]. In this work,

we propose a learning automata based threshold optimization algorithm which finds the

stochastic optimum values of the up/down thresholds efficiently in any stationary yet

potentially fast-varying random channel environment. The detailed implementations are

introduced next.

6.4.2 Achieving the Stochastic Optimal Thresholds

We consider an IEEE 802.11 station as the decision maker which adjusts the values

of 0, and Od. Without loss of generality, we assume that the maximum value of 0, and

Od are m, and md, i.e., the feasible set of 0, is A, = {1, m} and that of Od is given
by Ad = {1, md}. The station maintains two probability vectors P, and Pd for A,
and Ad, respectively. The k-th element in P,, i.e., P,, denotes the probability that u, is

set to k. pk is defined analogously. Simply put, at each decision instance, the threshold

optimization algorithm randomly determines the values of L, and Od according to P, and

Pd. Next, the probability vectors, i.e., P, and Pd, are updated and the iteration continues
until convergence, i.e., Prm = 1 and Pn = 1 where m* and n* denote the stochastic

optimal values of O, and Od, respectively.

Define a time series denoted by t = [to, t1, .. ] where to denotes the starting time

and other elements represent the exact time instances when the data rate is changed.

145

Denote

T(j) = [tj_, tj], j = 2, ..

as the j-th time period, or time duration. It is worth noting that within a particular time

period, say j, the value of the data rate, the values of thresholds, i.e., 0, and Od are
all fixed numbers. We denote these period-dependent parameters as3 r(j), Ou(j) and

Od(j). With this observation, we utilize the time series t as the time series of decision
instances. More specifically, for example, at time tj, a rate change is triggered either by

Ou(j) consecutive successful transmissions or Od(j) consecutive failed transmissions.
In tandem with the data rate up/down shift, new values of the thresholds, i.e., 0,(j + 1)
and d(j + 1) are determined by the threshold optimization algorithm according to the

probability vectors Pu and Pd.

Besides Pu and Pd, we introduce additional vectors for A, and Ad, namely, the
counting vectors Cu, Cd, the surplus vectors S,, Sd, the estimation vectors Du, Dd and

the comparison vectors Z,, Zd. The definitions of parameters of the algorithm are
provided as follows.

Algorithm:
Parameters4:

- Pu (Pd): The probability vector for 0, (Od) over Au (Ad).

- m. (md): The maximum value of O, (0d), or equivalently, the cardinality of Au (Ad).

- CL (Cd): The counting vector of O, (0d) where the k-th element, i.e., Ck (Cdk),
denotes the times that k has been selected as the value of 9, (Od).

Su (Sd): The surplus vector of 9, (0d) where the k-th element, i.e., SI (S), denotes
the accumulated throughput with Ou = k (Od = k).

3 Note that with a slight abuse of notation, we use j to denote the j-th time duration.
4 We present the analogous definitions in the parenthesis.

146

SD (Dd): The estimation vector of 0, (Od) where the k-th element, i.e., Dk (Dk), is
calculated by DI = (Dk = ).

R: The resolution parameter which is a positive integer and is tunable by the
station.

6 (6d): The stepsize parameter of Q, (Od) and is given by 6, = R (bd = -R).

c ((cd): The perturbation vector of L, (Od) where the k-th element, i.e., k ( k)), is
a zero mean random variable which is uniformly distributed in [- P +P ] P
where p is a system parameter and is controllable by the station. The notation of
[a, b]y represents [max(a, y), min(b, x)].

SZ (Zd): The comparison vector of L, (Od) where the k-th element, i.e., Zk (Zd), is
given by Zk = Dk + -Pk (Zd = Dk + k).

B: The predefined convergence threshold, e.g., 1, which is determined by the
station.

J: A running parameter which records the updated maximum achieved throughput
during one time period.
At time to:
Initialization:

The station sets Pu = [pi, pi,,* pm] where pi = for all 1 < < m.
Similarly, Pd is given by [pi, .. pi,- Pm] where pi = for all 1 < i < md.

Initializes C,, Cd, Su, Sd, DL and Dd to zeros.

Randomly selects the values of O6(1) and Od(1), say m, n, according to Pu(1) and
Pd(l).
Transmits with ,O = m and Od = n until T(1) ends, i.e., a data rate change is
triggered.

Records the average throughput within T(1) as J.
At time tj(j > 1):
Do:

Records the average throughput during T as 0(j). If O(j) > J, sets J = 0(j) and
remains J unchanged otherwise.

147

Updates the m-th element in the surplus vector S, and the n-th element in Sd by
adding the measured normalized throughput of the last time period, as Sm
SM +0) and S, = S, + )

Updates the counting vectors by adding one to the m-th counter in C, and the n-th
counter in Cd, as CL = C, + 1 and C = Cn + 1.

Updates the m-th element in DL and the n-th element in Dd by D, = s and
d,- c "-

For every element in Z, and Zd, updates Zk = Dk + P, k = 1, me and
Zk = Dk + k = 1, ... md.

Finds the element in Z, with the highest5 value of ZLk, k = 1, m me, say, the if-th
element in A,.

Similarly, finds the element in Zd which has the highest Zk, k = 1, .. md, say, the

Updates the probability vectors of Pu and Pd as

Pk = max(PL- ,,0) ifkkf/m,k= 1,.. ,m.
Pk = 1- Pk if k = (6-24)

and

Pdk = max(P k- d,0) if k/n,k= 1,-- ,md
Pd = 1-- Pdifk= (6-25)

where = 6 1 and d= 1
muxR mdxR
With the updated probability vectors Pu and Pd, new values of Ou and Od, i.e.,
Ou(j + 1) and Od(J + 1), are selected.

Starts the transmissions in T(j + 1) with O,(j + 1) and Od(j + 1).

Until:

max(PP) > B and max(Pd) > B where B is the predefined convergence threshold.

5 Note that a tie can be easily broken by a random selection.

148

End

The proposed threshold optimization algorithm is similar to the stochastic estimator

learning automata proposed in [126]. The key feature of this genre of learning automata

is the randomness deliberately introduced by p and Note that although p and O

are zero mean random variables, their variances are dependent on the values of C,

and Ck, respectively. Specifically, the variances approach to zeros with the increase

of the number of times that the corresponding values of 9, and Od are selected. As a

consequence, the threshold optimization algorithm inclines to more reliable stochastic

estimates and thus possesses a faster convergence behavior than other learning

algorithms [126]. The values that have been selected less frequently still have the

chance of being considered as optimal. However, the missing probability diminishes to

zero along iterations. In the algorithm, the resolution parameter R controls the stepsize

of probability adjustment in the algorithm. A smaller value of R produces a fine-grained

probability adjustment yet unavoidably prolongs the convergence time. The convergence

threshold B determines the stopping criteria of the algorithm. Therefore, a tradeoff

between optimality and convergence rate can be adjusted by tuning the value of B.

The steady state behavior of the proposed threshold optimization algorithm is

provided in the following theorem.

Theorem 6.2. The proposed threshold optimization algorithm is c-optimal for any

stationary channel environment with arbitrary distribution. Mathematically, for any

arbitrarily small e > 0 and y > 0, there exists a t' satisfying

Pr{|1 P | < e} > 1 7 Vt > t' (6-26)

and

Pr{ll- P*| < e} > 1- yVt > t' (6-27)

where m* and n* are the stochastic optimal values of O, and Od, respectively.

149

The proof of Theorem 6.2 follows similar lines as in [126] and is omitted for brevity.

For more discussions on the stochastic estimator algorithms, refer to [127] and [128]. In

the next section, we will demonstrate the efficacy of our proposed threshold optimization

algorithm via simulations.

6.5 Performance Evaluation

In this section, we evaluate the performance of the proposed threshold optimization

algorithm with simulations. For comparisons, we first implement the heuristics-based

threshold adjustment algorithms in [93] and [94]. In [93], the downshift threshold Od is

fixed to 1 and the default value of 0, is 10. After 0, successful transmissions, a data

upshift is triggered and if the first transmission after the rate upshift is successful, the

algorithm assumes that the link quality is improving rapidly [93]. Consequently, 0, is

set to a small number, e.g., 0, = 3, in order to capture the fast improving channel.

Otherwise, the algorithm assumes that the channel is changing slowly and thus a larger

value of 0, is desired, e.g., 0, = 10. We denote this threshold adaptation scheme

as DLA in our simulations. Another well-known threshold adjusting scheme, namely,

AARF, is proposed in [94]. Similarly, the rate downshift threshold is fixed to Od = 2

empirically. However, for O,, a binary exponential backoff scheme is applied. If the first

transmission after a rate upshift failed, the data rate is switched back to the previous rate

and the value of 0, is doubled with a maximum value of 50. The value of 0, is reset to 10

whenever a rate downshift is triggered.

To simulate the indoor office environment for IEEE 802.11 WLANs, we simulate

a Rayleigh fading channel environment. In other words, we assume a flat fading

environment. However, it could be either a fast fading or slow fading channel. The

Doppler spread (in Hz) corresponds to the channel fading speed where a large Doppler

slow-varying channel. It should be noted that the Doppler spread value describes the

time dispersive nature of the wireless channel [129], which is inversely proportional to

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the channel coherence time. More specifically, we relate the channel coherence time Tc

and the Doppler spread value fm as [129]

0.423
T =423 (6-28)
fm

Therefore, a larger Doppler spread value indicates a small channel coherence time

which represents a fast fading scenario. We conduct the simulations under various

IEEE 802.11 b PHY specification, i.e., the available data rates are 1 Mbps, 2 Mbps,

5.5 Mbps and 11 Mbps and the RTS/CTS signalling scheme is turned off. Since the

objective of our proposed threshold optimization algorithm is to combat with the channel

variation, the collision effect is omitted. Therefore, we consider a WLAN with one

station consistently transmitting packets to the AP However, note that [112] provides a

complementary solution to mitigate the collision effect and thus our algorithms can work

collectively as a joint solution. We emphasize that this simplification does not induce

any loss of generality since although seemingly simple, it produces all the challenging

problems involved in rate adaptation algorithms in a time-varying stochastic channel

environment. The data traffic is generated using constant bit rate UDP traffic sources

and the frame size is set to 1024 octets. The power of the transmitter and the power of

thermal noise are set to 50 mW and 1 mW, respectively. The SNR-BER relation is given

by Table 6-1 which is derived from [130].

We vary the Doppler spread value to simulate the various channel fading speeds.

For each value of Doppler spread value, the simulation is executed for 1000 seconds and

the average throughput of the following algorithms6 which are commonly based on the

6 The performance comparison of ARF and other open-loop rate adaptation
algorithms such as SampleRate and ONOE has been studied extensively in [96] and
[98] for various channel conditions.

threshold-based up/down mechanism, are compared: (1), OP the optimum throughput

attained by assuming an oracle which foresees the variation of channel and adapts the

data rate optimally. This curve is attained as a performance comparison benchmark; (2),

ARF the threshold-based rate adaptation algorithm with Ou = 10 and Od = 2; (3), DLA -

the dynamic threshold adjustment algorithm proposed in [93]; (4), AARF the threshold

adjustment algorithm proposed in [94]; (5), TOA the threshold optimization algorithm

proposed in this work. The algorithms are compared with each other in terms of the

average system throughput (in Mbps). For TOA, without loss of generality, we assume

that m, = md = 10 and the resolution parameter R is 1. The convergence threshold B is

0.999 and p is 1. The performance curves of the aforementioned algorithms are plotted

in Figure 6-1.

In Figure 6-1, we observe that except the OP curve, all other rate adaptation

Recall that a large Doppler spread value indicates a fast fading channel environment

and hence the average throughput deteriorates due to the incompetency of capturing

short channel fluctuations. Among which, ARF and AARF provide worst performance

with a slight difference. DLA performs better due to the capability of switching between

Table 6-1. SNR v.s. BER for IEEE 802.11 b data rates
SNR BPSK QPSK CCK CCK
(dB) (1Mbps) (2Mbps) (5.5 Mbps) (11 Mbps)
1 1.2E-5 5E-3 8E-2 1E-1
2 1E-6 1.2E-3 4E-2 1E-1
3 6E-8 2.1E-4 1.8E-2 1E-1
4 7E-9 3E-5 7E-3 5E-2
5 2.3E-10 2.1E-6 1.2E-3 1.3E-2
6 2.3E-10 1.5E-7 3E-4 5.2E-3
7 2.3E-10 1E-8 6E-5 2E-3
8 2.3E-10 1.2E-9 1.3E-5 7E-4
9 2.3E-10 1.2E-9 2.7E-6 2.1E-4
10 2.3E-10 1.2E-9 5E-7 6E-5

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2 4 6 8 10 12 14

Figure 6-1. Average throughput v.s. Doppler spread values

the large value and the small value of 0, for different channel conditions. Our proposed

threshold optimization algorithm, as demonstrated in Figure 6-1, consistently outperforms

other open-loop rate adaptation algorithms and bridges the performance gap with the

OP curve. This superiority becomes remarkably significant in fast fading stochastic

channel environments, i.e., scenarios with large Doppler spread values. While other

algorithms are jeopardized by the unmature rate changes due to the uncertainty

caused by the random and fast-varying channel environment, our proposed threshold

optimization algorithm, based on the learning automata techniques, is able to find

the stochastic optimum values of 0, and Od which maximize the expected system

performance. Therefore, our scheme is particularly suitable for fast changing yet

statistically stationary random channel environments where other open-loop rate

adaptation solutions usually provide unsatisfactory performance.

To illustrate the process of finding the stochastic optimum values of ~, and Od, in

Figure 6-2, we provide the trajectories of 0, and Od in a sample simulation run with a

fixed Doppler spread value of 10. It is observable that starting from the initial point, the

threshold optimization algorithm adapts the values of 0, and Od on the fly along with the

rate changes. The algorithm soon finds the stochastic optimum values and 0, and Od

converge to the optimum solutions effectively. The evolutions of the probability vectors,

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7
6
5
4
3- 3

0 20 40 60 80 100
Number of Transmissions

Figure 6-2. The trajectories of 0, and Od in achieving the stochastic optimum values

i.e., P, and Pd, are demonstrated in Figure 6-3 and Figure 6-4, respectively. Note that,

in Figure 6-3, the sixth curve which represents the value of P6 soon excels others and

approaches to 1. Correspondingly, the value of O, converges to 6 rapidly as depicted

in Figure 6-2. In Figure 6-4, the second curve approaches to unity gradually while

others diminish to zeros. As a consequence, in Figure 6-2, the value of Od converges

to the stochastic value, i.e., 2. In this sample run of simulation, the stochastic optimum

values of 9, and Od, which maximizes the expected throughput of the system, is given

by 8, = 6 and Od = 2. The proposed threshold optimization algorithm finds these values

effectively and efficiently, while providing superior performance than other non-adaptive

threshold-based rate adaptation algorithms, as demonstrated in Figure 6-1.

9o P1

--P S
-, .

Figure 6-3. Evolution of the probability vector Pu

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Figure 6-4. Evolution of the probability vector Pd

6.6 Conclusions and Future Work

In this chapter, we investigate the threshold-based rate adaptation algorithm which

is predominantly utilized in practical IEEE 802.11 WLANs. Although widely deployed, the

obscure objective function of this type of rate adaptation algorithms, commonly based

on the heuristic up/down mechanism, is less comprehended. In this work, we study

the threshold-based rate adaptation algorithm from a reverse engineering perspective.

The implicit objective function, which the rate adaptation algorithm is maximizing, is

unveiled. Our results provide, albeit approximate, an analytical model from which the

threshold-based rate adaptation algorithm, such as ARF, can be better understood.

the up/down thresholds, based on the learning automata techniques. Our algorithm

provably converges to the stochastic optimum solutions of the thresholds in arbitrary

stationary yet potentially fast changing random channel environment. Therefore, by

combining our work with the threshold adaptation scheme in [112], where the thresholds

are adjusted to mitigate the collision effects, a joint collision and random channel fading

resilient solution can be attained.

In this work, to emphasize on the impact of random channel variations, we restrict

ourselves to the scenario where all stations have a fixed and known transmission

probability p. The interaction of the transmission probabilities and the data rates seems

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Number of Transmissions

interesting and needs further investigation. Due to the competitive nature of channel

access, a stochastic game formulation may be utilized.

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CHAPTER 7
STOCHASTIC TRAFFIC ENGINEERING IN MULTI-HOP COGNITIVE WIRELESS
MESH NETWORKS

7.1 Introduction

The past decade has witnessed the emergence of new wireless services in daily

life. One of the promising techniques is the metropolitan wireless mesh networks

(WMN), which are envisioned as a technology which advances towards the goal of

ubiquitous network connection. Figure 7-1 illustrates an example of wireless mesh

network. The wireless mesh network consists of edge routers, intermediate relay routers

as well as the gateway node. Edge routers are the access points which provide the

network access for the clients. The relay routers deliver the traffic aggregated at the

edge routers to the gateway node, which is connected to the Internet, in a multi-hop

fashion.

*................ ... ...........
Figure 7-1. Architecture of wireless mesh networks

While the current deployed wireless mesh networks provide flexible and convenient

services to the clients, the performance of a mesh network is still constrained by several
each other and thus the network performance is devastated. To address this problem,
Figure 7-1. Architecture of wireless mesh networks

While the current deployed wireless mesh networks provide flexible and convenient

services to the clients, the performance of a mesh network is still constrained by several

limitations. The first barrier is due to the multi-hop nature of the wireless mesh network,

where the nodes in geographic proximity generate severe mutual interference among

each other and thus the network performance is devastated. To address this problem,

several scheduling schemes have been proposed in the literature [131]. Recently, a

novel coding-based scheme which may produce an interference-free wireless mesh

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network, is proposed [132]. Another example of the interference-free network is the

CDMA-based wireless mesh networks [133] where by assigning orthogonal codes for

each link, the network throughput is remarkably improved.

The second hinderance for the network performance is the limited usable frequency

resource. In current wireless mesh networks, the unlicensed ISM bands are most

commonly adopted for backbone communications. Not surprisingly, the wireless

mesh network is largely affected by all other devices in this ISM band, e.g., nearby

WLANs and Bluetooth devices. Moreover, the limited bandwidth of the unlicensed

band cannot satisfy the increasing demand for the bandwidth due to the evolving

network applications. Ironically, as shown by a variety of empirical studies [134], the

current allocated spectrum is drastically under-utilized. As a consequence, the urge

to explore the unused whitespace of the spectrum, which can significantly enhance

the performance of the wireless mesh networks, attracts tremendous attention in the

community [135-139].

Cognitive radios are proposed as a viable solution to the frequency reuse problem

[131]. The cognitive devices are capable of sensing the environment and adjusting

the configuration parameters automatically. If the primary user, i.e., the legitimate

user, is not using the primary band currently, the cognitive devices, namely, secondary

users, will utilize this whitespace of the spectrum. Incorporating with the established

interference-free techniques such as [132] and [18], the throughput of the wireless

mesh network can be dramatically enhanced. The protocol design for cognitive wireless

mesh networks (CWMN), or more generally, multi-hop cognitive radio networks, is

an innovative and promising topic in the community [140] and has been less studied

in the literature. In this work, we consider a cognitive wireless mesh network where

the unlicensed band, e.g., ISM band, is utilized by the mesh routers for the backbone

transmission. Moreover, each router is a cognitive device and hence is capable of

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sensing and exploiting the unused primary bands for transmissions whenever the

primary users are absent.

In this work, we investigate an important yet unexplored issue in the cognitive

wireless mesh networks, namely, the stochastic traffic engineering problem. More

specifically, we are particularly interested in how the traffic in the multi-hop cognitive

radio networks should be steered, under the influence of random behaviors of primary

users. It is worth noting that given a routing strategy, the corresponding network's

performance, e.g., the average queueing delay encountered, is a random variable. The

reason is that the available bandwidth for a particular link depends on the appearance

of all the affecting primary users. If all the primary users are vacant, a link can utilize all

available frequency trunks collectively by utilizing advanced physical layer techniques,

e.g., OFDMA. However, if all the primary users are present, the only available frequency

space is the unlicensed ISM band and thus the traffic on this link will experience longer

delay than the previous case. In other words, the performance of a traffic engineering

solution hinges intensely on the unpredictable random behaviors of the primary users.

We emphasize that in multi-hop cognitive radio networks, this distinguishing feature of

randomness, induced by the random behaviors of primary users, must be taken into

account in protocol designs. Due to the location discrepancy, it is possible that some

node is affected by many primary users while others are not. As a consequence, if we

route the traffic via this particular node, the transmissions are more likely to be corrupted

by the returns of the primary users. Apparently, a favorable solution is more inclined to

steer the traffic from those "severely-affected area", to the paths which are less affected

by the primary users. We will make this intuitive approach precise and rigorous in

this work. To our best knowledge, this work is the first work on the traffic engineering

problem in multi-hop cognitive radio networks, with a special focus on the impact of

random behaviors from the primary users.

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The rest of this chapter is organized as follows. The related work is reviewed in

Section 7.2 and Section 7.3 provides the system model of our work. The stochastic

traffic engineering problem with convexity is investigated in Section 7.4. In Section 7.5,

we extend our framework to the non-convex stochastic traffic engineering problem.

Performance evaluation is provided in Section 7.6, followed by concluding remarks in

Section 7.7.

7.2 Related Work

Traditional traffic engineering (TE) algorithms are proposed as the solution to

the traffic management of the network in a cost-efficient manner. Different from the

traditional QoS routing, the traffic engineering solution not only guarantees a certain

QoS level for each flow, but also optimizes a global performance metric over the whole

network, by splitting the ingress traffic optimally among several available paths. The

multi-path routing is usually supported by the Multi-Protocol Label Switching (MPLS)

techniques where the explicit routing path for a packet is predetermined rather than

being computed in a hop-by-hop fashion. For a pair of source and destination nodes,

the set of available paths, a.k.a., label switched paths (LSP), are established and

managed by signalling protocols such as RSVP-TE [141] and CR-LDP [142] or manually

configuration. The traditional traffic engineering solution evolves to the stochastic

traffic engineering (STE) solution when uncertainty exists in the network, e.g., the

random returns of the primary users in our scenario. TE solutions require consistent

route changes which are unfavorable in that the network will be overwhelmed by the

oscillations induced by the unpredictable behaviors of the primary users. In light of this

stability concern, STE solution alternatively pursues an optimum multi-path routing

strategy such that the expected utility of the network is maximized. The stochastic traffic

engineering with uncertainties are discussed in the literature such as in [143] and [144].

However, the previous works usually assume a probability distribution of the uncertainty,

while in our scenario, the behaviors of the primary users are completely unpredictable

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from the mesh network's point of view. Distinguishing from the previous works, we

propose an algorithmic solution which requires no prior knowledge about the distribution

of the uncertainty, in a stochastic network utility maximization framework. It is worth

noting that our work differ from the traditional state-dependent traffic engineering

solution as well. For example, in [145], a state dependent traffic engineering solution is

proposed. However, the authors assume that the system state, i.e., the current value

of uncertainty, is fully observable. In our approach, we do not assume that the ingress

node has the perfect knowledge of the current appearance of the network. We will

discuss this issue in detail in Section 7.4.

Recently, cognitive wireless mesh networks (CWMN) have attracted great

attention in the literature. In [135], the channel assignment is discussed in a CWMN.

In [136], a cluster-based cognitive wireless mesh network framework is proposed.

The infrastructure-based cognitive network is discussed in [137] with a focus on the

cooperative mobility and the channel selection schemes. The spectrum sensing and

channel selection are jointly considered in a unified framework in [138]. In addition, the

IEEE 802.16h is in the process of incorporating the cognitive radios into the WiMAX

mesh networks [139]. However, none of the previous works considers the stochastic

traffic engineering problem. Therefore, a systematic study of the impacts of the random

returns from the primary users, on the network routing performance is lacking in the

existing literature. In [146], the joint congestion control and traffic engineering problem

is considered. He et.al. propose a distributed algorithm to balance the user's utility and

the system's objective. However, the authors assume the environment is fixed and does

not consider the randomness which is the distinguishing yet usually overlooked feature

in cognitive radio networks. [147] and [148] discuss the routing issue in cognitive radio

networks yet the impact of random returns of primary users is not investigated. Hou

et.al. [149] [150] [151] [152] formulate the joint routing, power and subband allocation

problem in cognitive radio networks as a mixed-integer programming. However, the

channels' bandwidths are assumed to be fixed, i.e., the random behaviors of primary

users are still neglected. Our work is partially inspired by [153]. However, our work

differs from theirs in three crucial aspects. First, by targeting the stochastic traffic

engineering problem, our model differs from the joint power scheduling and rate control

work in [154]. Secondly, in [153, 154], the authors only consider a single-path scenario

while our work extends to a multi-path routing network where the network traffic can

be steered. Thirdly and most importantly, [153, 154] require that the current system

state is fully observable at the decision maker. To achieve this, the authors assume

a centralized mechanism which knows all the channel states of all the links over the

network. However, our work differs from [153, 154] significantly in that we do not require

that the current system's state is known, which is of great practical interest since in

multi-hop cognitive wireless mesh networks, the decision makers, i.e., the edge routers

in our scenario, cannot be aware of the appearance of all primary users in the whole

network as a priori. Moreover, our schemes enjoy a decentralized implementation, in

contrast to centralized mechanisms in [153, 154], by utilizing the feedback signals and

local information only. In our previous work of [20], we proposed a routing optimization

scheme to combat with the randomness of instantaneous traffic in non-cognitive

wireless mesh networks. With respect to [20], this work differs in the following ways.

First, in the wireless mesh networks considered in [20], the capacity of each wireless

link is assumed to be fixed, i.e., time-invariant. However, in cognitive wireless mesh

networks, due to the unpredictable appearance of primary users, the bandwidth of each

wireless link is random. Secondly, in [20], the quality of service (QoS) requirement is

not considered. Nevertheless, in this chapter, we particularly address the QoS concern

of each user, e.g., the expected accumulated delay on the paths cannot exceed a

user-specific delay tolerance, as will be elaborated in Section 7.4. Thirdly and most

importantly, the analysis in [20] was based upon the assumption that the users all have

convex utility functions. In this work, we extend the techniques to address the scenarios

162

with non-convex utility functions. We will discuss the aforementioned issues further in

the following sections.

7.3 System Model

We consider a multi-hop wireless mesh network illustrated in Figure 7-1 where

an uplink traffic model is considered, i.e., all edge routers aggregate the traffic from

clients and deliver to the gateway node via the intermediate relay routers. To ensure

connectivity, we utilize the ISM 2.4G band as the underlying common channel for the

wireless mesh network. In addition, each link can utilize the opportunistic channels,

i.e., secondary bands to increase the link's achievable data rate whenever the primary

user is vacant. We assume that there exists1 IMI primary users. Each primary user

possesses a licensed frequency channel and each mesh router is a cognitive node

which has the capability of sensing the current wireless environment. We model the

multi-hop cognitive wireless mesh network as a directional graph g where the vertices

are the nodes. We also denote link (i,j) as link e, e c E where t(e) = / and r(e) = j

We first consider a particular link denoted by (m, n). The instantaneous available

frequency bands, at time t, for a node / is denoted by I,(t), which is determined by

the current presence of the primary users. Besides the underlying ISM band, the

communication between m and n can further utilize all secondary bands within

Im(t) q In(t), if available. The current cognitive radio devices benefit largely from

capabilities. For example, by utilizing the multi-carrier modulation, e.g., OFDMA, a

cognitive radio device can utilize all the disjoint available frequency band simultaneously

[149, 150, 155-157]. At the transmitter, a software based radio combines waveforms

for different sub-bands and thus transmit signal at these sub-bands simultaneously.

1 The symbol of |X| represents the cardinality of the set X.

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While at the receiver, a software based radio decomposes the combined waveforms

and thus receives signal at these sub-bands simultaneously [151, 152, 156, 157]. In

this work, we assume a spectrum sensing scheme available that each node can sense

the presence of the primary users in range, such as [131, 158], although the time of

random returns cannot be predicted. A link will utilize all the available vacant bands and

that the cognitive radios are full-duplex and can transmit at different bands concurrently

[151, 152, 156, 157]. We further assume that some scheduling mechanism is in place or

some physical layer mechanisms are utilized such that the nodes cannot interfere with

each other during the transmissions. For example, in a multi-channel multi-radio wireless

mesh network, the channels can be assigned properly that the transmissions do not

interfere with the neighboring nodes [132, 159]. Other examples are the OFDMA/CDMA

based wireless mesh networks [133, 160] where the interference among nodes can

be eliminated by assigning orthogonal subcarriers/codes. We emphasize that this

assumption is only for the sake of modeling simplicity and does not incur any loss of

generality, as will be clarified shortly.

It is worth noting that the available bandwidth of each link in the cognitive wireless

mesh network is a random variable. For example, at time instance tl, node m has three

secondary bands available, i.e., Im(ti) = {/0, /1, 2, Is} and In(t1) = {/0, 2, ,/, 14 } due

to the location discrepancy, where band 0 is the underlying unlicensed ISM band and

1, 2, 3, 4, 5 are the licensed bands of primary users. The current bandwidth of link (m, n)

is represented by Wm,n(tl) = BWo + BW2 3 where BW, is the bandwidth of

band i. At another time instance t2, the primary user 2 returns and the bandwidth of link

(m, n) becomes Wmn(t2) = BWo + BW3. In other words, the bandwidth of links are

random variables which are determined by the unpredictable appearance of the primary

users. We model this randomness induced by the primary users as a stationary random

process with arbitrary distribution. The system is assumed to be time-slotted. In each

time slot n, the system state is assumed to be independent and is denoted by a state

vector s = {51, ... 61m }, s e S, where 5, = 1 denotes the absence of the i-th primary

user and 0 otherwise. We denote the stationary probability distribution of state s as

Ts. For the ease of exposition, we assume that the primary users are static. However,

we emphasize that our model can be extended to mobile primary users scenarios

straightforwardly. For example, if a primary user is moving following a Markovian walk

model with well defined steady state distribution, the following analysis still applies.

Without loss of generality, we express the link capacity in the form of CDMA-based

networks, i.e., the capacity of a wireless link e e E, given the system state s, is denoted

by cs, which is given by [4, 161] c W = Ws l Iog2(1 + Ky~), where Ws is the bandwidth of

link e in state s and 7y is the current SINR value of link e. The constant T is the symbol

period and will be assumed to be one unit without loss of generality [4]. The constant

K = iog(ER) where 01 and 02 are constants depending on the modulation scheme
and BER denotes the bit error rate. We will assume K = 1 in this work for simplicity

[161]. Note that our network model can be incorporated into other types of networks

such as MIMO, OFDM with TDMA or CSMA/CA based MAC protocols by modifying the

form of the capacity accordingly, which represents the achievable data rate in general.

For example, if we consider a scheduling-based MAC protocol where each link obtains

a time share of the channel access, the achievable data rate is given by c c = cq x ,'

where be is the fraction of time that the link is active following the scheduling scheme

and ce is the nominal Shannon capacity of the link.

There are |LI unicast sessions in the network, denoted by set L, where each

session / has a traffic demand di. We associate each session with a unique user.

Therefore, we will use session I and user I interchangeably. For each session I e L,

we denote the source node and destination node as 5(I) and D(I), respectively. Recall

that we assume an uplink traffic model and thus all the source nodes are edge routers

and the destination node is the gateway. Furthermore, to improve the reliability and

165

dependability, we allow multi-path routing schemes. We denote the available2 set

of acyclic paths from 5(1) to D(I) by P/ and the k-th path is represented by P /. We

introduce a parameter rk as the flow allocated in the k-th path of session I. The overall

flow of user I, represented by x1, is given as

x= r (7-1)
k= 1
0

where [x]b denotes max{min{b, x}, a}. Define an |E|-by-I|Pi matrix H1 where the element

H".k = 1 if link e is on the k-th path of P/ and 0 otherwise. Hence, H = {H1, [ HIL}

represents the network topology. Note that the traffic splitting and the source routing are

executed on the source node 5(1).

For each link e e E, there is an associated cost function, denoted by I(fe, cs)

where fe is the accumulated flow on link e. We assume the function I/ is an increasing,

differentiable and convex function of fe for a fixed ce. For example, if we assume

Il(fe, ci) = -1 when cs > fe, the cost essentially represents the delay for a unit flow

on link e under the M/M/1 assumption. Note that in our scenario, even the accumulated

flow fe is fixed, the value of cost function is random due to the state-dependent variable

ci. From the network's perspective, the stochastic traffic engineering solution will

distribute the aggregated flow among multiple paths optimally, in the sense that the

overall network utility is maximized. In next section, we will formulate the stochastic

traffic engineering problem in a stochastic network utility maximization framework [4] and

provide a distributed solution which requires no priori information about the underlying

probability distribution, i.e., 7s, of the system states.

2 The available set of multiple paths can be obtained by signalling mechanisms such
as RSVP-TE[141] or pre-configured manually. In this work, we assume a predetermined
set of acyclic paths. The protocol design for acquiring such paths is beyond the scope of
this work.

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7.4 Stochastic Traffic Engineering with Convexity

7.4.1 Formulation

In the standard network utility maximization framework, each user has a utility

function Ul(xl), which reflects the degree of satisfaction of user / by transmitting at a rate

of x1, e.g., Ul(x/) = log(x/). In this section, we assume the utility functions to be concave

and differentiable. The non-convex utility functions are considered in Section 7.5. Note

that the fairness issue can be embodied in the utility functions [4]. For example, in the

seminal paper [113], the log-utility functions are adopted to achieve the proportional

fairness among different flows.

Define a feasible stochastic traffic engineering solution as r = [rl, .. riI] where

r/ [r/1, r' ]. We can formulate the stochastic traffic engineering problem as

P:

max U (Y rk)
ICL kGcP,

s.t.

r1k < dl V/ E L (7-2)
kcG

7, r/k e, c's) < bi VI e L (7-3)
sGS k
fe < 7,csVe E (7-4)
scS

fe = S ',krk Ve GE (7-5)
/EL keP,

cs = We 1og2(1 + Kys) Ve E (7-6)
T

where e e P represents the links along the k-th path of user I. The variable in P, is

the vector of r. The first set of constraints reflect that the overall data rates of all paths

cannot exceed the traffic demand dl. The second set of constraints indicate that for

167

each user /, the expected cost has to be no more than a predefined constraint bi. The

third set of constraints represent that the aggregated flow on link e cannot exceed the

average link capacity. Apparently, if the underlying probability distribution of each state

7s is known as a priori, Pi is a deterministic convex optimization problem and thus easy

to solve. However, in practice, the accurate measurement of probability distribution is

a non-trivial task. In [20], we utilized a stochastic approximation based approach to

circumvent the difficulty of estimating the probability distribution. In the following, we will

extend this technique and develop a tailored distributed algorithm to address the issues

of time-varying link capacities as well as the user-specific QoS requirements, which are

of particular interest in multi-hop cognitive wireless mesh networks.

First, define the Lagrangian function of P1 as

L(r, A, v)

/IL keP I kePL

Svi bi T (s e (f, Cs))
I/EL sGS kEPI eEPk

Y.ie(fe -SscD)
eEGE sES

e6]E sGS
= s Y Ul( r) + A (d- r )+ vl bl
sIc Ik kk kc

sGS /GL \ keP, keP
-5r ( ik (vi(fe, cPe))}
kIEPE ) +eG P E) v

k< eP, / eE )J

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Define

M5(A, p, v)

= sup Y U (U/(Z rk) + A/(d r/) + vib
r>0
ILO kEIP, keP,

rk(l(fe, Cs )+ le)) + Jes (7-7)
kEPi eP, / eGE

Let f be the optimum solution of (7-7). We will discuss how to obtain f shortly. The dual

function of P1 is obtained by

g(A, p, v) = TsMA(A,, v). (7-8)
scS

Thus, the dual problem of P1 is given by

P2
min g(A,j, v). (7-9)
A,'P,v>O

7.4.2 Distributed Algorithmic Solution with the Stochastic Primal-Dual Approach

In this subsection, we propose a distributed algorithmic solution of Pi, or equivalently

p2, based on the stochastic primal-dual method. In order to reach the stochastic
optimum solution, the dual variables A, p and v are updated according to the following

dynamics

A(n + 1) = [Ai(n) ai(n)(C(n)] V/E IL (7-10)

pe(n+ 1) = [Pe(n) ae(nn)e(n)] Ve E (7-11)

vl(n + 1) = [v(n) b(n)pi(n)]+ V/ E L (7-12)

where [x]+ denotes max (0, x) and n is the iteration number. a (n), ae(n) and ab(n) are

the current stepsizes while (1(n), _e(n) and p/(n) are random variables. More precisely,

they are named the stochastic subgradient of the dual function g(A, p) and the following

169

requirements need to be satisfied

({(,(n)lA(1), A(n)} = Ox,g(A, p, v) V/I L (7-13)

{(e(n)|i(1),' ,~ (n)} = 9,g(A, p, v) Ve E (7-14)

({p/(n)|v(1), ... v(n)} = O,g(A, p, v) V/I L (7-15)

where ((.) is the expectation operator and A(1), A(n), p(1), p(n) and v(1), v(n)
denote the sequences of solutions generated by (7-10), (7-11) and (7-12), respectively.
By Danskin's Theorem [162], we can obtain the subgradients as

(,(n) = d TIk(n) V/I L (7-16)
kEP

)e(n) = c(n)- fe(n) Ve E (7-17)

p,(n) = bi Tk(n) I I(fe(n), c|(n)) V/ E L (7-18)
kIEPI eGeP

where /k is the optimum solution of (7-7). Note that ce(n) denotes the instantaneous

channel capacity on link e at iteration n.

We next show how to calculate Ms(A, p, v) in (7-7), i.e., finding the optimum

solution, denoted by r, which maximizes

( r + A(di rk) + vib,
/IL \ kEP, kElP,

r/k(V/(fec )+ e)) +-ece (7-19)
kIEPi eEPk eGE

Note that when updating the primal variable, i.e., r, the link costs are deterministic which
are obtained via the feedback signal, e.g., ACK messages. Therefore, by utilizing the

same stochastic subgradient approach, we have

rkn + 1) = [rk(n) + a(n)l(n)] d(7-20)
YI \"I/- "/ O 7-20

170

where

(n) k rIk( A (le vlle(fe, cS)) (7-21)
ee pk

is the stochastic subgradient measured at time n.
Theorem 7.1. The proposed algorithm converges to the global optimum of Pz with
probability one, if the following constraints of stepsizes are satisfied: (1) a(n) > 0, (2)

Z0 a0(n) = oo, and (3) 0(a(n))2 < oo, V/I L and e E, where a represents ae,
a/, ab and ar generally.

Proof. First, let us revisit the updating equations of (7-10), (7-11) and (7-12). Note that
in the stochastic subgradient approach, the measured values of (1(n), e(n) and pl(n)
are considered as the instantaneous observation of the real gradients, denoted by (1(n),

(e(n) and p/(n), respectively. We consider the relationship of (1(n) and (/(n) for instance.
The observation value, i.e., (1(n), can be rewritten as

(1(n) = (i(n) 6(i(n)) + 6((n)) i(n) + (n)

= i(n) + ((/(n)) (/(n) + (/(n) (( (n))

= (/(n) + C/(n) + (/(n) (7-22)

where is the expectation operator and

(n) = ((i/(n)) (<(n) (7-23)

1(n) = /(n) ((C/(n)). (7-24)

Note that /(n) is the difference between the expectation of the observations and the real
gradient. Hence, it is the biased estimation error term. Next, we examine that

Q(V(n)1/(" 1), (0)) = 0 Vn. (7-25)

Therefore, the series of ((n) is a martingale difference sequence [163]. The relationship

of (7-22) indicates that the observation value is the real gradient disturbed by a biased

estimation error as well as a martingale difference noise. We next investigate the

convergence conditions of the stochastic primal-dual approach. For C/(n), the following

requirement

ae(n)| (Ci(n)) ((n)| < co (7-26)
n=0
is satisfied due to the stationary assumption. Similarly,

2
(((/(n) ) = (((/(n) ((i/(n)))2) (7-27)

is bounded as well. The similar analysis can be extended to _e(n), pi(n) and rT(n) in

(7-11), (7-12) and (7-20) straightforwardly. Therefore, the standard conditions are

satisfied and the convergence result of Theorem 1 follows [8].

It is worth noting that the aforementioned distributed algorithm enjoys the merit of

distributed implementation from an engineering perspective. With the current values of

dual variables, each source node 5(I) optimizes (7-19) according to (7-21) and (7-20).

The information needed is either locally attainable or acquirable by the feedback along

the paths. For example, the channel states of the intermediate nodes along paths can

be piggybacked by the end-to-end acknowledgement messages from the destination

node, i.e., the gateway node in our scenario. The source node updates the A/ and v1

according to (7-10) and (7-12) where the needed information is calculated by (7-16)

and (7-18), respectively. For each link e, the current status of (7-17) is measured. Next,

the value of /i is updated following (7-11). The iteration continues until an equilibrium

point is reached. Note that our framework can incorporate the wireless lossy network

scenarios by replacing the flow rate with the effective flow rate in the leaky-pipe flow

model [164].

172

7.5 Stochastic Traffic Engineering without Convexity

Thus far, we have considered the scenarios where all the users have concave

utility functions. However, in practice, several network applications may possess a

non-concave utility function. For example, in a data streaming application, the user is

satisfied if the achieved data rate exceeds a threshold, where the utility function is a step

function and thus the convexity does not preserve. Therefore, the proposed stochastic

primal-dual approach in Section 7.4 cannot be applied here. It is worth noting that we

can still formulate the stochastic traffic engineering problem as in Pi except that the

optimization problem is a stochastic non-convex programming, which is NP-hard in

general, and computationally prohibitive to solve even in a centralized fashion [165].

In the following section, we will propose an algorithmic solution to the non-convex

stochastic traffic engineering problem, based on the learning automata techniques.

Moreover, we analytically show that the proposed algorithm will converge to the global

optimum solution asymptotically, in a decentralized fashion.

We first convert the compact strategy space of each user into a discretized set

denoted by R. More specifically, each user, say /, maintains a probability vector p/,k for

each path k e P/. The segment of [0, rm] is quantized into Q sections where rm is the

maximum allowed transmission rate on any path. In other words, the continuous variable

rk is transformed into a discrete random variable, r within a discretized set R with

Q 1 elements. The data rate is randomly selected from R according to the probability

vector of p/,k where the q-th element, pk, q = 0, .. ,Q, denotes the probability that the

/-th user transmits with a rate of r/k = q x r on the k-th path of P/. Associated with

each probability vector P/,k, there is a weighting vector w/,k with the same dimension of

1 x (Q + 1). The probability vector p/,k is uniquely determined by the weighting vector

w/,k by the softmax function [166],

e Vk
pq Q V, k, q (7-28)
q=0 ewl,k

173

where wqk is the q-th element of w/,k, q = 0,... Q.

Next, we formulate an identical interest game where the players are the |L|

source nodes and the common objective function is the overall network utility, i.e., the

summation of the utility functions. In addition, for each source node, a team of learning

automata [167] is constructed. At each time step, every source node picks the data

rates on its own paths according to the probability vectors, which are determined by the

weighting vectors. Based on the feedback signal E, which will be defined shortly, each

source node adjusts the weighting vectors and the iteration continues. The executed

algorithm on every source node, say /, is provided as follows.

Algorithm:

Repeat:

For every path, say k, randomly selects a transmission rate rik from R, according
to the current probability vector plk(n) where n denotes the current time slot.

After receiving the feedback signal E(n) from the gateway node, if the cost
constraint is satisfied, the weighting vector w/,k is updated as

e /k ,

W,k(n 1)= [(n) + (n)wk(n)] for q for q(7-29)
wpk(n -- )=[ Wk(n) + /(0 J;k(n)

Otherwise, the weighting vector remains the same.

The probability vector pl,k is then updated, following equation (7-28).

Until:

max(plk(n + 1)) > B where B is a predefined convergence threshold.

In the algorithm, r(n) is the learning parameter of the algorithm satisfying 0 <

r(n) < 1. C is a sufficiently large yet finite number which keeps the weighting vector
bounded. The sequence of ~k(n) is a set of i.i.d. random variables with zero mean and

a variance of o-2(n). The global feedback signal E(n) is calculated by the gateway node

and sent back to all source nodes, as

-(n) = ) (7-30)

where J is a number to normalize the feedback signal. For example, we can set J to

the maximum value of overall utility till n and update this value on the fly. Therefore, the

value of E(n) lies within [0, 1]. Ul is the non-convex utility function for the /-th user. Note

that the utility functions of all users are assumed to be truly acquired by the gateway

node [168]. In practice, the value of E(n) can be circulated efficiently by established

multicast algorithms such as [169]. Based on the feedback, the learning automata team

adjusts the weighting vector in a decentralized fashion. In addition, note that B is the

predefined convergence threshold, e.g., B = 0.999, which provides a tradeoff between

the performance of the algorithm and its convergence speed.

Before analyzing the steady state behavior of the proposed algorithm, we first

discuss the following concepts.

Denote the maximum network utility of the original traffic engineering problem, i.e.,

Pi, as O*. Next, we define the final outcome of the proposed algorithm as 0'. We say

that the algorithm provides an c-accurate solution, if for any arbitrarily small C > 0, there

exists a Q' such that

I* O'I < VQ > Q' (7-31)

A potential game [67] is defined as a game where there exists a potential function V

such that

V(a', a-_) V(a", a_/) = Ul(a', a_/) Ul(a", a-_) V/, a', a" (7-32)

where UL is the utility function for player I and a', a" are two arbitrary strategies in its

strategy space. The notation of a_/ denotes the vector of choices made by all players

other than /.

175

A weighted potential game [67] is defined as a game where there exists a potential

function V such that

(V(a', al) V(a", al)) x hi = Ul(a', a_) Ul(a", al) (7-33)

for all /, a', a" where hi > 0.

According to the definitions, it is apparent that the formulated identical interest game

is a special case of weighted potential games. In the following theorem, we will provide

the convergence behavior of a more general setting for weighted potential games, and

hence the result applies to our specific scenario naturally.

Theorem 7.2. For an N-person weighted potential game where each person represents

a team of learning automata, the proposed algorithm can converge to the global

optimum solution asymptotically, which is an c-accurate solution to the original stochastic

traffic engineering problem, for sufficient small value of r and o.

Proof. The proof follows similar lines as in [167]. However, we extend the result to a

more general setting where the underlying N-person stochastic game is a weighted

potential game. Therefore, the proof in [167] can be viewed as a special case. Define V

as the potential function of the game. Note that the selected rate is determined by the

probability vector which is generated uniquely by the weighting vector. Therefore, we

can view the weighting vector as the variable in this case and the objective function is

given by z = ((V|w). In the updating procedure of (7-29), signal E is replaced by V.

We first verify that

(Oz )((V w) e wq pkVw ( kn), r k ).
aWk SW kWq =0e Yk Q e WfJ

176

Note that

eWk
(v(1- )w(n))
Q4 e Wfuk
q=0 ,k

S= epk e )(Vw(n), r)k
q q=0 e ,k
Wq W

q q=oe0 we q=0 e k
az

Next, it is straightforward to verify that the standard conditions in [170] (Chap.6,

Theorem 7) are satisfied. We omit the verifications since they are the similar procedures

as in [167]. Thus, we conclude that the above dynamic weakly converges to the

following SDE [170, 171]

dw = Vz + adW (7-34)

for a sufficiently small 7 -> 0 where a is the standard deviation of the i.i.d. random

variables 'fk and W is a standard Wiener Process. Note that the SDE (7-34) falls

into the category of Langevin equation [172] which is well-known that the probability

measure concentrates on the global maximum solution of z for a sufficiently small a

[167, 172]. Therefore, we conclude that in the weighted potential game scenario, the

proposed algorithm will converge to the global optimum of the objective function, for the

quantized data rate setting. The association of the c-accurate solution to the original

stochastic traffic engineering problem of P, follows the result of [173].

Note that Theorem 2 establishes the convergence behavior of the aforementioned

algorithm with no additional requirement for the problem structure. In contrast,

we propose a stochastic primal-dual approach in Section 7.4, which requires the

underlying problem to be a stochastic convex programming. The stochastic primal-dual

approach cannot be applied efficiently otherwise. However, the aforementioned

learning-based algorithm is suitable for almost every aspect such as stochastic

177

non-convex programming and stochastic mixed-integer programming. The asymptotically

convergence result still holds. Therefore, in this work, the proposed algorithms

provide two different exemplifying methods for protocol designs under the stochastic

environment. However, it is worth noting that for the latter approach, the tradeoff for

general applicability is the convergence speed. In other words, in order to achieve an

accurate result, i.e., when c is small, the convergence speed may be slow. The actual

convergence speed depends on the values of c, Q, 7-, a as well as the inherent structure

of the problem and hence is difficult to quantify. Fortunately, in practical applications,

achieving a "good enough" result is sometimes satisfactory. This tradeoff can be

achieved by utilizing diminishing values of 7 and a, as demonstrated in the simulated

annealing literature [78] and [174]. To sum up, if the stochastic traffic engineering

problem possesses a nice property of convexity, the algorithm based on the stochastic

primal-dual approach in Section 7.4 is recommended due to its nice decomposed

structure and computationally efficient solution. However, if the problem is non-convex in

nature, the learning automata based algorithm can be utilized to achieve an approximate

solution. The tradeoff between the accuracy and convergence speed can be tuned by

adjusting the values of 7 and a.

7.6 Performance Evaluation

In this section, we present a simple yet illustrative example to demonstrate the

theoretical results. We consider a cognitive wireless mesh network3 depicted in Figure

7-2. There are three edge routers as the source nodes, denoted by A, B, C, which

transmit to the gateway node GW via the relay routers X, Y and Z. Among all feasible

3 Figure 7-2 only shows the links on the available paths obtained by the signalling
mechanisms or manually configurations. The actual physical topology of the network
can be potentially larger.

178

paths, we select the following available paths for edge routers, as summarized in Table

7-1.

Relay Routers

Edge Routers

Figure 7-2. Example of cognitive wireless mesh network

Table 7-1. Available paths for edge routers
A P,: {A X -GW}
P : {A -> X -- Y GW}
P: {A X Y Z GW
B PA: {B- X GW}
P: {B Y -GW}
P: { B Y -X GW}
4: {B Y Z GW}
P : {B Z -GW}
C PI: {C Z- GW}
P : {C- Z- Y GW}
P : {C- Z -- Y X GW}

There are five primary users in the area, denoted by 1, 2, 3, 4 and 5 where each

one has a primary band of 10MHz. The common ISM band is assumed to be 10MHz.

The return probability of the primary users is given as w = [0.2, 0.3, 0.4, 0.3, 0.3]. The

transmitting power of each node is fixed as 100mW and the noise power is assumed

to be 3mW. We consider a model where the received power is inversely proportional

to the square of the distance. Note that the transmitting power is uniformly spread on

all available bands. In addition, we explicitly specify the affecting primary users for a

particular node. We use {i,j, k, } to represent that a particular node is affected by

primary user i,j, k, For example, node X is labeled with {1, 2} which indicates that

179

the transmission of node X will devastate the transmissions of primary user 1 and 2 if

the corresponding primary band is utilized. Note that the central node, namely, Y, is

most severely affected by all primary users. Intuitively, to achieve an expected optimum

solution, the stochastic traffic engineering algorithms are inclined to steer the traffic

away from Y. We will demonstrate this detour effect next.

We first consider the cognitive wireless mesh network with convexity, e.g., Ul(xl) =

log x1 to achieve a proportional fairness among the flows [161]. The link cost is assumed

to be in the form of I((fe, ce) = which reflects the delay experienced for a unit flow

on link e under the M/M/1 assumption [76]. Note that if fe > Ce, the cost is +o. We set

the traffic demand of all edge routers as dl = 30Mbps while the cost budget is bl = 5.

The step sizes are chosen as a = 1/n where n is the current iteration step. Figure 7-3A

illustrates the trajectories of the rate variables and Figure 7-3B shows the convergence

of the network overall utility as well as the individual utility functions4 We observe that

while the rate variables converge as the iterations go, the overall objective, i.e., the sum

of the individual utilities, approaches to the global optimum indicated by the dashed

line, which is attained by calculating the steady state distribution following the return

probability w.

In addition, Table 7-2 provides the rate on each path after convergence for a sample

run of the algorithm. For comparison, we provide the convergence rates when node Y is

switched from the most affected node to the least affected node, i.e., node Y can utilize

all five primary bands all the time, in Table 7-3. From Table 7-2 and 7-3, it is interesting

to note that, in the first scenario, each user allocates a relatively small amount of flow

on the paths which traverse through node Y. On the contrary, when node Y is less

affected, all the flows allocate noticeably larger data rates on paths that traverse through

4 Note that Figure 7-3B also reflects the evolution of the throughput of each edge
router logarithmically.

180

Convergence of Rate Variables
11

--

S- loalOp....... riu
11 -r
---- r

0 10 20 30 40 50 60 70 80 90 100
Iteration

A Trajectories of Rate Variables

Convergence of Overall Utility
10------------
Overall Utility

-U,
7 F

2

0 10 20 30 40 50 60 70 80 90 100
Iteration

B Trajectories of Utility Functions

Figure 7-3. Cognitive wireless mesh networks with convexity

Y despite the fact that node Y is the central node which is least favorable by traditional

traffic engineering solutions. Therefore, our proposed stochastic traffic engineering

algorithm is of particular interest for multi-hop cognitive wireless mesh networks due to

the capability of steering the traffic away from the severely affected areas automatically,

without a prior knowledge of the underlying probabilistic structure, in a distributed

fashion.

181

Table 7-2. Convergence rates when Y is affected by all five primary users

P1.
P':

P2.
P :

P :
P1.
PB.

15.3035
3.2458
2.7362
7.3725
1.9584
2.1752
2.6458
5.3489
17.3051
2.3792
2.5675

Table 7-3. Convergence rates when Y is not affected by
A P : 9.6413
PA: 11.1335
P : 9.1861
B P : 5.0101
PB: 6.2625
PB: 4.9711
PI: 4.9921
PB: 8.7878
C P: 15.3118
P : 9.1042
PF: 5.6121

any of the primary users

We next consider a cognitive wireless mesh network with non-convex utility

functions. Specifically, we consider the utility function as

1, if x > 2Mbps
U(x) 2Mbps.
0, if xi < 2Mbps.

while other settings are the same as in the previous scenario. Additionally, we utilize

diminishing values of r and a as 7 -

Without loss of generality, we set

1/n and o = 1/n, where n is the iteration step5 .

100 and the quantization level Q

20. The

5 By utilizing diminishing parameters, a tradeoff between the performance and the
convergence speed can be achieved by tuning the decreasing speed [78].

182

(7-35)

maximum allowed rate rm is assumed to be 10. Therefore, the discretized data rate set

is given by R = [0.5, 1.0, 1.5, 9.0, 9.5, 10.0]. Figure 4 illustrates the evolution of the

probability vector of PA,1. Note that as the iterations evolve, the probability of pA i.e.,

the probability that router A chooses the twentieth data rate (rm in this case), excels

others and approaches to 1 asymptotically. We plot the evolutions of the probability

vectors of other paths in Figure 5 collectively. For each path, the probability of selecting

one particular data rate soon excels others. We observe that router A selects the

twentieth, the first and the fourth date rate on its three paths asymptotically. Meanwhile,

router B inclines to choose the eighth, the second, the first, the fourth and the twentieth

data rate on its paths. The steady state data rates for router C is the twentieth, the first

and the first element in R, as depicted in Figure 5. It is interesting to notice that all the

routers automatically detour the traffic from the severely affected node Y by allocating

more data rate on other paths.

Router A (Path 1)
20
A,1

4-o
02
0

Iteration

Figure 7-4. Trajectory of the probability vector of router A's first path

7.7 Conclusions

In this chapter, we investigate the stochastic traffic engineering (STE) problem

in cognitive wireless mesh networks. To harness the randomness induced by the

unpredictable behaviors of primary users, we formulate the STE problem in a stochastic

network utility maximization framework. For the cases where convexity holds, we derive

a distributed algorithmic solution via the stochastic primal-dual approach, which provably

converges to the global optimum solution. For the scenarios where convexity is not

183

Router A (Path 3)

Router B (Path 3) Router B (Path 4) Router B (Path 5) Router C (Path 1)

Iteration Iteration Iteration Iteration
Router C (Path 2) Router C (Path 3)
1,2 P4C,
8o 2o
50 100 150 20 0 50 100 150 200 o 50 100 150 2000

Iteration Iteration

Figure 7-5. Trajectories of the probability vectors on other paths

attainable, we propose an alternative decentralized algorithmic solution based on the

learning automata techniques. We show that the algorithm converges to the global

optimum solution asymptotically, under certain conditions.

In our work, we restrict ourself in a single gateway scenario. The extension to

the multiple gateway scenario seems interesting and needs further investigation.

In addition, in this work, we consider a cooperative case where all the edge routers

attempt to maximize the overall network performance. In the cases where the edge

routers are non-cooperative, each player is interested in its own utility rather than

the social welfare. Stochastic game theory provides a feasible tool to address the

non-cooperative case, which remains as future research. We also assume a negligible

delay for the feedback signal while in a more general case, the impact of feedback

delay needs further investigation. One feasible solution is to utilize the distributed robust

optimization framework [175] where the worst case performance is maximized given that

the feedback delay/error is within a reasonable range. Our work initiates a first step to

investigate the impact of unpredictable returns of primary users, on the stochastic traffic

engineering problem in cognitive wireless mesh networks.

Router A (Path 2)

Router B (Path 1)

Router B (Path 2)

CHAPTER 8
CONCLUSIONS

In this dissertation, we provide efficient and effective solutions to a number of

dynamic resource allocation and optimization problems in wireless networks with

time-varying channel conditions.

For wireless ad hoc networks, we first develop a dynamic pricing scheme which

can achieve a network-wide stability while maximizing the accumulated revenue with

a controllable tradeoff with the average delay. In addition, a service differentiation

solution can be achieved by prioritizing the flows according to their QoS requirements.

We also propose a minimum energy scheduling algorithm which minimizes the overall

energy consumption in the network without decreasing the throughput region. For

wireless mesh networks, we investigate the joint frequency and power allocation

problem among multiple access points in a game theoretical framework. The proposed

negotiation-based algorithm can achieve the optimal frequency and power allocation

with arbitrarily high probability. The price of anarchy in non-cooperative wireless mesh

networks is also studied. For wireless sensor networks, we propose a constrained

queueing model to capture the complex cross layer interactions among multiple sensor

nodes. We also study the stochastic network utility maximization problem and propose

a cross layer solution which solves the problems of admission control, load balancing,

dynamic scheduling and routing in a collective fashion. For wireless local area networks,

we study the thresholds-based rate adaptation algorithms from a reverse engineering

perspective, where the implicit objective function is revealed. We also investigate the

stochastic traffic engineering problem in multi-hop cognitive radio networks, where

distributed algorithms are developed to maximize the overall network utility.

185

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197

BIOGRAPHICAL SKETCH

Yang Song received his Bachelor and Master of Science degrees from Dalian

University of Technology and the University of Hawaii in 2004 and 2006, respectively.

Since August 2006, he has been working as a Research Assistant with the Wireless

Networks Laboratory in the Department of Electrical and Computer Engineering at the

University of Florida. His research topics include routing, scheduling and cross layer

computing.

198

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2REVENUEMAXIMIZATIONINMULTI-HOPWIRELESSNETWORKS ..... 17 2.1Introduction ................................... 17 2.2RelatedWork .................................. 19 2.3RevenueMaximizationwithQoS(QualityofService)Requirements ... 22 2.3.1SystemModel .............................. 22 2.3.2ProblemFormulation .......................... 25 2.4QoS-AwareDynamicPricing(QADP)Algorithm ............... 27 2.5PerformanceAnalysis ............................. 30 2.5.1ProofofRevenueMaximization .................... 31 2.5.2ProofofNetworkStability ....................... 39 2.5.3ProofofQoSProvisioning ....................... 40 2.6Simulations ................................... 41 2.6.1Single-HopWirelessCellularNetworks ................ 41 2.6.2Multi-HopWirelessNetworks ..................... 44 2.7Conclusions ................................... 47 3ENERGY-CONSERVINGSCHEDULINGINWIRELESSNETWORKS ..... 48 3.1Introduction ................................... 48 3.2SystemModel ................................. 50 3.3MinimumEnergySchedulingAlgorithm ................... 54 3.3.1AlgorithmDescription ......................... 54 3.3.2Throughput-Optimality ......................... 57 3.3.3AsymptoticEnergy-Optimality ..................... 62 3.4Simulations ................................... 64 3.5Conclusions ................................... 67 4CHANNELANDPOWERALLOCATIONINWIRELESSMESHNETWORKS 70 4.1Introduction ................................... 70 4.2SystemModel ................................. 72 4.3CooperativeAccessNetworks ........................ 74 5

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4.3.1CooperativeThroughputMaximizationGame ............ 75 4.3.2NETMA-Negotiation-BasedThroughputMaximizationAlgorithm 78 4.4Non-CooperativeAccessNetworks ...................... 84 4.5AnExtensiontoAdaptiveCodingandModulationCapableDevices .... 90 4.6PerformanceEvaluation ............................ 93 4.6.1LegacyIEEE802.11Devices ..................... 93 4.6.1.1Exampleofsmallnetworks ................. 94 4.6.1.2Exampleoflargenetworks ................. 96 4.6.2ACM-CapableDevices ......................... 96 4.7Conclusions ................................... 98 5CROSSLAYERINTERACTIONSINWIRELESSSENSORNETWORKS ... 100 5.1Introduction ................................... 100 5.2RelatedWork .................................. 102 5.3AConstrainedQueueingModelforWirelessSensorNetworks ...... 104 5.3.1NetworkModel ............................. 104 5.3.2TrafcModel .............................. 105 5.3.3QueueManagement .......................... 108 5.3.4Session-SpecicRequirements .................... 109 5.4StochasticNetworkUtilityMaximizationinWirelessSensorNetworks .. 110 5.4.1ProblemFormulation .......................... 111 5.4.2TheANRACrossLayerAlgorithm ................... 112 5.4.3PerformanceoftheANRAScheme .................. 114 5.5PerformanceAnalysis ............................. 118 5.6CaseStudy ................................... 125 5.7ConclusionsandFutureWork ......................... 128 6THRESHOLDOPTIMIZATIONFORRATEADAPTATIONALGORITHMSINIEEE802.11WLANS ................................ 130 6.1Introduction ................................... 130 6.2RelatedWork .................................. 133 6.3ReverseEngineeringfortheThreshold-BasedRateAdaptationAlgorithm 135 6.4ThresholdOptimizationAlgorithm ....................... 143 6.4.1LearningAutomata ........................... 144 6.4.2AchievingtheStochasticOptimalThresholds ............ 145 6.5PerformanceEvaluation ............................ 150 6.6ConclusionsandFutureWork ......................... 155 7STOCHASTICTRAFFICENGINEERINGINMULTI-HOPCOGNITIVEWIRELESSMESHNETWORKS ................................. 157 7.1Introduction ................................... 157 7.2RelatedWork .................................. 160 7.3SystemModel ................................. 163 7.4StochasticTrafcEngineeringwithConvexity ................ 167 6

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7.4.1Formulation ............................... 167 7.4.2DistributedAlgorithmicSolutionwiththeStochasticPrimal-DualApproach ................................ 169 7.5StochasticTrafcEngineeringwithoutConvexity .............. 173 7.6PerformanceEvaluation ............................ 178 7.7Conclusions ................................... 183 8CONCLUSIONS ................................... 185 REFERENCES ....................................... 186 BIOGRAPHICALSKETCH ................................ 198 7

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LISTOFTABLES Table page 2-1Averageadmittedratesformultimediaows .................... 44 2-2Averageadmittedratesformultimediaows .................... 47 4-1Dataratesv.s.SINRthresholdswithmaximumBER=10)]TJ /F5 7.97 Tf 6.59 0 Td[(5 ........... 90 6-1SNRv.s.BERforIEEE802.11bdatarates .................... 152 7-1Availablepathsforedgerouters ........................... 179 7-2ConvergencerateswhenYisaffectedbyallveprimaryusers ......... 182 7-3ConvergencerateswhenYisnotaffectedbyanyoftheprimaryusers ..... 182 8

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LISTOFFIGURES Figure page 2-1Anexampleofmulti-hopwirelessnetworks .................... 23 2-2Aconceptualexampleofthenetworkcapacityregionwithtwomultimediaows.Theminimumdataraterequirementsreducethefeasibleregionoftheoptimumsolution ........................................ 27 2-3Asingle-hopwirelesscellularnetworkwiththreeusers .............. 42 2-4ImpactofdifferentvaluesofJontheperformanceofQADP ........... 43 2-5ImpactofdifferentvaluesofJontheaverageexperienceddelays ........ 43 2-6Queuebacklogdynamicsforallusers ....................... 45 2-7PricedynamicsinQADPforallusers ....................... 45 2-8VirtualqueuebacklogupdatesinQADPforJ=50000 .............. 46 2-9AveragequeuebacklogsinthenetworkforJ=50000 .............. 46 3-1Networktopologywithinterconnectedqueues ................... 65 3-2ComparisonoftheenergyconsumptionsoftheMaxWeightalgorithmandtheMESalgorithm .................................... 66 3-3TheaveragequeuebacklogsinthenetworkforJ=50and150 ......... 67 3-4TheaveragequeuebacklogsinthenetworkforJ=350and500 ........ 68 3-5ComparisonofthelifetimeoftheMaxWeightalgorithmandtheMESalgorithm 68 4-1Hierarchicalstructureofwirelessmeshaccessnetworks ............. 70 4-2AnillustrativeexampleofmultipleNashequilibria ................. 77 4-3MarkovianchainofNETMAwithtwoplayers .................... 81 4-4PerformanceevaluationofthewirelessmeshaccessnetworkwithN=5andc=3 .......................................... 94 4-5ThetrajectoryoffrequencynegotiationsinNETMAwhenN=5andc=3 ... 95 4-6ThetrajectoryofpowernegotiationsinNETMAwhenN=5andc=3 ..... 95 4-7PerformanceevaluationofthewirelessmeshaccessnetworkwithN=20andc=3 .......................................... 97 4-8Performanceevaluationofthewirelessmeshaccessnetworkwith/withoutthepricingscheme .................................... 98 9

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4-9ThetrajectoriesofpowerutilizationpricesofeachACM-capableAP ...... 98 4-10ThetrajectoriesofactualpowerutilizedbyeachACM-capableAP ....... 99 5-1Topologyofwirelesssensornetworks ....................... 106 5-2Examplenetworktopology ............................. 125 5-3AveragenetworkutilityachievedbyANRAfordifferentvaluesofJ ....... 126 5-4AveragenetworkqueuesizebyANRAfordifferentvaluesofJ ......... 127 5-5SampletrajectoriesoftheadmittedratesoftwosessionsforJ=5000 ..... 127 5-6SampletrajectoriesofthevirtualqueuesforJ=5000 .............. 128 5-7SampletrajectoriesofthehopselectionsforJ=5000 .............. 129 6-1Averagethroughputv.s.Dopplerspreadvalues .................. 153 6-2Thetrajectoriesofuanddinachievingthestochasticoptimumvalues .... 154 6-3EvolutionoftheprobabilityvectorPu ........................ 154 6-4EvolutionoftheprobabilityvectorPd ........................ 155 7-1Architectureofwirelessmeshnetworks ...................... 157 7-2Exampleofcognitivewirelessmeshnetwork ................... 179 7-3Cognitivewirelessmeshnetworkswithconvexity ................. 181 7-4TrajectoryoftheprobabilityvectorofrouterA'srstpath ............. 183 7-5Trajectoriesoftheprobabilityvectorsonotherpaths ............... 184 10

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Theinterconnectedqueuenetworkmodelhasattractedsignicantattentionsincetheseminalworkof[ 12 ]wherethewell-knownMaxWeightschedulingalgorithmisproposed.Neelyextendstheresultsintoageneraltimevaryingsettingin[ 13 14 ],basedonwhichapioneeringstochasticnetworkoptimizationtechniqueisdeveloped[ 15 16 ].Foracomprehensivetreatmentonthisarea,referto[ 17 ]andthereferencestherein.Unfortunately,intheliterature,fewworkhasbeendevotedtoaddressingtheissueofQoSprovisioninginpracticalwirelesssettings,whichisofspecialinterestinsupportingmultimediatransmissionsinmulti-hopwirelessnetworks.Forexample,[ 18 20 ]investigatethedelayconstraintsbyassumingthatthequeuebehavesasanM/M/1queue.However,thismaynotberealisticduetothecomplexinteractionofqueuesinducedbytheschedulingalgorithm.In[ 2 21 22 ],delayguaranteedschedulingalgorithmsarestudied.However,thederiveddelayboundisuptoalogarithmicfactoroftheproposeddelayrequirement,giventhatthedelayrequirementssatisfycertainper-serverandper-sessionconditions[ 2 ].Findingageneralpolicythatisabletoachievearbitraryfeasibledelayrequirementsisstillanopenproblem.Inaddition,whileexistingsolutionsonprioritizedtransmissionsareavailable,e.g.,[ 23 24 ],thequestionofhowtoachieveaminimumdatarateguaranteeconcurrently,aswellasmaximizingasystem-widerevenue,isunspecied.In[ 25 26 ],lazypacketschedulingalgorithmsareproposed.However,theyrequiretheknowledgeonfuturestochasticarrivalsasapriori,whileinourscheme,a.k.a.,QADPalgorithm,suchinformationisnotneeded. Revenuemaximizationproblemhasbeenstudiedextensivelyintheliteratureformanydifferentsettings.Nevertheless,ingeneral,eitherQoSprovisioningisnotparticularlyaddressed[ 27 30 ],orthenetworkmodelisrestrictedtowirednetworkswherethechannelconditionsofthesystemareassumedtobetime-invariantandremainunchanged[ 31 32 ].Ourworkisinspiredby[ 33 ].However,ourworkdiffersfrom[ 33 ]inthefollowingcrucialaspects.First,[ 33 ]studiesasinglehopnetworkwithonlyoneaccesspoint(AP)whileourworkinvestigatesageneralmulti-hopwirelessad 21

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orpeakpowerlimitations.Forawirelesslink(a,b),thelinkdataratea,b(t)isafunctionofI(t)andS(t).Wedenote(t)asthevectoroflinkratesofalllinksattimeslott. Figure2-1. Anexampleofmulti-hopwirelessnetworks ThereareCcommodities,a.k.a.,ows,inthenetwork,whereeachcommodity,sayc,c=1,2,,C,isassociatedwitharoutingpathPc=fc(0),c(1),,c(c)gwherec(0)andc(c)arethesourceandthedestinationnodeofowcwhilec(j)denotesthej-thhopnodeonitspath.Withoutlossofgenerality,weassumethateverynodeinthenetworkinitiatesatmostoneow4.However,multipleowscanintersectatanynodeinthenetwork.Eachnodemaintainsaseparatequeueforeveryowthatpassesthroughit.Foreachowc,denoteAc(t)astheexogenousarrivaltothetransportlayerofnodec(0)duringtimeslott.Weassumethatthestochasticarrivalprocess,i.e.,Ac(t),hasanexpectedaveragerateofc.Morespecically,wehave limt!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E(Ac())=c8c.(2) Forasinglequeue,denetheoverowfunction[ 13 ]as g(B)=limsupt!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0Pr(Q()>B)(2) 4Ifanodeinitiatesmorethanoneowinthenetwork,wecanreplacethisnodewithmultipleduplicatenodesandthefollowinganalysisstillholds. 23

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whereQ()isthequeuebacklogattime.Wesaythequeueisstableif[ 13 14 ] limB!1g(B)!0.(2) Anetworkisstableifallthequeuesinthisnetworkarestable. Denote=f1,,Cgasthearrivalratevectorofthenetwork.Notethatallthearrivalratevectorsaredenedinanaveragesense.Aowcontrolmechanismisimplementedwhereduringtimeslott,anamountofRc(t)trafcisadmittedtothenetworklayerforowc.Apparently,wehaveRc(t)Ac(t).Forsimplicity,weassumethattherearenoreservoirstoholdtheexcessivetrafc,i.e.,incomingpacketsaredroppedifnotadmitted.However,weemphasizethatouranalysiscanbeappliedtogeneralcaseswheretransportlayerreservoirsaredeployedtobuffertheun-admittedtrafc.Thenetworkcapacityregion,a.k.a.,thenetworkstabilityregion,denotedby,isdenedasalltheadmissionratevectorsthatcanbesupportedbythenetwork,inthesensethatthereexistsapolicythatstabilizesthenetworkunderthisadmissionrate.Moreover,weareparticularlyinterestedinmultimediatransmissionswhereeachowchasspecicQoSrequirements.Tobespecic,eachowchasaminimumdataraterequirementcaswellasanapplication-dependentprioritizedservicelevelrequest,denotedbyc,1cL.Denote=f1,,Cgand=f1,,Cgastheminimumratevectorandtheservicelevelrequestvectorofthenetworkwhere1hasthehighestpriorityintermsofthedelayguaranteesprovidedbythenetwork.Inaddition,weassumethattheminimumratevector,i.e.,,isinsideofthenetworkcapacityregion.Sincethatifisinherentlynotfeasible,wecannotexpecttondanypolicytomeetthosedemandsandtheonlysolutionistoincreasethenetwork'sinformation-theoreticcapacitybytraditionalmethodssuchasaddingmorechannels,radios,enablingnetworkcoding,orutilizingMIMOtechniqueswithmultipleantennas.Nevertheless,inthiswork,werestrictourselvestothespecicquestionofhowtondasimpleyetoptimalpolicyforthenetworkifsuchrequirementsareindeedtheoreticallyattainable.We 24

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emphasizethat,however,theanswertothisquestionisbynomeansstraightforwardduetotheintractabilityofquantifyingtheunderlyingnetworkcapacityregion.Moreover,timevaryingchannelconditionsandstochasticexogenousarrivalsmaketheproblemevenmorechallenging.Itisourmainobjectivetodevelopanoptimumpolicywithoutknowingthenetworkcapacityregionandthestatisticalcharacteristicsofrandomarrivalprocessesaswellastimevaryingchannels.WewillformulatetheQoS-awarerevenuemaximizationproblemnext. 2.3.2ProblemFormulation DenotethequeuebacklogofnodenforowcasQcn(t).NotethatQcc(c)0sincewheneverapacketreachesthedestination,itisconsideredasleavingthenetwork.ThequeueupdatingdynamicofQcn(t)isgivenasfollows.Forj=1,,c)]TJ /F3 11.955 Tf 11.96 0 Td[(1,wehave Qcc(j)(t+1)[Qcc(j)(t))]TJ /F8 11.955 Tf 11.96 0 Td[(outc(j),c(t)]++inc(j),c(t) (2) andforj=0, Qcc(j)(t+1)=[Qcc(j)(t))]TJ /F8 11.955 Tf 11.96 0 Td[(outc(j),c(t)]++Rc(t) (2) where[x]+denotesmax(x,0)andinn,c(t),outn,c(t)representtheallocateddatarateoftheincominglinkandtheoutgoinglinkofnoden,bytheschedulingalgorithm,withrespecttoowc.Notethat( 2 )isaninequalitysincetheprevioushopnodemayhavelesspacketstotransmitthantheallocateddatarateinc(j),c(t). Duringtimeslott,pc(t)ischargedforowcastheperunitowprice.Thefunctionalityofthepriceisnotonlytocontroltheadmittedows,butalso,moreimportantly,tobuildupasystem-widerevenuefromthenetwork'sperspective.Wefurtherassumethateachowisassociatedwithaparticularuserandthuswewilluseowanduserinterchangeably.Everyusercisassumedtohaveaconcave,differentiableutilityfunctionOc(Rc(t))whichreectsthedegreeofsatisfactionbytransmittingwithdatarateRc(t).Attimeslott,usercselectsadataratewhich 25

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optimizesthenetincome,a.k.a.,surplus,i.e., Rc(t)=argmaxr(Oc(r))]TJ /F6 11.955 Tf 11.95 0 Td[(rpc(t))8c=1,,C.(2) Withoutlossofgenerality,weassumethat Oc(r)=log(1+r).(2) However,westressthatthefollowinganalysiscanbeextendedtootherheterogeneousformsofutilityfunctionsstraightforwardly.Notethatthefairnessissueofmultipleowscanbesolvedbychoosingutilityfunctionsproperly.Forexample,autilityfunctionoflog(r)representstheproportionalfairnessamongcompetitiveows.Formorediscussions,referto[ 4 ]and[ 10 ]. Fromthenetworkadministrator'sperspective,theoverallnetwork-widerevenueisthetargettobemaximized.Meanwhile,thestabilityofthenetworkaswellastheQoSrequirementsfrommultimediaowsneedtobeaddressed.Formallyspeaking,theobjectiveofthenetworkistondanoptimalpolicyto QoS-AwareRevenueMaximizationProblem: maximizeD=liminft!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0O() (2) s.t. (a) thenetworkisstable, (b) theminimumdataraterequirements,i.e.,,aresatised, (c) theguaranteedmaximumend-to-enddelaysformultiplemultimediaowsareprioritizedaccordingtotheservicelevelsof, where O(t)=E(XcRc(t)pc(t))(2) istheexpectedoverallnetworkrevenueduringtimeslott,withrespectedtotherandomnessofarrivalprocessesandchannelvariations. 26

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byc,whichiscalculatedby c=maxjj cc,8c=1,,C.(2) PartII:DynamicPricing Foreverytimeslott,thesourcenodeofowc,i.e.,c(0),measuresthevalueofQcc(0)(t)andYc(t).IfQcc(0)(t)>Yc(t),theinstantaneousadmissionpriceissetas pc(t)=vuut cQcc(0)(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Yc(t) J(2) andpc(t)=0otherwise. PartIII:Scheduling Foreverytimeslott,ndalinkscheduleI(t),fromthefeasiblesetS(t),whichsolves maxI(t)2S(t)X(a,b)2Ea,b(t)a,b(2) where a,b=maxc:(a,b)2Pc(c(Qca(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qcb(t)))(2) if9c,suchthat(a,b)2Pc,anda,b=0otherwise. END ItisworthnotingthatPartIofQADPcanbeprecalculatedbeforeactualtransmissions.TheweightassignmentcanbeimplementedeitherbythenetworkcontrollerwhichknowstheQoSrequirementsofallows,orbymutualinformationexchangesamongmultiplemultimediaows,inadecentralizedfashion.ThevalueofcrepresentstheQoS-wiseimportanceofowcandremainsunchangedunlesstheQoSrequirementsfromows,i.e.,(,),areupdated,bywhichanewweightcalculationistriggered. ThedynamicpricingpartisthecrucialcomponentofQADP.Byfollowing( 2 ),notonlytheincomingadmittedratescanberegulatedeffectively,butalsotheoverallaveragenetworkrevenuecanbemaximized,aswillbeshownshortly.Notethataftertheweightassignment,inordertocomputepc(t),thesourcenodeofowc,i.e.,c(0),whichisconsideredastheedgenodeofthenetwork,requiresonlylocalinformation,i.e.,currentbacklogsofthesourcedataqueueandthevirtualqueue.Itisinteresting 29

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(a) RevenueMaximization liminft!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0O()D)]TJ /F6 11.955 Tf 13.15 8.09 Td[(K J(2) whereKisaconstantandisgivenby K=Xccc(2) andcisdenedin( 2 ). (b) NetworkStability ThenetworkisstableunderQADPalgorithm,i.e.,foreveryqueueinthenetwork,( 2 )issatised. (c) QoSProvisioning ByfollowingQADPalgorithm,anyfeasibleminimumdataraterequirementscanbesatised.Inaddition,theguaranteedmaximumaverageend-to-enddelaysformultimediaowswithserviceleveljarejtimeslargerthanthatoflevelonetransmissions. Itisofgreatimportancetoobservethatin( 2 ),theachievedperformanceofQADPalgorithmcanbepushedarbitrarilyclosetotheoptimumsolutionDbyselectingasufcientlylargevalueofJ.TheproofofTheorem 2.1 isprovidedintherestofthissection. 2.5.1ProofofRevenueMaximization Recallthattheminimumratevectorisassumedtolieinsidethecapacityregion.Therefore,thereexistsasmallpositivenumber~>0suchthat+~12where1isaunityvectorwithdimensionC. Lemma1. Foranyfeasibleinputratevector###,thereexistsastationary6randomizedpolicy,denotedbyRAND,whichgenerates E)]TJ /F8 11.955 Tf 5.47 -9.69 Td[(outn,c)]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c)]TJ /F8 11.955 Tf 11.95 0 Td[(#cn(t)=08n,c,t(2) 6Stationarymeansthattheprobabilisticstructureoftherandomizedpolicydoesnotchangewithdifferentvaluesofqueuebacklogs. 31

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Deneasystem-widepotentialfunction(PF)as PF(ZZZ(t))=XcPFc(ZZZ(t))(2) where PFc(ZZZ(t))=1 2 Xnc(Qcn(t))2+c(Yc(t))2!.(2) NotethatPF(ZZZ(t))isascalar-valuednonnegativefunction.Dene (ZZZ(t))=E(PF(ZZZ(t+1)))]TJ /F6 11.955 Tf 11.96 0 Td[(PF(ZZZ(t))jZZZ(t))(2) asthedriftofthepotentialfunctionPF(ZZZ(t)). Forowc,wetakethesquareofbothsidesof( 2 )and( 2 )andobtain7 )]TJ /F6 11.955 Tf 5.48 -9.69 Td[(Qcc(j)(t+1)2)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(Qcc(j)(t)2+)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(outc(j),c(t)2+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(inc(j),c(t)2)]TJ /F3 11.955 Tf 11.95 0 Td[(2Qcc(j)(t))]TJ /F8 11.955 Tf 5.48 -9.68 Td[(outc(j),c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inc(j),c(t) forj=1,,c)]TJ /F3 11.955 Tf 11.95 0 Td[(1,and )]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Qcc(j)(t+1)2)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Qcc(j)(t)2+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(outc(j),c(t)2+(Rc(t))2)]TJ /F3 11.955 Tf 11.95 0 Td[(2Qcc(j)(t))]TJ /F8 11.955 Tf 5.48 -9.69 Td[(outc(j),c(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Rc(t) (2) forj=0.Similarly,for( 2 ),wehave (Yc(t+1))2(Yc(t))2+(Rc(t))2+(c)2)]TJ /F3 11.955 Tf 9.3 0 Td[(2Yc(t)(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(c). (2) 7Weusethefactthatfornonnegativerealnumbersa,b,c,d,ifa[b)]TJ /F6 11.955 Tf 12.5 0 Td[(c]++d,thena2b2+c2+d2)]TJ /F3 11.955 Tf 11.96 0 Td[(2b(c)]TJ /F6 11.955 Tf 11.96 0 Td[(d),asgiveninLemma4.3on[ 17 ]. 33

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Bycombiningtheinequalitiesabove,wehave PFc(ZZZ(t+1)))]TJ /F6 11.955 Tf 11.95 0 Td[(PFc(ZZZ(t))c+cQcc(0)(t)Rc(t))]TJ /F8 11.955 Tf 11.95 0 Td[(cYc(t)(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(c))]TJ /F13 11.955 Tf 11.29 11.36 Td[(XncQcn(t))]TJ /F8 11.955 Tf 5.48 -9.68 Td[(outn,c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inn,c(t) (2) where c=cN(max)2+(Rmaxc)2+1 2(c)2.(2) Notethat( 2 )issummedoverthewholenetwork.Ifnodenisnotonthepathofowc,inn,c(t)=outn,c(t)=0.Moreover,inn,c(t)=0forthesourcenodeofowcandoutn,c(t)=0forthedestinationnodeofowc.Next,wesumoverallowstoderivethenetwork-widepotentialfunctiondifferenceas PF(ZZZ(t+1)))]TJ /F6 11.955 Tf 11.95 0 Td[(PF(ZZZ(t))K)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xn,ccQcn(t)(outn,c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inn,c(t))+XccQcc(0)(t)Rc(t))]TJ /F13 11.955 Tf 11.96 11.36 Td[(XccYc(t)(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(c) whereK=Pcc.Therefore,wehave (ZZZ(t))K)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xn,ccQcn(t)E)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(outn,c(t))]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c(t)jZZZ(t)+XccQcc(0)(t)E(Rc(t)jZZZ(t)))]TJ /F13 11.955 Tf 19.26 11.36 Td[(XccYc(t)E(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(cjZZZ(t)). (2) 34

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Next,forapositiveconstantJ,wesubtractbothsidesof( 2 )byJE)]TJ 5.48 -.72 Td[(PcRc(t)pc(t)jZZZ(t)andhave (ZZZ(t)))]TJ /F6 11.955 Tf 11.96 0 Td[(JE XcRc(t)pc(t)jZZZ(t)!K)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xn,ccQcn(t)E)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(outn,c(t))]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c(t)jZZZ(t)+XccQcc(0)(t)E(Rc(t)jZZZ(t)))]TJ /F13 11.955 Tf 11.29 11.36 Td[(XccYc(t)E(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(cjZZZ(t)))]TJ /F6 11.955 Tf 9.3 0 Td[(JE XcRc(t)pc(t)jZZZ(t)!. (2) Notethat( 2 )isgeneralandholdsforanypossiblepolicy. Foranarbitrarilysmallpositiveconstant0
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thatthereexistsarandomizedpolicy,denotedbyRAND,whichyields E(outn,c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inn,c(t))]TJ /F6 11.955 Tf 11.96 0 Td[(r,c(t))=8c,n=c(0)(2) and E(outn,c(t))]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c(t))=8c,n6=c(0)(2) and E(r,c(t)+)c+~8c.(2) DenotetheRHSof( 2 )as.Withoutlossofgenerality,weassume~.Therefore,forrandomizedpolicyRAND,wehave RANDK)]TJ /F8 11.955 Tf 11.95 0 Td[( Xn,ccQcn(t)+XccYc(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE Xc(r,c(t)+)^p,c(t)jZZZ(t)! (2) where^p,c(t)isthecorrespondingpricewhichinducesarateofr,c(t)+by( 2 ). Lemma2. QADPalgorithmminimizestheRHSof( 2 )overallpossiblepolicies. Proof. TheschedulingpartofQADPalgorithminSection 2.4 isaweightedversionofMaxWeightalgorithm[ 12 13 34 ]whichmaximizesPn,ccQcn(t)(outn,c(t))]TJ /F8 11.955 Tf 12.45 0 Td[(inn,c(t))=P(a,b)2Ea,b(t)a,bif9c,suchthat(a,b)2Pc,foreverytimeslott.ThedynamicpricingpartofQADPisessentiallymaximizing Xc)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(c(Yc(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qcc(0)(t))Rc(t)+JRc(t)pc(t)(2) foreverytimeslot.By( 2 )and( 2 ),weseethatQADPndsanoptimumpricepc(t)whichmaximizes M=c(Yc(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qcc(0)(t))(1 pc(t))]TJ /F3 11.955 Tf 11.96 0 Td[(1)+J(1)]TJ /F6 11.955 Tf 11.96 0 Td[(pc(t)).(2) DeneW=c(Qcc(0)(t))]TJ /F6 11.955 Tf 11.96 0 Td[(Yc(t)). 36

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Case1:IfW>0,wehave M=J+W)]TJ /F3 11.955 Tf 11.96 0 Td[((Jpc(t)+W1 pc(t)).(2) Taketherstorderderivative,wehave M0=W (pc(t))2)]TJ /F6 11.955 Tf 11.95 0 Td[(J.(2) Thesecondorderderivativebringsus M00=)]TJ /F6 11.955 Tf 24.36 8.09 Td[(W (pc(t))3<0.(2) Therefore,Misaconcavefunctionandtheoptimumvalueisachievedby pc(t)=r W J.(2) Case2:IfW0,Misadecreasingfunctionwithrespecttopc(t).Therefore,pc(t)=0. FollowingLemma 2 ,weconcludethatforQADPalgorithm, (ZZZ(t)))]TJ /F6 11.955 Tf 11.95 0 Td[(JE XcRc(t)pc(t)jZZZ(t)!QADPRANDK)]TJ /F8 11.955 Tf 11.96 0 Td[( Xn,ccQcn(t)+XccYc(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE Xc(r,c(t)+)^p,c(t)jZZZ(t)!. (2) WetakeexpectationwiththedistributionofZZZ(t),onbothsidesof( 2 ).BythefactthatEY(E(XjY))=E(X),wehave E(PF(ZZZ(t+1))))]TJ /F6 11.955 Tf 11.95 0 Td[(E(PF(ZZZ(t)))K+JO(t))]TJ /F8 11.955 Tf 11.95 0 Td[(E Xn,ccQcn(t)+XccYc(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE Xc(r,c(t)+)^p,c(t)!. (2) 37

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Notethat( 2 )issatisedforalltimeslots.Therefore,wetakeasumontimeslots=0,,T)]TJ /F3 11.955 Tf 11.96 0 Td[(1andhave E(PF(ZZZ(T))))]TJ /F6 11.955 Tf 11.95 0 Td[(E(PF(ZZZ(0)))TK+T)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0JO())]TJ /F9 7.97 Tf 11.29 14.94 Td[(T)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E Xn,ccQcn()+XccYc()!)]TJ /F9 7.97 Tf 11.29 14.95 Td[(T)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JE Xc(r,c()+)^p,c()!. (2) Divide( 2 )byTandrearrangetermstoobtain 1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E Xn,ccQcn()+XccYc()!+1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JE Xc(r,c()+)^p,c()!K+1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JO()+E(PF(ZZZ(0))) T (2) wherethenonnegativityofthepotentialfunctionisutilized.WersttakeliminfT!1onbothsidesof( 2 )andhave8 liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E Xn,ccQcn()+XccYc()!+liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JE Xc(r,c()+)^p,c()!K+liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JO(). (2) 8Weassumethattheinitialqueuesizesofrealdataqueuesandvirtualqueuesarebounded. 38

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Notethatthersttermof( 2 )isnonnegativeandthuswehave liminfT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0O()liminfT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0E Xc(r,c()+)^p,c()!)]TJ /F6 11.955 Tf 13.15 8.08 Td[(K J. Recallthattheanalysisaboveappliestoarbitrarilysmall0
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wheremaxisthemaximumpossibledatarateonalink.Moreover,by( 2 )and( 2 ),wehaveOmax=CwhereCisthenumberofowsinthenetwork.Therefore,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0E Xn,ccQcn()+XccYc()!K+JC max. Usingthefactthat X(t)Y(t)8t)limsupX(t)limsupY(t),(2) wehave,foreveryowc,theaveragequeuelengthonitsroutingpathisboundedby limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0E cXj=0Qcc(j)()!K+JC cmax. (2) Finally,byapplyingMarkovInequality,weconcludethatalldataqueuesinthenetworkarestable,basedonthefactthattheRHSof( 2 )isbounded. 2.5.3ProofofQoSProvisioning Similarto( 2 ),foreveryvirtualqueueYc(t),wecanobtain limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E(Yc())K+JC cmax. (2) Therefore,bysimilaranalysisandthedenitionofvirtualqueues,weconcludethatthevirtualqueuesarestableandthustheminimumdataraterequirementsimposedbymultimediaowsareachieved. Next,weshowthatQADPindeedprovidesaservicedifferentiationsolutionontheguaranteedmaximumend-to-enddelaysforallmultimediaows.Denotetheactualexperiencedaveragedelayofowcas!c.ByLittle'sLaw,!cisapproximated9by 9Notethatweconsideraheavyloadednetwork.Propagationdelaysareassumedtobenegligiblecomparedtoqueueingdelaysandthusareomitted. 40

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!c=averageoverallqueuelengthonthepathofowc averageincomingrateofowc=limsupT!11 TPT)]TJ /F5 7.97 Tf 6.59 0 Td[(1=0EPcj=0Qcc(j)() limsupT!11 TPT)]TJ /F5 7.97 Tf 6.58 0 Td[(1=0E(Rc()). (2) InlightofthestabilityofvirtualqueueYc(t),wehavelimsupT!11 TPT)]TJ /F5 7.97 Tf 6.59 0 Td[(1=0E(Rc())c.Hence,wecanobtain !cK+JC cmaxc.(2) Equivalentlyspeaking,todifferentiatetheguaranteedmaximumdelaybound,weneedtondasetofweightssuchthat ccc=dd`d(2) issatisedforanypairofmultimediaowscandd.Therefore,itisstraightforwardtoverifythattheweightassignmentalgorithminQADPindeedprovidesaservicedifferentiationsolutionwheretheguaranteedmaximumend-to-enddelaysofmultimediaowsaredistinguishedaccordingtoapplication-dependentservicelevelrequests,whichcompletestheproofofTheorem 2.1 .TheperformanceofQADPalgorithmwillbeevaluatednumericallyinSection 2.6 2.6Simulations 2.6.1Single-HopWirelessCellularNetworks Werstconsiderasingle-hopwirelesscellularnetworkwithdownlinkmultimediatransmissions,asshowninFigure 2-3 .Thebasestation(BS)isassociatedwiththreeuserswithinnitebackloggedtrafc.Aseparatequeueismaintainedbythebasestationforeveryuser.Inaddition,ateachtimeslot,BScanonlytransmittooneparticularuser.Awirelesslinkisassumedtohavethreeequallypossiblechannelstates,i.e.,Good,Medium,Bad.Thecorrespondingtransmissionratesforthreechannelstatesare20,15and10bitsperslot,respectively.Therefore,thebasestationencountersacomplex 41

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Table2-1. Averageadmittedratesformultimediaows Flow1 Flow2 Flow3 J=50 4.01 2.71 3.07 J=100 4.04 2.72 3.09 J=500 4.05 2.91 3.11 J=1000 4.31 3.07 3.16 J=5000 4.51 3.57 3.42 J=10000 4.63 3.94 3.62 J=20000 4.75 4.28 4.09 J=50000 5.09 4.71 4.61 J=100000 5.13 4.88 4.85 Notethat,inFigure 2-7 ,thepricesimposedforthreeusersaredynamicallyadjustedateverytimeslot.ItisworthnotingthatwheneveradataqueueinFigure 2-6 hasatendencytobuildup,thepriceimposedbyQADPalgorithm,asshowninFigure 2-7 ,risescorrespondingly,whichinturndiscouragestheexcessiveadmittedrateandthusallqueuesinthenetworkremainbounded.Asaresult,thestabilityofthenetworkisachieved. 2.6.2Multi-HopWirelessNetworks Wenextconsideramulti-hopwirelessnetworkwithatopologyshowninFigure 2-1 .Therearethreemultimediaowsexistinthenetwork,denotedbyFlow1,2,3.TheroutingpathsofowsarespeciedbyP1=fA,B,C,Dg,P2=fF,G,C,DgandP3=fE,F,G,Hg.Withoutlossofgenerality,weassumeatwo-hopinterferencemodelwhichrepresentsthegeneralIEEE802.11MACprotocols[ 36 37 ].Othercongurationsarethesameasthesingle-hopscenariodescribedaboveexceptthatthepossiblelinkratesareassumedtobe40,30,20bitsperslotforthreechannelconditions.Weobservethatinthisnetworktopology,linkC!DandlinkF!Garesharedbytwodifferentows.Therefore,intheschedulingpartofQADP,theparticularowwithalargerweightedqueuebacklogdifferenceshouldbeselected. InFigure 2-8 ,wespecicallydepictthedynamicsofthreevirtualqueuesfortherst400timeslotswithJ=50000.Unlikethesingle-hopcaseinFigure 2-6 ,thevirtual 44

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J,aswellasthenetworkservicedifferentiationintermsofdelaysareanalogoustothesingle-hopscenariodiscussedabove.Duplicatedsimulationguresareomitted. Table2-2. Averageadmittedratesformultimediaows Flow1 Flow2 Flow3 J=50 3.04 2.08 3.88 J=100 3.12 2.03 3.98 J=500 3.17 2.04 4.60 J=1000 3.30 2.03 5.04 J=5000 4.13 2.01 6.45 J=10000 4.36 2.02 7.32 J=20000 5.08 2.03 8.12 J=50000 6.43 2.01 9.18 J=100000 7.63 2.01 9.89 2.7Conclusions Weconsideramulti-hopwirelessnetworkwheremultipleQoS-specicmultimediaowssharethenetworkresourcejointly.Tomaximizetheoverallnetworkrevenue,weproposeadynamicpricingbasedalgorithm,namely,QADP,whichachievesasolutionthatisarbitrarilyclosetotheoptimum,subjecttonetworkstability.Moreover,aweightassignmentmechanismisintroducedtoaddresstheservicedifferentiationissueformultimediaowswithdifferentdelaypriorities.Intandemwiththevirtualqueuetechniquewhichprovidesminimumaveragedatarateguarantees,theQoSrequirementsofmultiplemultimediastreamsareaddressedeffectively. Inthiswork,theapplication-dependentservicelevelrequestsandtheusers'utilityfunctionsareassumedtobeattainedtruthfully.Themechanismdesignforstrategy-proofuserinformationacquisitionseemsinterestingandneedsfurtherinvestigation.Moreover,inthiswork,weassumethattheinformationofchannelstates,i.e.,S(t),isavailableforQADPalgorithm,whichiseitheracquiredbythecentralbasestationorapproximatedbythedistributedlocalschedulingalgorithms.Asafuturework,anonlinechannelprobingmechanismneedstobeincorporatedintoQADP,assuggestedin[ 39 ]. 47

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ofowf,i.e.,thenumberofnewpacketsthatareinitiatedbynodenf0,isdenotedbyAf(t).Fortheeaseofexposition,weassumethatAf(t)isi.i.d.foreverytimeslotwithanaveragerateoff.Inaddition,thearrivalprocessesofallowsareassumedtobeindependent.Wefurtherassumethatthemaximumnumberofnewpacketsgeneratedbyaowduringonetimeslotisupperbounded,i.e.,Af(t)Amax,8f,t.Weemphasizethatthei.i.d.assumptionincursnolossofgeneralityandourmodelcanbeextendedtocaseswhereAf(t)isnon-stationaryinastraightforwardfashion,asin[ 14 ]. Everynodeinthenetworkmaintainsaseparatequeueforeachowthatpassesthroughit.DenoteQfn(t)asthequeuebacklogattimet,fornoden,wherefisoneoftheowsthattraversesthroughn,i.e.,n2Rf.WeassumethatQfn(t)=0,8tifn=nff.Thatistosay,ifapacketreachesthedestination,weconsiderthepacketasleavingthenetworkimmediately.DenetheoverowfunctionofQfnas O(M)=limsupt!11 tt)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0Pr(Qfn()>M).(3) Thequeueisstableif[ 14 ] limM!1O(M)!0(3) andthenetworkisstableifallindividualqueuesarestableconcurrently. Fornotationbrevity,wedenethenetworkadmissionratevectoras=fn,f,8n,fgwheren,fistheaverageexogenousarrivalrateofowfonqueue1Qfn.Wehaven,f=fifn=nf0andn,f=0otherwise.Denoteasthenetworkcapacityregion[ 12 ],namely,thesetofallfeasibleadmissionratevectors,i.e.,,thatthenetworkcansupport,inthesensethatthereexistsaschedulingalgorithmwhichstabilizesthenetworkundertrafcload.Itisshownin[ 12 ]and[ 14 ]thatisconvex,closedandbounded. 1Notethatwithaslightabuseofnotation,weuseQfntodenoteboththequeueitselfandthenumberofpacketsinthequeue. 51

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schedulingalgorithms,includingthosewithaprioriknowledgeonthefuturisticarrivalsandchannelconditions. 3.3MinimumEnergySchedulingAlgorithm 3.3.1AlgorithmDescription Theminimumenergyscheduling(MES)algorithmisgivenasfollows. MESALGORITHM: Ateverytimeslott: Everylink(a,b)2Lndstheowfwhichmaximizes maxf:(a,b)2Rf2Qfa(t))]TJ /F6 11.955 Tf 16.77 8.09 Td[(Ja,b pa,b(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Qfb(t))]TJ /F6 11.955 Tf 11.96 0 Td[(Ja,b(3) whereJisapositiveconstantwhichistunableasasystemparameter. Everylink(a,b)2Lcalculatesthelinkweightas Ha,b(t)=2Qfa(t))]TJ /F6 11.955 Tf 16.77 8.09 Td[(Ja,b pa,b(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Qfb(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Ja,b+(3) where[x]+denotesmax(x,0). Forthenetwork,alinkscheduleI(t)isselectedwhichsolves maxI(t)2(t)X(a,b)2Lua,b(t)Ha,b(t)(3)END NotethattheproposedMESalgorithmisdifferentfromtheMaxWeightalgorithmproposedin[ 12 13 ].IntheoriginalMaxWeightalgorithm,theweightofaparticularlink(a,b)isthequeuedifferencebetweennodeaandb.However,intheMESalgorithm,asindicatedby( 3 ),thelinkweightHa,b(t)isrelatedtothepotentialenergyconsumptionsonthislinkunderthecurrentchannelconditionaswell.Morespecically,thelinkweightintheMESalgorithmisthequeuedifferencesubtractedbyaweightedenergyconsumptionfactor,i.e.,Ja,b pa,b(t)+a,b,whereJrepresentstheweight.Intuitively,ifthecurrentchannelisunfavorable,i.e.,pa,b(t)issmall,theenergyrequiredforasuccessful 54

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owf,wedeneavirtualqueueYf(t)whichisinitiallyempty,i.e., Yf(0)=0,8f(3) andthevirtualqueueupdatingdynamicisdenedas Yf(t+1)=[Yf(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Rf(t)]++f8f(3) whereRf(t)istheadmittedtrafcforowfattimeslottandfisafeasibleminimumraterequirementofowf.Intuitively,ifeachofthevirtualqueuesinthesystemisstable,theaveragearrivalrateshouldbelessthantheaveragedeparturerateofthevirtualqueueandhencewehave limT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Rf(t)f,8f(3) whichisexactlythedesiredminimumraterequirementforowf.Moreover,thevirtualqueuesareeasytoimplement.Forexample,thesourcenodeofowfcanmaintainasoftwarebasedcountertomeasurethebacklogupdatesofvirtualqueueYf(t).Therefore,theminimumraterequirementscanbeincorporatedintotheproposedschemestraightforwardly. 3.3.2Throughput-Optimality WerstshowthattheproposedMESalgorithmisthroughputoptimal,asgiveninthefollowingtheorem. Theorem3.1. TheproposedMESalgorithmisthroughputoptimal,i.e.,foranarbitrarynetworkadmissionratevectorwhichisinsideofthenetworkcapacityregion,MESstabilizesthenetworkunder. Proof. WerstprovidethequeueupdatingequationofQfn.Notethatapacketisremovedfromthetransmitter'squeueifandonlyifitisreceivedbythereceiversuccessfully.Therefore,thequeueupdatingdynamicisgivenby Qfn(t+1)[Qfn(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uoutn,f(t)]++uinn,f(t)+An,f(t)(3) 57

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whereuoutn,f(t)anduinn,f(t)aretheallocateddatarateontheoutgoinglinkandtheincominglinkofnoden,withrespecttoowf.Notethatuoutn,f(t)=guoutn,f(t)poutn,c(t)andpoutn,c(t)isthecurrentsuccessfultransmissionprobabilityonthislink.Notethatuoutn,f(t)=0,8tifnodenisthedestinationnodeofowfanduinn,f(t)=0,8tifnodenisthesourcenodeofowf.Furthermore,An,f(t)=Af(t)ifnisthesourcenodeofowfandAn,f(t)=0otherwise. From( 3 ),wehave (Qfn(t+1))2)]TJ /F3 11.955 Tf 11.96 0 Td[((Qfn(t))2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[((umax)2+(umax+Amax)2)]TJ /F3 11.955 Tf 9.3 0 Td[(2Qfn(t))]TJ /F6 11.955 Tf 5.48 -9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(An,f(t). (3) Next,wesum( 3 )overthewholenetworkonalldataqueuesandobtain Xn,f(Qfn(t+1))2)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xn,f(Qfn(t))2B)]TJ /F3 11.955 Tf 11.95 0 Td[(2Xn,fQfn(t))]TJ /F6 11.955 Tf 5.48 -9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t) (3) where B=jNjjFj)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((umax)2+(umax+Amax)2(3) isaconstant. DenoteQQQ(t)=fQfn(t),8f,ngastheinstantaneousqueuebacklogsinthenetwork.WetaketheconditionalexpectationwithrespecttoQQQ(t)on( 3 )andhave E Xn,f(Qfn(t+1))2jQQQ(t)!)]TJ /F6 11.955 Tf 11.95 0 Td[(E Xn,f(Qfn(t))2jQQQ(t)!B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.47 -9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(An,f(t)jQQQ(t). Dene GMESQ(t)=E Xn,fhoutn,f(t)outn,f poutn,f(t)+hinn,f(t)inn,fjQQQ(t)! 58

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whereoutn,f(inn,f)denotestheenergyneededforapackettransmission(reception)ontheoutgoing(incoming)linkofnoden,withrespecttoowf. Inaddition,wedene GMES(t)=E Xn,fhoutn,f(t)outn,f poutn,f(t)+hinn,f(t)inn,f! astheexpectednetwork-wideenergyconsumptionduringtimeslott,byfollowingtheproposedMESalgorithm.Apparently,wehave E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(GMESQ(t)= GMES(t). Next,weaddbothsidesbyJGMESQ(t)whereJisapositiveconstant,andhave E Xn,f(Qfn(t+1))2jQQQ(t)!)]TJ /F6 11.955 Tf 11.96 0 Td[(E Xn,f(Qfn(t))2jQQQ(t)!+JGMESQ(t)B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t)jQQQ(t)+JGMESQ(t)B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t)jQQQ(t)+JE Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,fjQQQ(t)!. (3) DenotetheR.H.S.of( 3 )as.Itisofgreatimportancetoobservethat,fromthealgorithmdescriptionoftheMESalgorithmabove,ateverytimeslott,theMESalgorithmessentiallyminimizestheR.H.S.of( 3 )overallpossibleschedulingalgorithms. Sinceliesintheinteriorofthenetworkcapacityregion,itimmediatelyfollowsthatthereexistsasmallpositiveconstant>0suchthat +=f(1,1+),,(n,f+),g2.(3) 59

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ByinvokingCorollary3.9in[ 17 ],weclaimthatthereexistsarandomizedschedulingpolicy,denotedbyRA,whichstabilizesthenetworkwhileprovidingadatarateof E uoutn,f(t))]TJ ET q .478 w 197.33 -60.4 m 215.53 -60.4 l S Q BT /F6 11.955 Tf 197.33 -71.73 Td[(uinn,f(t)jQQQ(t)=n,f+,8t(3) where uoutn,c(t), uinn,c(t)arethelinkdataratesinducedbytherandomizedpolicyRA.Therefore,wehave RA=B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Pn,fQfn(t)+JEPn,f uoutn,f(t)outn,f poutn,f(t)+ uinn,f(t)inn,fjQQQ(t). (3) Notethatthelasttermin( 3 )istheactualenergyconsumptionbyRAalgorithmduringtimeslott.Following( 3 ),wehave RAB)]TJ /F3 11.955 Tf 11.95 0 Td[(2Xn,fQfn(t)+JGmax.(3) Inlightof( 3 ),wehave E)]TJ 5.48 -.72 Td[(Pn,f(Qfn(t+1))2jQQQ(t))]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 -.72 Td[(Pn,f(Qfn(t))2jQQQ(t)+JGMESQ(t)MESRAB)]TJ /F3 11.955 Tf 11.95 0 Td[(2Pn,fQfn(t)+JGmax. (3) Next,wetaketheexpectationwithrespecttoQQQ(t)on( 3 )andhave E)]TJ 5.48 -.71 Td[(Pn,f(Qfn(t+1))2)]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 -.71 Td[(Pn,f(Qfn(t))2+J GMES(t)B)]TJ /F3 11.955 Tf 11.95 0 Td[(2E)]TJ 5.48 -.72 Td[(Pn,fQfn(t)+JGmax. (3) 60

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Notethattheaboveinequalityholdsforanytimeslott.Hence,wesumoverfromtimeslot0toT)]TJ /F3 11.955 Tf 11.96 0 Td[(1andobtain E)]TJ 5.48 -.72 Td[(Pn,f(Qfn(T))2)]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 -.72 Td[(Pn,f(Qfn(0))2+JPT)]TJ /F5 7.97 Tf 6.59 0 Td[(1t=0 GMES(t)TB)]TJ /F3 11.955 Tf 11.95 0 Td[(2PT)]TJ /F5 7.97 Tf 6.59 0 Td[(1t=0Pn,fE)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Qfn(t)+TJGmax. (3) Next,werearrangetermsanddivide( 3 )byTandhave 21 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0Xn,fE)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(Qfn(t)B+JGmax+E)]TJ 5.47 -.71 Td[(Pn,f(Qfn(0))2 T)]TJ /F6 11.955 Tf 9.3 0 Td[(J1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t))]TJ /F6 11.955 Tf 13.15 9.03 Td[(E)]TJ 5.48 -.71 Td[(Pn,f(Qfn(T))2 T. (3) Notethatthelasttwotermsof( 3 )arebothnon-positive.BytakinglimsupT!1onbothsidesof( 3 ),weattain limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Xn,fE)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Qfn(t)B+JGmax 2<1.(3) Therefore,foreveryindividualqueueQfn,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Qfn(t)B+JGmax 2<1.(3) Finally,byinvokingMarkovinequality,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Pr)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Qfn(t)>M
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3.3.3AsymptoticEnergy-Optimality Inthissection,weshowthattheMESalgorithmyieldsanasymptoticoptimalsolutionto( 3 ),i.e.,theaverageenergyconsumptioninducedbytheMESalgorithmcanbearbitrarilyclosetotheglobalminimumsolutionof( 3 ),byselectingasufcientlylargevalueofJ. Theorem3.2. DeneCMESandCastheaverageenergyconsumptioninducedbytheMESalgorithmandtheoptimal(minimum)solutionof( 3 ),respectively.TheperformanceoftheMESalgorithmisgivenby CMESC+B J(3) whereBisdenedin( 3 ).Therefore,bychoosingJ!1,theperformanceoftheMESalgorithmcanbepushedarbitrarilyclosetotheoptimumsolutionC. Proof. First,denotetheoptimalsequenceoflinkrates,whichgeneratestheoptimumsolutionC,asuuu(0),uuu(1),,uuu(t),.Next,letusconsideradeterministicpolicy,denotedbyDE,whichallocatesexactlytheoptimumlinkdataratesoneverytimeslott.Similarto( 3 ),wehave E)]TJ 5.48 -.71 Td[(Pn,f(Qfn(t+1))2jQQQ(t))]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 -.71 Td[(Pn,f(Qfn(t))2jQQQ(t)+JGMESQ(t)MESDE (3) where DE=B)]TJ /F3 11.955 Tf 11.95 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t)jQQQ(t)+JE Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,fjQQQ(t)!. (3) Itisworthnotingin( 3 ),onlythedataratesarereplacedbytheonesgeneratedbytheDEalgorithmwhereasallothervaluesremainthesame.Notethatinthiscase,the 62

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optimaldataratesuoutn,f(t)anduinn,f(t)areknownasaprioribytheDEalgorithmandthusareconstantswithrespecttoQfn(t). Wetaketheexpectationonbothsidesof( 3 )andhave E Xn,f(Qfn(t+1))2!)]TJ /F6 11.955 Tf 11.95 0 Td[(E Xn,f(Qfn(t))2!+J GMES(t)B)]TJ /F3 11.955 Tf 9.3 0 Td[(2E Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F8 11.955 Tf 11.95 0 Td[(n,fjQQQ(t)!+JE Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,f!. (3) Weemphasizethat,however,inthisscenario,atarbitrarytimeslott,thevalueof uoutn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uinn,f(t))]TJ /F8 11.955 Tf 11.96 0 Td[(n,f(3) couldbeeithernon-positiveornonnegative.Inotherwords,therelationshipof( 3 )doesnothold.Tocircumventthis,werstsum( 3 )overtimeslotst=0,,T)]TJ /F3 11.955 Tf 12.3 0 Td[(1anddivideitbyT.Thus,weattain J1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t)B+J1 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0E Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,f!+E)]TJ 5.48 -.72 Td[(Pn,f(Qfn(0))2 T)]TJ /F3 11.955 Tf 11.96 0 Td[(21 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0(t) where (t)=E Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uinn,f(t))]TJ /F8 11.955 Tf 11.96 0 Td[(n,fjQQQ(t)!.(3) 63

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BytakinglimsupT!1,wehave JlimsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t)B+JC)]TJ /F3 11.955 Tf 9.3 0 Td[(2limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0(t). (3) Notethat lim(AB)=lim(A)lim(B)(3) iflim(A)andlim(B)existandarebounded.Recallthatintheprevioussection,wehaveshownthat limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Xn,fE)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Qfn(t)(3) existsandisnite.Moreover,sinceuoutn,f(t)anduinn,f(t)aretheoptimumsolutionto( 3 ),thenetworkstabilityisachievedunder.Therefore,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0uoutn,f(t)limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0uinn,f(t)+n,f sinceotherwise,thenetworkstabilitycannotbeensuredbyuuu.Finally,weconcludethat limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0(t)(3) isnonnegativeandthusthelasttermof( 3 )canbeomitted.Consequently,bydividingJonbothsidesof( 3 ),wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t)B J+C.(3) Therefore,Theorem 3.2 holds. 3.4Simulations Weconsideramulti-hopwirelessnetworkillustratedinFigure 3-1 .Therearethreeowsinthenetwork,denotedbyFlow1,2,3.TheroutingpathsofowsarespeciedbyR1=fA,B,C,Dg,R2=fF,G,C,DgandR3=fE,F,G,Hg.Theexogenous 64

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pushedarbitrarilyclosetotheglobalminimumsolution.Moreover,theimprovementontheenergyefciencyisachievedwithoutlosingthethroughput-optimality.Therefore,theproposedMESalgorithmisofgreatimportancefornetworkprotocoldesignsinenergy-constrainedmulti-hopwirelessnetworkssuchaswirelesssensornetworks. 69

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CHAPTER4CHANNELANDPOWERALLOCATIONINWIRELESSMESHNETWORKS 4.1Introduction Metropolitanwirelessmeshnetworksgainenormouspopularityrecently[ 51 ].Thedeploymentofwirelessmeshnetworksnotonlyfacilitatesthedatacommunicationbyremovingcumbrouswiresandcables,butalsoprovidesameansofInternetaccessscheme,whichisafurthersteptowardsthegoalofcommunicatinganywhereanytime.Nomatterwherethelocationisorthepurposethatthewirelessmeshnetworkisdeployed,thesameconceptuallayeredarchitectureisutilized.Figure 4-1 illustratesthehierarchicalstructureofwirelessmeshnetworks.Theperipheralnodesaretheaccesspoints(AP)whichprovidewirelessaccessfortheendusers,orclients.EachAPisassociatedwithameshrouter.Theycanbemanufacturedinasingledevicewithtwoseparatefunctionalradios[ 52 ][ 53 ],orsimplyconnectedwithEthernetcables[ 54 ].Themeshroutersarecapableofcommunicatingwitheachotherviathewirelessbackbone.ThecentralnodeisagatewaymeshrouterwhichfunctionsasaninformationexchangebetweenthewirelessmeshaccessnetworkandothernetworkssuchastheInternet.Boththeroutingalgorithmicdesignandchannelassignmentforbackbonemeshroutersareinterestingissuesandattracttremendousattention[ 55 57 ]. Figure4-1. Hierarchicalstructureofwirelessmeshaccessnetworks Inthiswork,weinvestigateanimportantissuewhichneedstobesolvedinwirelessmeshnetworks.AsinFigure 4-1 ,theAPanditsassociatedclientsformaregularWLANcell,whichoperateswiththedefactoIEEE802.11standards. 70

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probabilityofthetransmissionchannel,whichisafunctionofSINRvalueprovidingthemodulationandcodingscheme[ 64 ].Apparently,ifthereisonlyonecellinthemeshaccessnetwork,theAPwillboostthepowerasmuchaspossibletoincreasethevalueofandthusthethroughputismaximized. WenowconsiderthecaseswhereNcellscoexistinthewirelessmeshaccessnetwork.Letpiandfidenotethepowerandfrequencyforthei-thAP,respectively.Weusep=[p1,p2,,pN]andf=[f1,f2,,fN]torepresentthepowerandfrequencyassignmentvectorforallNAPs.Therefore,foreachcelli,thevalueofSINR4,i.e.,i,isafunctionof(p,f).Thethroughputofonecelldependsnotonlyonthepowerlevelandfrequencyofitself,butalsothoseofotherAPsinthenetwork.Therefore,thethroughputmaximizationproblemiscoupledandbynomeansstraightforward. Inthefollowingtwosections,wewilldiscussthescenarioswheretheAPsarecooperativeandnon-cooperative,respectively,undertheassumptionthatthemodulationandcodingschemesofAPsarepre-loadedandxed.InSection 4.5 ,wewillextendouranalysisbyconsideringthescenariowheretheAPsarecapableofadaptivecodingandmodulation.TheperformanceevaluationofallscenariosareprovidedbysimulationsinSection 4.6 4.3CooperativeAccessNetworks Inthissection,weconsiderthescenarioswhereallAPsinthewirelessmeshaccessnetworkarecooperative.ThetransmissionpowerofAPsarequantizedintodiscretepowerlevelsforsimplicity.Fromthesystempointofview,wewanttondajointfrequencyandpowerlevelassignmentsuchthattheoverallthroughputinthewhole 4Throughoutthechapter,thetermSINRofthecellrepresentstheaverageSINRamongalltheclientsinthecell,whichcanbeobtainedbyamovingaverageofthereportedSINRvalue. 74

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networkismaximized.Ourobjectivefunctioncanbewrittenas Unetwork(p,f)=NXi=1Ri(i)=NXi=1Ri(p,f)(4) whereRiisdenedin( 4 ). However,ndingtheoptimalfrequencyandpowerassignmentwhichmaximizes( 4 )isnon-trivial.Theinterdependencymakestheproblemcoupledanddifculttosolvebytraditionaloptimizationmethods[ 65 ].Acombinationof(p,f)isnamedaproleandanaiveapproachtosolvetheproblemistoinvestigateallprolesexhaustively.However,thisisimpossibleinpractice.Forexample,inamedium-sizewirelessmeshaccessnetworkwith20APswhereeachhas3frequencychannelsand10powerlevels,thesearchspaceis(310)20proles!Obviously,thecentralizedalgorithmsarenotfavorableinthewirelessmeshaccessnetworkduetothescalabilityconcern.Next,wewillintroduceadecentralizednegotiation-basedthroughputmaximizationalgorithm,fromagame-theoreticalperspective. 4.3.1CooperativeThroughputMaximizationGame TheAPsinthewirelessmeshaccessnetworksareconsideredasplayers,i.e.,decisionmakersofthegame.WemodeltheinteractionamongAPsasaCooperativeThroughputMaximizationGame(CTMG),whereeachplayerhasanidenticalobjectivefunctionUi,as Ui(p,f)=Unetwork(p,f)=NXj=1Rj(p,f),8i.(4) Foreachplayeri,allpossiblefrequencyandpowerlevelpairsformastrategyspaceiwhichhasasizeofcl,wherecisthenumberoffrequencychannelsavailableandlisthenumberoffeasiblepowerlevels.Dene =12N.(4) Then,theNplayersautonomouslynegotiateaboutthejointfrequency-powerproleininordertondtheoptimalprolewhichmaximizes( 4 ).However,duetothe 75

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interdependencyamongNplayerscausedbymutualinterference,onequestionofinterestisthatwhetherthisnegotiationwilleventuallymeetanagreement,a.k.a.,aNashequilibrium.TheimportanceofNashequilibrialiesinthatapossiblesteadystateofthesystemisguaranteed.IfthegamehasnoNashequilibrium,thenegotiationprocessneverstopsandoscillatesinaneverlastingfashion.Inaddition,weareconcerningaboutwhattheperformanceofthesteadystateswouldbe,ifexist,intermsofoverallthroughputofthewholenetwork.Weprovideanswerstothesequestionsinthefollowing. Lemma3. TheCTMGisapotentialgame. ApotentialgameisdenedasagamewherethereexistsapotentialfunctionPsuchthat P(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i))]TJ /F6 11.955 Tf 11.96 0 Td[(P(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i)=Ui(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i))]TJ /F6 11.955 Tf 11.95 0 Td[(Ui(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(i)8i,a0,a00(4) whereUiistheutilityfunctionforplayerianda0,a00aretwoarbitrarystrategiesini.Morespecically,wehavea0=[p0i,f0i]anda00=[p00i,f00i].Thenotationofa)]TJ /F9 7.97 Tf 6.59 0 Td[(idenotesthevectorofchoicesmadebyallplayersotherthani.Potentialgameshavebeenbroadlyappliedinmodelingtheinteractionsincommunicationnetworks[ 66 ].Thepopularityisonaccountofthenicepropertiesofpotentialgames,suchas PotentialgameshaveatleastoneNashequilibrium. AllNashequilibriaarethemaximizersofthepotentialfunction,eitherlocallyorglobally. ThereareseverallearningschemesavailablewhichareguaranteedtoconvergetoaNashequilibrium,suchasbetterresponseandbestresponse[ 67 ][ 68 ]. Fordetaileddescriptionaboutpotentialgames,readersarereferredto[ 67 ]and[ 69 ],whichinvestigatethepotentialgametheoryinengineeringcontext. Weobservethatinthecooperativecase,eachplayerhasthesameutilityfunctionasin( 4 ),whichistheoverallthroughputofthenetwork.Apparently,onepotential 76

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functionofthegameisthecommonutilityfunctionitself,i.e., P=U1=U2==UN.(4) Infact,thegameswhereallplayerssharethesameutilityfunctionarecalledidenticalinterestgames[ 70 ],whichisaspecialcaseofpotentialgamesandhenceallthepropertiesofpotentialgamescanbeapplieddirectly. Intheliterature,bothbestresponseandbetterresponsearepopularlearningmechanismsthathavebeenutilizedinpotentialgames[ 71 73 ].Ateachstepofthebestresponseapproach,oneoftheplayersinvestigatesitsstrategyspaceandchoosestheonewithmaximumutilityvalue.Thisupdatingprocedureiscarriedoutsequentially.Theprimarydrawbackofthebestresponseisthecomputationalcomplexity,whichgrowslinearlywiththecardinalityofthestrategyspace.Animprovementofthebestresponseistheso-calledbetterresponse,whereateachstep,theplayerupdatesaslongastherandomlyselectedstrategyyieldsabetterperformance.Thedramaticallyreducedcomputationisthetradeoffwiththeconvergencespeed.BoththebestresponseandthebetterresponsedynamicsareguaranteedtoconvergetoaNashequilibriuminpotentialgames[ 66 ].However,theremaybemultipleNashequilibriainapotentialgameandtheperformanceofdifferentequilibriamayvarydramatically.Therefore,althoughthebestresponseandthebetterresponsecouldguaranteetheconvergence,theymayreachanundesirableNashequilibriumwithinferiorperformance. Figure4-2. AnillustrativeexampleofmultipleNashequilibria LetusconsideranillustrativeexampleinFigure 4-2 .TherearefourlabeledAPsinthenetwork.AandBareclosetoeachother,andsoareCandD.Withoutlossof 77

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generality,weassumethattheAPshavethesamepowerandonlyadjusttheoperatingfrequenciesinanorderofA!B!C!Dtoavoidtheinterference.TheadaptationcontinueswiththebestresponsemechanismuntilaNashequilibriumisreached.Supposethattherearetwofrequencychannelsavailable,say1and2.First,Arandomlyselectsonechannel,say1.Bwillpick2.Next,Chasthechancetoupdate.SinceCisclosertoBthanA,channel1willbeselected.Finally,Dwillchoosechannel2.Byinspection,weclaimthatprole1)]TJ /F3 11.955 Tf 12.45 0 Td[(2)]TJ /F3 11.955 Tf 12.44 0 Td[(2)]TJ /F3 11.955 Tf 12.44 0 Td[(1isaNashequilibriumsincenoplayeriswillingtoupdateitsstrategyunilaterally.Meanwhile,weobservethatanotherprole1)]TJ /F3 11.955 Tf 13 0 Td[(2)]TJ /F3 11.955 Tf 13 0 Td[(1)]TJ /F3 11.955 Tf 13 0 Td[(2isalsoaNashequilibrium.Obviously,thesecondNashequilibriumgeneratesmuchlessinterferencethantherstNashequilibriumandhenceyieldssuperiorperformanceintermsofoverallthroughput.However,thebestresponseonlyleadstothelessdesirableNashequilibrium. Infact,theexistenceofmultipleNashequilibriaisobservedin[ 71 ]bysimulations.However,theauthorsfailtospecifywhichonewouldbethesteadystateoftheirgameduetothelimitationofthebestresponse,eveninastatisticalfashion.RecallthattheNashequilibriaarethemaximizersofthepotentialfunctioninpotentialgames,convergingtoaninferiorNashequilibriumanalogouslyindicatesbeingtrappedatalocaloptimumofthepotentialfunction.However,itistheglobaloptimum,i.e.,theoptimalNashequilibrium,thatisthedesirablesteadystatewhichweareyearningfor. Next,weintroduceanegotiation-basedthroughputmaximizationalgorithm(NETMA)whichcanconvergetotheoptimalNashequilibriumwitharbitrarilyhighprobability. 4.3.2NETMA-Negotiation-BasedThroughputMaximizationAlgorithm WeassumethattheAPsarehomogeneousandeachhasauniqueIDforroutingpurpose.EachAPmaintainstwovariablesDpreandDcur.TheAPhastheknowledgeofitscurrentthroughputandrecordsitinDcur.Wheneverthereisachangeofthroughput 78

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5. Forasmoothingfactor>0,thek-thAPkeepsf0andp0withprobability ePcur= ePcur=+ePpre==1 1+e(Ppre)]TJ /F9 7.97 Tf 6.59 0 Td[(Pcur)=(4) 6. Thek-thAPbroadcastsanothershortnotifyingmessage,whichindicatestheendofupdatingprocessandaspecicnumber,toalltheotherAPs. Until: Thestoppingcriteria)]TJ /F1 11.955 Tf 10.09 0 Td[(ismet. Notethatinstep 6 ,thespecicformatofdependsonthepredenedstoppingcriterion)]TJ /F1 11.955 Tf 6.78 0 Td[(.Forexample, Ifthestoppingcriterionisthemaximumnumberofnegotiationsteps,isacounterwhichaddsoneaftereachupdatingprocess. IfthestoppingcriterionisthatnoAPhasupdatedforacertainnumberofsteps,isabinarynumberwhere1meansupdating. Ifthestoppingcriterionisthatthedifferencebetweensumthroughputobtainedinconsecutivestepsarelessthanapredenedthreshold,isthecalculatedsumthroughputaftereachupdatingprocess. Wecanhaveotherstoppingcriteria)]TJ /F1 11.955 Tf 6.77 0 Td[(sandcorrespondingformatsofaswell. TheNETMAalgorithmisinspiredbytheworkin[ 74 ],whereasimilaralgorithmwasrstintroducedinthecontextofstreamcontrolinMIMOinterferencenetworks.Thedistinguishingfeatureofthistypeofnegotiationalgorithms,fromthebetterresponseandthebestresponse,istherandomnessdeliberatelyintroducedonthedecisionmakinginstep 5 .TherationalecanbeillustratedinFigure 4-2 intuitively.Ifthereisnorandomnessindecisionmaking,i.e.,=0,thefourAPsmaygettrappedatalowefciencyNashequilibrium1)]TJ /F3 11.955 Tf 11.91 0 Td[(2)]TJ /F3 11.955 Tf 11.9 0 Td[(2)]TJ /F3 11.955 Tf 11.91 0 Td[(1.However,withtherandomnesscausedbynonzero,theymayreachanintermediatestate1)]TJ /F3 11.955 Tf 12.42 0 Td[(2)]TJ /F3 11.955 Tf 12.42 0 Td[(2)]TJ /F3 11.955 Tf 12.42 0 Td[(2andarriveattheoptimumNashequilibrium1)]TJ /F3 11.955 Tf 12.15 0 Td[(2)]TJ /F3 11.955 Tf 12.16 0 Td[(1)]TJ /F3 11.955 Tf 12.16 0 Td[(2eventually.Moreover,theupdatingruleinstep 5 alsoimpliesthatiff0andp0yieldabetterperformance,i.e.,Ppre)]TJ /F6 11.955 Tf 12.3 0 Td[(Pcur<0,thek-thAPwillkeepthemwithhighprobability.Otherwise,itwillchangewithhighprobability. ThesteadystatebehaviorofNETMAischaracterizedinthefollowingtheorem. 80

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Theorem4.1. NETMAconvergestotheoptimalNashequilibriuminCTMGwitharbitrarilyhighprobability. Proof. TheproofofTheorem 4.1 followssimilarlinesoftheproofin[ 74 ]and[ 75 ]. Figure4-3. MarkovianchainofNETMAwithtwoplayers First,weobservethatthejointfrequency-powernegotiationgeneratesanN-dimensionalMarkovianchain.Figure 4-3 illustratestheMarkovianchainintroducedbyNETMAwithtwoplayers,sayAandB.Letxandybethechoicesforeachplayer,wherex2Aandy2B.Inotherwords,playerAcanchooseafrequency-powerpairfrom[x1,,xcl]andplayerBcanchoosefrom[y1,,ycl].Notethatatanarbitrarytimeinstant,onlyoneoftheplayerscanupdate.InFigure 4-3 ,forexample,state(x1,y1)canonlytransittoastateeitherinthesameroworthesamecolumn,notanywhereelse.ThisistrueforeverystateintheMarkovianchain.LetSi,jdenotethestateof(xi,yi).Wehave Pr(Sm,njSi,j)=8>>>>>>><>>>>>>>:eP(Sm,n)= 2cl(eP(Sm,n)=+eP(Si,j)=),ifm=iorn=j0,otherwise.(4) 81

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whereisthesmoothingfactorinstep 5 ofNETMAandP(Si,j)isthevalueofthepotentialfunction,i.e.,( 4 ),atthestateofSi,j. LetusderivethestationarydistributionPrforeachstate.Weexaminethebalancedequations.Writingthebalanceequations[ 76 ]atthedashedline,weobtain clXk=2Pr(S1,1)Pr(S1,kjS1,1)=clXk=2Pr(S1,k)Pr(S1,1jS1,k).(4) Bysubstituting( 4 )with( 4 ),wehave Pclk=2Pr(S1,1)eP(S1,k)= eP(S1,1)=+eP(S1,k)==Pclk=2Pr(S1,k)eP(S1,1)= eP(S1,1)=+eP(S1,k)=. (4) Observingthesymmetryofequation( 4 )aswellastheMarkovianchain,wenotethatthesetofequationsin( 4 )areallbalancedifforarbitrarystate~Sinthestrategyspace,thestationarydistributionis Pr(~S)=KeP(~S)=(4) whereKisaconstant.Byapplyingtheprobabilityconservationlaw[ 77 ][ 76 ],weobtainthestationarydistributionfortheMarkovianchainas Pr(~S)=eP(~S)= PSi2eP(Si)=(4) forarbitrarystate~S2. Inaddition,weobservethattheMarkovianchainisirreducibleandaperiodic.Therefore,thestationarydistributiongivenin( 4 )isvalidandunique. LetSbetheoptimalstatewhichyieldsthemaximumvalueofpotentialfunctionP,i.e., S=argmaxSi2P(Si).(4) 82

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erroneousoperationhappens,forexample,twoAPsupdateatthesametimeinourcase,itonlyprolongstheconvergencetimeforNETMAyetdoesnotaffectthenaloutputofNETMA.Thisisbecausethatsuchanerror,asveriedin[ 72 ]viaextensivesimulations,hasnoinuenceonthestatisticallymonotonic-increasingtendencyofthepotentialfunction. 4.4Non-CooperativeAccessNetworks Intheprevioussection,wediscussthescenarioswhereallAPsinthewirelessmeshaccessnetworkarecooperative,andtheoverallthroughputismaximizedbynegotiationsamongautonomousAPsusingtheNETMAmechanism.However,cooperationisnotalwaysattainable.Althoughthefunctionalityofrelayingpacketsforeachothercanbeachievedbyincentivemechanismssuchas[ 79 ],theadjustableparametersinsideeachcellcannotbeenforcedandeffectivelycontrolled.TheNAPsmaybelongtodistinctself-interestedusersandtheycareaboutexclusivelytheirownthroughputratherthantheoverallaggregatedthroughput.Inotherwords,theutilityfunctionofeachselshuseris Ui=Ri(i)(4) whereRiisthethroughputofthei-thcell,denedin( 4 ).AnalogoustoCTMG,wecanformulatetheinteractionamongNselshAPsasaNon-cooperativeThroughputMaximizingGame(NTMG)whereeachAPisattemptingtondthefrequency-powerpairwhichmaximizesitsownSINRvalueaswellasthecorrespondingthroughput.Asinthecooperativecase,eachplayer'sutilityfunctiondependsonthefrequencyandpowerofitselfaswellasthoseofothers.However,NTMGisnolongeranidenticalinterestgame. Lemma4. InNTMG,alltheAPswilltransmitwiththemaximalpowerattheNashequilibrium,ifexists. 84

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Theorem4.2. ThereexistsatleastoneNashequilibriuminNTMG. Proof. Letusrstconsiderthesimpliedgame.Foreachplayer,theutilityfunctionisgivenas Ui=)]TJ /F3 11.955 Tf 9.3 0 Td[((Xk2Fi(fi)pkgki+Ni) (4) =)]TJ /F3 11.955 Tf 9.3 0 Td[((Xk6=i,k2Npkgki(fi)]TJ /F6 11.955 Tf 11.95 0 Td[(fk)+Ni) (4) where (k)=8><>:1,ifk=00,otherwise.(4) Weconjecturethatoneofthefeasiblepotentialfunctionsis P=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(1 2Xi2NXk2Fi(fi)pkgki.(4) Thevericationisasfollows. 2P=)]TJ /F13 11.955 Tf 11.29 11.36 Td[(Xi2NXk2Fi(fi)pkgki=)]TJ /F13 11.955 Tf 11.29 24.51 Td[(8<:Xk2Fi(fi)pkgki+Xj6=i,j2NXk2Fj(fj)pkgkj9=;=)]TJ /F13 11.955 Tf 11.29 24.51 Td[(8<:Xk2Fi(fi)pkgki+Xj6=i,j2Nfpigij(fi)]TJ /F6 11.955 Tf 11.96 0 Td[(fj)+Xk2Fj(fj),k6=ipkgkjg9=;=)]TJ /F13 11.955 Tf 11.29 20.44 Td[((Xk6=i,k2Npkgki(fk)]TJ /F6 11.955 Tf 11.95 0 Td[(fi)+Xj6=i,j2Npigij(fi)]TJ /F6 11.955 Tf 11.96 0 Td[(fj)+Xj6=i,j2NXk2Fj(fj),k6=ipkgkj9=;. (4) 86

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Notethat pkgki(fk)]TJ /F6 11.955 Tf 11.96 0 Td[(fi)=pigik(fi)]TJ /F6 11.955 Tf 11.95 0 Td[(fk)(4) foranypairofi,k.Wehave P=)]TJ /F13 11.955 Tf 11.29 20.44 Td[((Xk6=i,k2Npkgki(fk)]TJ /F6 11.955 Tf 11.95 0 Td[(fi)+1 2Q()]TJ /F6 11.955 Tf 9.3 0 Td[(i))(4) where Q()]TJ /F6 11.955 Tf 9.3 0 Td[(i)=Xj6=i,j2NXk2Fi(fi),k6=ipkgkj(4) andQ()]TJ /F6 11.955 Tf 9.3 0 Td[(i)isindependentoffi.Therefore,forarbitrarytwofrequenciesa0anda00ofplayeri,wehave Qa0()]TJ /F6 11.955 Tf 9.29 0 Td[(i)=Qa00()]TJ /F6 11.955 Tf 9.3 0 Td[(i)(4) and P(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i))]TJ /F6 11.955 Tf 11.96 0 Td[(P(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i)=nPk2Fi(a0)pkgki+1 2Qa0()]TJ /F6 11.955 Tf 9.3 0 Td[(i)o)]TJ /F13 11.955 Tf 11.29 13.27 Td[(nPk2Fi(a00)pkgki+1 2Qa00()]TJ /F6 11.955 Tf 9.3 0 Td[(i)o=Ui(a0,a)]TJ /F9 7.97 Tf 6.58 0 Td[(i))]TJ /F6 11.955 Tf 11.95 0 Td[(Ui(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(i). (4) Therefore,accordingtothedenitionin( 4 ),thesimpliedgameisapotentialgameandhasatleastoneNashequilibrium.Thus,theexistenceofNashequilibriuminNTMGisobtainedfromtheequivalencederivedfromLemma 4 AsshowninLemma 4 ,attheequilibrium,thenon-cooperativeAPswillalwaystransmitatthemaximumpowerlevel.ThisseemtobethebestchoiceforeachoneoftheAPs.However,itisusuallynotafavorablestrategyfromasocial-welfarepointofview.Tobridgetheperformancegap,weproposealinearpricingschemetocombatwiththeselshbehaviors,i.e.,theplayersareforcedtopayataxproportionaltotheutilizedresources.Forexample,wecouldimposeapricetoallselshAPsforthepower 87

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theyutilize.HenceforeachAP,theutilityfunctionbecomes Ui=Ri(i))]TJ /F8 11.955 Tf 11.96 0 Td[(ippi(4) whereiprepresentsthepowerutilizationpricespecicforthei-thAPandpiisitstransmissionpower.Therefore,themorepowerAPuses,themoretaxithastopay.Byimposingpowerpricesproperly,amoredesirableequilibriummaybeinduced,fromasocial-welfarepointofview.WedenethecorrespondinggameasaNon-cooperativeThroughputMaximizationGamewithPricing(NTMGP). Letusrstinvestigatetheimpactofpricesonthebehaviorsofplayers.Ifip=0,wherenopriceisimposed,thei-thAPwilltransmitatthemaximumpowerandcausesextrainterferencetootherAPs.However,ifweimposeanunbearablyhighprice,saypi=1,theAPwouldrathernottotransmitatall.Basedontheseobservations,weproposeaheuristiclinearpricingschemetoimprovetheoverallthroughputinnon-cooperativewirelessmeshaccessnetworks. Toenforcethescheme,weintroduceapricingdictatorunit(PDU)intothenetworkwhichdeterminesthepricesforallAPsandinformsthemtimely.Inaddition,weassumethatthePDUhasthemonitoringcapabilityandisawareoftheoperatingfrequenciesofeachcell.TherearetwopriceschargedbythePDUforeachnon-cooperativeAP.Besidesthepowerutilizingpriceip,afrequencyswitchingpriceifisimposedonthei-thAPwheneveritchangestheoperatingfrequency.Thepricesettingprocessisdescribedasfollows. Pricesettingprocess: PhaseI: ThePDUsets1f=Nf=0and1p=Np=0andallAPsplayNTMGuntilconverges,i.e.,aNashequilibriumisreached. ThePDUcollectsthecurrentthroughputinformationfromeachcell,denotedbyMi,whereiistheindexofthecell. 88

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PhaseII: ThePDUsets1f=Nf=1. ForeachAPindexedbyi=1,,N: 1. ThePDUsetsip=1forthei-thAPandlettheAPsplaytheNTMGP.Uponconvergence,thePDUcollectstheoverallthroughput,sayVi,inthecurrentpricesetting. 2. Calculatethepowerutilizingpriceforthei-thAPas ~ip=Vi)]TJ /F13 11.955 Tf 11.96 8.97 Td[(PNj=1,j6=iMj pmax(4) 3. Resetip=0. Output: Powerutilizingpricevector~p=[~1p,,~Np] Frequencyswitchingpricevector~f=[1,,1] Inthepricesettingprocessabove,thePDUimposeszeropricesforallAPsinitially.Asaconsequence,allAPswilltransmitwithpmaxattheequilibrium,asshowninLemma 4 .Uponconvergence,thePDUxesthefrequencyswitchingpricetoinnitywhichdiscouragesthenon-cooperativeAPsfromswitchingchannelsthereafter.In( 4 ),PNj=1,j6=iMjisthesumthroughputforallcellsotherthani,whenthei-thAPtransmitswiththemaximalpowerduetothezeropowerprice.Similarly,Viisthesumthroughputofothercellswhenthei-thAPissilentduetotheunaffordablepowerprice.Therefore,in( 4 ),thepowerutilizationpricechargedforthei-thAP,a.k.a.,~ip,canbeviewedasacompensationtotheimpactitcausesontheoverallthroughputofothercells.Themorepoweritutilizes,themoresevereitaffectstheotherplayersandthusthemoreitpays,asillustratedin( 4 ).Hence,byimposingtaxesdeliberately,theselshbehaviorsofnon-cooperativeAPsareeffectivelydiscouragedandamoredesirableequilibriumcanbeinduced,intermofoverallthroughputofthewholenetwork.TheproofofexistenceofNashequilibriuminNTMGPisstraightforward.Notethattheutility 89

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functionof( 4 )isquasi-concavewithrespecttopower.TheexistenceofpurestrategyNashequilibriumfollowsdirectlyfromtheresultsof[ 80 ].WewillpresentthedetailedperformanceevaluationofCTMG,NTMGandNTMGPinSection 4.6 4.5AnExtensiontoAdaptiveCodingandModulationCapableDevices Sofar,wehaveassumedthatthethroughputofacellisgivenintheformof( 4 ),whichisnotacontinuousfunction.Tobeprecise,inbothSection 4.3 andSection 4.4 ,weareconnedtothetraditionalIEEE802.11familydeviceswherethemodulationandcodingschemesarepre-determinedandxed.Forexample,Table 4-1 providesamappingbetweenSINRvaluesandcorrespondingrates[ 81 ],thefeasibledatatransmissionrateisadiscretesetandisusuallymuchlessthanthetheoreticalchannelcapacity.WenamesuchdevicesaslegacyIEEE802.11devices. Table4-1. Dataratesv.s.SINRthresholdswithmaximumBER=10)]TJ /F5 7.97 Tf 6.59 0 Td[(5 Rate(Mbps) MinimumSINR(dB) 1 -2.92 2 1.59 5.5 5.98 11 6.99 However,thankstotheadvanceofcodingtechniques,themaximumdatatransmissionratecanbelargelyclosedtothetheoreticalShannoncapacityinAWGNchannels[ 82 ].Notethatachievingthisrequiresvariable-ratetransmissionsbymatchingtotheinstantaneousSINR,whichcanbeimplementedinpracticethroughadaptivecodingandmodulation(ACM)techniques[ 83 ].ThismotivatesustoextendourresultstoACM-capabledevices.Morespecically,weareconsideringmoreadvancedandpowerfulAPswhichattempttotunethefrequencyandpowerinordertomaximize Ci(i)=Wilog2(1+i)(4) 90

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whereCiistheShannoncapacityofthei-thcell,andWiisthebandwidth.Notethatinthisscenario,themaximumachievabletransmissionrate,asdenotedbytheShannoncapacity,isacontinuousvariablewithrespecttoi,ratherthandiscrete,asexempliedinTable 4-1 .ItisworthnotingthattheonlydifferenceinthisscenarioisthealternativeobjectivefunctionofeachAP.Therefore,alltheresultswehaveobtainedsofarareextendable8tothisspecialscenariowithmerelyachangeoftheobjectivefunction. Furthermore,duetothecontinuityoftheobjectivefunctionin( 4 ),theaforementionedheuristicpricingschemeinSection 4.4 canbeimproved.Withoutlossofgenerality,weassumethatallthecellshaveunitybandwidth.Restated,weassumethatthePDUisacentralizeddevicewhichknowsthechannelenvironmentsufcientlybygreedyacquiring.Inaddition,thePDUisassumedtohavemonitoringcapabilityandisawareoftheoperatingfrequencyofeachcell.ThetailoredpricingschemeforthisACM-capablescenarioisdescribedasfollows. First,thePDUsets1f=Nf=0and1p=Np=0andallAPsplayselshly,untilaNashequilibriumisachieved. ThePDUsets1f=Nf=1. ForeachAPindexedbyi=1,,N,thePDUsets 8Morespecically,inthecooperativecase,wereplace( 4 )withUnetwork(p,f)=PNi=1Ci(i)whereasinthenon-cooperativecase,( 4 )and( 4 )arereplacedbyUi=Ci(i)andUi=Ci(i))]TJ /F8 11.955 Tf 11.95 0 Td[(ippi,respectively. 91

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ip=jXk2Fi@Ck @pij=jXk2Fi)]TJ /F6 11.955 Tf 77.21 8.09 Td[(pkgkkgik ln2(1+k)(Pj2Fkpjgjk+Nk)2j=Xk2Fipkgkkgik ln2(1+i)(pkgkk k)2=Xk2Figik2k ln2(1+k)pkgkk. (4) ThenthePDUinformseachAPthecorrespondingprices,i.e.,if=1andipcalculatedin( 4 ). Notethatthemajordifferenceinthispricingschemeliesin( 4 ),wheretheipchargedforthei-thAPisthePigouvianpriceleviedtodiscourageitsrecklesspowerincrease. AfterobtainingthepricesimposedbythePDU,eachAPxestheoperatingfrequencyandcalculatestheoptimumpowerpi,whichmaximizesUi=Ci)]TJ /F6 11.955 Tf 12.87 0 Td[(piip,accordingtothefollowingsteps. @Ui @pi=@Ci @pi)]TJ /F8 11.955 Tf 11.95 0 Td[(ip(4) where @Ci @pi=gii ln2(Pk2Fipkgki+Ni)(1+i).(4) Bysetting( 4 )equaltozero,wehave pi=1 ln2ip)]TJ /F13 11.955 Tf 13.15 18.52 Td[(Pk2Fipkgki+Ni giipmaxpmin.(4) where[x]badenotesmaxfminfb,xg,agandpmax,pminrepresentthemaximumandminimumpowerofthedevice,respectively.NotethatPk2Fipkgki+Nicanbeeasilymeasuredwhenthei-thAPsetsitstransmissionpowertozero.Therefore,ndingthe 92

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optimumpowerofeachAPrequiresonlylocalinformationandcanbeimplementedinadistributedfashion. In( 4 ),weobservethatthepricesimposedonAPsdependonthepowervectorp,andviceversa.Therefore,aftereachAPoptimizesitspoweraccordingto( 4 ),thePDUmeasuresthenewvalueofpower,adjuststhepricefollowing( 4 ),andthenannouncesthenewpriceagain.Thepricingsettingprocessin( 4 )andtheutilitymaximizationprocessin( 4 )areexecutedinaniterativemanner,untilconvergence.Thewholepricingschemeissummarizedasfollows. PricingScheme: 1. ThePDUinitiallysetszeroprices.Asaconsequence,allAPswillconvergeataNashequilibriumwithmaximumpower. 2. ThePDUsetsinnityforthefrequencyswitchingpricewhichpreventsthewholenetworkfromunstablefrequencyoscillations. 3. ThePDUsetsandannouncesthepowerpricesforeachAPaccordingto( 4 ). 4. InformedbythePDU,eachAPoptimizesitstransmissionpowerfollowing( 4 ). 5. Gobacktostep 3 untiltheiterationconverges. Theperformanceevaluationofthistailoredpricingschemeisillustratedinthenextsection. 4.6PerformanceEvaluation 4.6.1LegacyIEEE802.11Devices Inthissubsection,werstinvestigatetheperformanceofNETMA,NTMGandNTMGPwithlegacydevices,i.e.,thescenariosweconsideredinSection 4.3 andSection 4.4 .WeassumethewirelessmeshaccessnetworkhasNhomogeneousAPs.Theothersimulationparametersaresummarizedasfollows. EachAPhasamaximumpowerpmax=100mWandaminimumpowerpmin=10mWand10differentpowerlevelsas[10mW,20mW,,100mW]. Thenoiseexperiencedateachreceiverisassumedidenticalandhasapowerof2mW. 93

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Figure4-5. ThetrajectoryoffrequencynegotiationsinNETMAwhenN=5andc=3 Figure4-6. ThetrajectoryofpowernegotiationsinNETMAwhenN=5andc=3 AsindicatedbytheOPcurve,theglobaloptimumobtainedbyenumerationapproachfunctionsastheupperboundoftheoverallthroughput.InFigure 4-4 ,weobservethatNETMAgraduallycatchesupwiththeglobaloptimumasnegotiationsgo.Asexpected,thenon-cooperativeAPsyieldremarkablyinferiorperformanceintermsofoverallthroughput,depictedbytheNTMGcurve.TheinefciencyisduetotheselshbehaviorthatAPstransmitatthemaximumpowerandareregardlessoftheinterference.TheexistenceofNashequilibriuminbothCTMGandNTMGaresubstantiatedbytheconvergenceofcurvesinFigure 4-4 .Figure 4-5 andFigure 4-6 depictthetrajectoriesoffrequencynegotiationsandpowerlevelnegotiationsinNETMA,respectively.Attheinitialization,eachAPrandomlypicksafrequencyandapowerlevel 95

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andnegotiateswitheachotherfollowingNETMAmechanism,untiltheoptimumNashequilibriumisachieved.NotethatwhenthefrequencyvectorandpowervectorconvergeinFigure 4-5 andFigure 4-6 ,thecorrespondingoverallthroughputobtainedbyNETMAcatchestheglobaloptimuminFigure 4-4 simultaneously. 4.6.1.2Exampleoflargenetworks Wenowconsideralargewirelessmeshaccessnetworkwith20APs.Theenumerationapproachisnolongerfeasibleinthisscenarioduetotheenormousstrategyspace.The20APsarerandomlyscatteredinad-by-dsquare,wherethesidelengthdisatunableparameterinsimulations.Weinvestigatebothcooperativeandnon-cooperativecasesrepresentedbyNETMAandNTMGcurves,wherethemaximumnumberofiterationsissetto!=1000.Figure 4-7 pictoriallydepictstheperformanceinefciencyofNTMGcausedbythenon-cooperativeAPswhichtransmitatthemaximumpower.TheaveragethroughputperAPiscalculatedbyaveragingtheresultsof50simulations,foreachvalueofthesidelengthd.InFigure 4-7 ,itisworthnotingthatasthesidelengthdgetsbigger,theperformancegapbetweenNETMAandNTMGreduces.Thereasonisthatwhentheareaislarge,theimpactofmutualinterferenceislesssevereandsoistheperformancedeterioration.However,whenthenetworkiscrowded,i.e.,dissmall,theselshbehaviorsareremarkablydevastating. Toalleviatethethroughputdegradationbythenon-cooperativeAPs,weimplementthelinearpricingschemeintroducedinSection 4.4 .ThethroughputimprovementisillustratedasNTMGPinFigure 4-7 .Itisnoticeablethatbyutilizingtheproposedpricingscheme,theefciencyofNashequilibriumisdramaticallyenhanced,especiallyforcrowdednetworks.Therefore,theselshincentivesofthenon-cooperativeAPshavebeeneffectivelysuppressed. 4.6.2ACM-CapableDevices Inthissubsection,weinvestigatetheperformanceofourmodelwithACM-capabledevices.Sincethemajordifferenceliesintheimprovedpricingschemetailoredforthis 96

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Figure4-7. PerformanceevaluationofthewirelessmeshaccessnetworkwithN=20andc=3 specicscenario,wewillonlyprovidetheperformanceevaluationofthetailoredpricingscheme,i.e.,NTMGP,inordertoavoidduplicateresults.Weconsiderapopulatednetworkwhere30APsarerandomlyscatteredinan100m-by-100msquare,i.e.,N=30.Allothersimulationparametersarethesameasintheprevioussubsectionexceptthatthepowerisacontinuousvariableinthisscenario.Werstinvestigatetheperformanceintermsofoverallachievablerateofthenetworkwhennopricingschemeisapplied,asaperformancebenchmark.Afterwards,thetailoredpricingschemeisimplementedtoimprovetheequilibriumefciency,a.k.a.,theoverallperformanceattheequilibrium.AsshowninFigure 4-8 ,theoverallachievablerateofthenetworkisdramaticallyimprovedbythepricingscheme.ThePDUadaptstheannouncedpriceforeachAPaccordingtotheoptimumpower,whichiscalculatedbythepreviousannouncedprice,inaniterativefashion.Theaggregatedachievablerateconvergesasthepricesettingiterationgoes,asdepictedbyFigure 4-8 Figure 4-9 showsthetrajectoriesofthepowerutilizationpricesforeachAP.ThecounterpartofactualutilizedpowerisillustratedinFigure 4-10 ,wherethecurvesstartatpmaxandevolvewiththeannouncedpriceinFigure 4-9 AsobservedinFigure 4-8 ,toachieveabetterperformance,thepricesettingprocessneedstobeexecutediteratively,comparingwiththeoneshotheuristic 97

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viewedasthegatewaysofthewirelesssensornetwork.DenotethesetofsinksasD=fd1,d2,,djDjg.Inthiswork,weconsiderageneralscenariowherethedatapacketsfromasourcenodecanbedeliveredtotheDPCviaanyofthesinknodeinD.Therefore,differentfromtheexistingliteraturesuchas[ 3 88 89 ],thesourcenodesdonotspecifytheparticulardestinationnodeforthegeneratedpackets.Theselectionofthedestinationnodeisachievedbythenetworkviadynamicroutingschemes.ThenetworktopologyconsideredinthisworkisillustratedinFigure 5-1 Figure5-1. Topologyofwirelesssensornetworks Foraparticularnodeinthenetwork,saynoden,wedenotednasthenumberofminimumhopsfromnodentothed-thdatasinkinsetD.Dene fn=mind(dn),d=1,,jDj(5) astheminimumvalueofdnfornoden,i.e.,theminimumnumberofhopsfromnodentoasinknodeinsetD.Weassumethatnodenisawareofthevalueoffnaswellasthosevaluesoftheneighboringnodes,whichareattainableviaprecalculationsbytraditionalroutingmechanismssuchasDijkstra'salgorithm. 106

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transmittingXs(t)numberofpackets.Itisworthnotingthatbyselectingproperutilityfunctions,thefairnessamongcompetitivesessionscanbeachieved.Forexample,ifUs(Xs(t))=log(Xs(t)),aproportionalfairnessamongmultiplesessionscanbeenforced[ 4 10 11 ]. 5.3.3QueueManagement Foreachnodeninthenetwork,therearejNj)]TJ /F13 11.955 Tf 19.91 3.16 Td[(fnnumberofqueuesthataremaintainedandupdated.ThequeuesaredenotedbyQn,h,whereh=fn,,jNj)]TJ /F3 11.955 Tf 18.66 0 Td[(1.NotethatjNj)]TJ /F3 11.955 Tf 19.53 0 Td[(1isthemaximumnumberofhopsforaloop-freeroutingpathinthenetwork.ThepacketsinthequeueofQn,hareguaranteedtoreachoneofthesinknodesinsetDwithinhhops,aswillbeshowninSection 5.4 .Itisinterestingtoobservethatforanewlygeneratedpacketbysessions,thesourcenode,i.e.,n0s,canplaceitinanyofthequeuesofQn0s,h,whereh=fn0s,,jNj)]TJ /F3 11.955 Tf 17.09 0 Td[(1,forfurthertransmission.Thatistosay,consecutivepacketsfromthesourcenoden0smaytraversethroughdifferentnumberofhopsbeforereachingadestinationsinknodeinsetD.Therefore,whenanewpacketisgenerated,thesourcenodeneedstomakeadecisiononwhichqueuethepacketshouldbeplaced,namely,trafcsplittingdecision.Inaddition,thedecisionshouldbemadepromptlyonanonlinebasiswithlowcomputationalcomplexity. Withaslightabuseofnotation,weuseQn,htodenotethequeueitselfandQn,h(t)torepresentthenumberofqueuebacklogs3intimeslott.Forasinglequeue,sayQn,h,itisstableif[ 13 14 ] limB!1g(B)!0(5) where g(B)=limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Pr(Qn,h(t)>B). Thenetworkisstableifalltheindividualqueuesinthenetworkarestable. 3Intheunitofpackets. 108

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Foralink(n,j)2L,werequirethatthepacketsfromQn,hcanbeonlytransmittedtoQj,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1,ifexists.Therefore,thequeueupdatingdynamicforQn,hisgivenby Qn,h(t+1)24Qn,h(t))]TJ /F13 11.955 Tf 16.97 11.36 Td[(X(n,j)2Lun,hn,j(t)35++X(m,n)2Lum,h+1m,n(t)+XsXhs(t)n=n0s(5) where[A]+denotesmax(A,0)andun,hn,j(t)representstheallocateddatarateforthetransmissionsofQn,h!Qj,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1onlink(n,j),attimeslott,and N)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xh=fnun,hn,j(t)=un,j(t) whereun,j(t)=n,j(t)ifIn,j(t)=1,i.e.,link(n,j)isscheduledtobeactiveduringtimeslott,andun,j(t)=0otherwise.ThenotationofXhs(t)denotesthenumberofpacketsthatareadmittedtothenetworkforsessionsandarestoredinqueueQn,hforfuturetransmissions.TheindicatorfunctionA=1ifeventAistrueandA=0otherwise.Notethattheinequalityin( 5 )incorporatesthescenarioswherethetransmitterofaparticularlinkhaslesspacketsinthequeuethantheallocateddatarate.Weassumethatduringonetimeslot,thenumbersofpacketsthatasinglequeuecantransmitandreceiveareupperbounded.Mathematicallyspeaking,wehave X(m,n)2Lum,h+1m,n(t)uin,8n,h,t, (5) and X(n,j)2Lun,hn,j(t)uout,8n,h,t. (5) 5.3.4Session-SpecicRequirements Inthiswork,weconsiderascenariowhereeachsessionhasaspecicraterequirements.Therefore,toensuretheminimumaveragerate,weneedtondapolicy 109

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that limT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0Xs(t)s,8s.(5) Inaddition,weassumethateachsessioninthenetworkhasanaveragehoprequirements.Morespecically,dene Ms(t)=jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1Xh=gn0shXhs(t) (5) where jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1Xh=gn0sXhs(t)=Xs(t),8s,t. Werequirethatforeachsessions, limT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Ms(t)s,8s.(5) Notethattheaveragehopforaparticularsessionsisrelatedtotheaveragedelayexperiencedandtheaverageenergyconsumedforthepackettransmissionsofsessions.Therefore,byassigningdifferentvaluesofsands,aprioritizedsolutionamongmultiplecompetitivesessionscanbeachievedforthenetworkresourceallocationproblem. 5.4StochasticNetworkUtilityMaximizationinWirelessSensorNetworks Intheprevioussection,weproposeaconstrainedqueueingmodeltoinvestigatetheperformanceofmulti-hopwirelesssensornetworks.Themodelconsistsofseveralimportantissuesfromdifferentlayers,includingtherateadmissioncontrolproblem,thedynamicroutingproblemaswellasthechallengeofadaptivelinkscheduling.Tobetterunderstandtheproposedconstrainedqueueingmodel,inthissection,wewillexaminethestochasticnetworkutilitymaximizationproblem(SNUM)inmulti-hopwirelesssensornetworks.Asacrosslayersolution,anAdaptiveNetworkResourceAllocation(ANRA)schemeisproposedtosolvetheSNUMproblemjointly.TheproposedANRAschemeisanonlinealgorithminnaturewhichprovablyachievesanasymptoticallyoptimalaverage 110

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dynamicroutingaswellasadaptivelinkschedulingcomponents.Moreover,theANRAalgorithmcanachieveanautomaticloadbalancingsolutionbyutilizingdifferentsinknodescorrespondingtothevariationsofthenetworkconditions.TheANRAalgorithmisanonlinealgorithminnaturewhichrequiresonlythestateinformationofthecurrenttimeslot.WeshowthattheANRAalgorithmachievesa(1)]TJ /F8 11.955 Tf 12.21 0 Td[()optimalsolutionwherecanbearbitrarilysmall.Therefore,theproposedANRAalgorithmisofparticularinterestfordynamicwirelesssensornetworkswithtimevaryingenvironments. 5.4.2TheANRACrossLayerAlgorithm BeforepresentingtheproposedANRAscheme,weintroducetheconceptofvirtualqueues[ 16 17 34 ]tofacilitateouranalysis.Specically,foreachsessions,wemaintainavirtualqueueYs,whichisinitiallyempty,andthequeueupdatingdynamicisdenedas Ys(t+1)=[Ys(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Xs(t)]++s,8s,t.(5) Similarly,wedeneanothervirtualqueueforeverysessions,denotedbyZs,andthequeuedynamicisgivenby Zs(t+1)=[Zs(t))]TJ /F8 11.955 Tf 11.96 0 Td[(s]++Ms(t),8s,t,(5) whereMs(t)isdenedin( 5 ).Notethatthevirtualqueuesaresoftwarebasedcounterswhichareeasytomaintain.Forexample,thesourcenodeofeachsessioncancalculatethevaluesofvirtualqueuesYs(t)andZs(t)andupdatethevaluesaccordinglyfollowing( 5 )and( 5 ).Inaddition,weintroduceapositiveparameterJwhichistunableasasystemparameter.TheimpactofJonthealgorithmperformancewillbediscussedshortly.TheproposedANRAcrosslayeralgorithmisgivenasfollows.ADAPTIVENETWORKRESOURCEALLOCATION(ANRA)SCHEME: JointRateAdmissionControlandTrafcSplitting(attimet): 112

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Foreachsourcenode,sayn0s,thereareanumberofqueues,i.e.,Qn0s,h,h=fn0s,,jNj)]TJ /F3 11.955 Tf 17.93 0 Td[(1.Findthevalueofhwhichminimizes Zs(t)h+Qn0s,h(t)(5) wheretiesarebrokenarbitrarily.Denotetheoptimumvalueofhash.Thesourcenoden0sadmitsanumberofnewpacketsas Xs(t)=min]Xs(t),As(t)(5) where ]Xs(t)="J 2)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(Zs(t)h+Qn0s,h(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Ys(t)#+.(5) Fortrafcsplitting,thesourcenoden0swilldepositallXs(t)packetsinQn0s,h. JointDynamicRoutingandLinkScheduling(attimet): Foreachlink(m,n)2L,denealinkweightdenotedbyWm,n(t),whichiscalculatedas Wm,n(t)="maxh=fm,,jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1(Qm,h(t))]TJ /F6 11.955 Tf 11.96 0 Td[(Qn,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1(t))#+.(5) NotethatifQn,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1doesnotexist,thetransmissionsfromqueueQm,htoQn,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1areprohibited.Attimeslott,thenetworkselectsaninterference-freelinkscheduleI(t)whichsolves maxI(t)2(t)m,n(t)Wm,n(t).(5) Iflink(m,n)isactive,i.e.,Im,n(t)=1,thequeueofQm,hisselectedfortransmissionswhere h=argmaxh=fm,,jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1(Qm,h(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qn,h)]TJ /F5 7.97 Tf 6.58 0 Td[(1(t)).(5) END Notethat( 5 )issimilartotheoriginalMaxWeightalgorithm,introducedin[ 12 ]andgeneralizedin[ 13 14 34 ].Thedynamicroutingandlinkschedulingareaddressedjointlybysolving( 5 ),whichrequirescentralizedcomputation.However,following[ 12 ],manyworkhavebeenfocusedonthedistributedsolutionsof( 5 ).Althoughthedistributedcomputationissueisnotthefocusofthiswork,weemphasizethatourproposedANRAschemecanbeapproximatedwellbyexistingdistributedsolutionssuchas[ 36 38 41 42 45 46 48 ]. 113

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Aa,a=1,,jAj,wehave0AaAmaxwheredenotestheelement-wisecomparison.WeuseatorepresentthesteadystatedistributionofAa. Theorem5.1. IftheconstraintsintheSNUMproblemaresatised,themaximumnetworkutility,denotedbyU,canbeachievedbyaclassofstationaryrandomizedpolicies.Mathematically,thevalueofUisthesolutionofthefollowingoptimizationproblem,withtheauxiliaryvariablespkaandRka,as maxXaaXsUs(jSj+1Xk=1pkaRka) (5) s.t. Theconstraintsin( 5 )aresatised. 0RkaAa. pka0,8a,k. PjSj+1k=1pka=1,8a. Proof. WeproveTheorem 5.1 byshowingthatforarbitrarypolicywhichsatisestheconstraintsintheSNUMproblem,theoverallnetworkutilityisatmostU,whichistheoptimumutilityattainedbyaclassofstationaryrandomizedpolicies.Inotherwords,weneedtoshowthat UP=limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 XsUs(Xs(t))!U(5) whereUPistheaveragenetworkutilityunderapolicyP. ForeachstateinA,sayAa,deneRaasthesetofnonnegativeratevectorsthatareelement-wisesmallerthanAa.DeneCRaastheconvexhullofsetRa.Therefore,anypointinCRacanbeconsideredasafeasiblenetworkadmittedratevectorgiventhatthecurrentarrivalratevectorisAa.NotethateverypointinCRaisavectorwithadimensionofjSj-by-1.Therefore,itcanberepresentedbyaconvexcombinationofatmostjSj+1points,denotedbyRka,k=1,,jSj+1,accordingtoCaratheodory'stheorem.Inlightofthis,werstconsideratimeintervalfrom0toT)]TJ /F3 11.955 Tf 12.19 0 Td[(1.DenoteNa(T) 115

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asthesetoftimeslotsthatA(t)=Aa.Therefore,wecanrewrite( 5 )as UP=limsupT!1XajNa(T)j TXsUs0@jSj+1Xk=1pkaRka1A. Duetothestationaryassumption,wehave UP=XaaXslimsupT!1Us0@jSj+1Xk=1pkaRka1A. Notethatwealsoassumethattheutilityfunctioniscontinuousandbounded.Therefore,thecompactnessoftheutilityfunctionsisassured.Next,wefocusonasubsequenceoftimedurations,denotedbyTi,i=1,,1.Denote UPnet(Ti)=XaaXsUs0@jSj+1Xk=1pka(Ti)Rka1A. Itisstraightforwardtoverifythat UP=limsupi!1UPnet(Ti). Duetothecompactnessoftheutilityfunctions,followingBolzano-Weierstrasstheorem[ 90 ],weclaimthatthereexistsasubsequenceofTi,i=1,,1,suchthat limi!1Us0@jSj+1Xk=1pka(Ti)Rka1A!fUas. DenoteepkaasthevalueswhichgeneratefUas,i.e., fUas=Us0@jSj+1Xk=1epkaRka1A. Wehave UP=limsupi!1Unet(Ti)=XaaXsUs0@jSj+1Xk=1epkaRka1A. AccordingtothedenitionofUin( 5 ),weconcludethatUPU. 116

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Intuitively,Theorem 5.1 indicatesthattheglobalmaximumnetworkutilitycanbeachievedbycertainrandomizedstationarypolicies.However,tocalculateU,thestationarypolicyneedstoknowthesteadystatedistributionswhicharedifculttoobtaininpractice.Inlightofthis,weproposeanadaptivenetworkresourceallocationscheme,namely,ANRA,whichisanonlinesolutionanddoesnotrequiresuchstatisticalinformationasapriori.Fornotationsuccinctness,denote UA(t)=XsE(Us(Xs(t))) astheexpectednetworkutilityinducedbytheANRAscheme.TheperformanceoftheANRAalgorithm,withaparameterJ,isgivenasthefollowingtheorem. Theorem5.2. ForagivensystemparameterJ,wehave liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0UA(t)U)]TJ /F3 11.955 Tf 14.6 10.74 Td[(B J(5) whereBisaconstantandisgivenby B=jNj(jNj)]TJ /F3 11.955 Tf 17.94 0 Td[(1))]TJ /F3 11.955 Tf 5.48 -9.69 Td[((uout)2+(uin+Amax)2+Xs)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((s)2+(s)2+jSj(Amax)2(jNj)]TJ /F3 11.955 Tf 17.94 0 Td[(1)4+1. Inaddition,theconstraintsintheSNUMproblem,i.e.,( 5 ),aresatisedsimultane-ously. Proof. TheproofofTheorem 5.2 isdeferredtoSection 5.5 ItisworthnotingthatifweletJ!1,theperformanceinducedbytheANRAalgorithmcanbearbitrarilyclosetotheglobaloptimumsolutionU.However,asatradeoff,alargervalueofJalsoyieldsalongeraveragequeuesizeinthenetwork.AccordingtoLittle'sLaw,alargerqueuesizecorrespondstoalongeraveragedelayexperiencedinthenetwork.Therefore,byselectingthevalueofJproperly,atradeoff 117

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betweenthenetworkoptimalityandtheaveragedelayinthenetworkcanbeachieved.Wewilldiscussmoreaboutthisissueinthenextsection. 5.5PerformanceAnalysis Inthissection,weprovideaprooftoTheorem 5.2 intheprevioussection.Recallthatin( 5 )and( 5 ),weintroducetwovirtualqueues,i.e.,Ys(t)andZs(t)foreachsessions.Therefore,theaveragerateandtheaveragehoprequirementsfromallsessionsareconvertedintothestabilityrequirementsforthevirtualqueues.Forexample,thevirtualqueueupdateofYs(t)isgivenby( 5 ).IfthevirtualqueueYsisstable,theaveragearrivalrateshouldbelessthantheaveragedeparturerateofthequeue,i.e., slimT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0Xs(t), whichisexactlytheminimumaverageraterequirementimposedbysessions.Bythesametoken,theaveragehoprequirementofsessionsisconvertedtothestabilityproblemofthevirtualqueueZs.Dene QQQ(t)=(QQQ(t),YYY(t),ZZZ(t)),namely,allthedataqueuesandthevirtualqueuesinthenetwork.Ourobjectiveistondapolicywhichstabilizesthenetworkwithrespectto QQQwhilemaximizingtheoverallnetworkutility. Wersttakethesquareof( 5 )andhave Qn,h(t+1)2Qn,h(t)2+0@X(n,j)2Lun,hn,j(t)1A2+0@X(m,n)2Lum,h+1m,n(t)+XsXhs(t)n=n0s1A2)]TJ /F3 11.955 Tf 9.3 0 Td[(2Qn,h(t)0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.35 Td[(X(m,n)2Lum,h+1m,n(t))]TJ /F13 11.955 Tf 11.95 11.35 Td[(XsXhs(t)n=n0s1A. Sinceweassumethateachnodegeneratesatmostonesession,wehave XsXhs(t)n=n0sAmax,8t,n. 118

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Notethatifweallowthatanodecaninitiatemultiplesessions,wehave XsXhs(t)n=n0sjSjAmax,8t,n wherejSjisthenumberofsessionsinthenetwork. Inlightof( 5 )and( 5 ),wehave Qn,h(t+1)2)]TJ /F13 11.955 Tf 11.96 13.27 Td[(Qn,h(t)2uout2+uin+Amax2)]TJ /F3 11.955 Tf 19.26 0 Td[(2Qn,h(t)0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXhs(t)n=n0s1A. (5) Wenextsum( 5 )overallthedataqueuesinthenetwork,i.e.,Qn,h,andhave Xn,hQn,h(t+1)2)]TJ /F13 11.955 Tf 11.96 11.35 Td[(Xn,hQn,h(t)2B1)]TJ /F3 11.955 Tf 19.26 0 Td[(2Qn,h(t)0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXhs(t)n=n0s1A (5) where B1=jNj(jNj)]TJ /F3 11.955 Tf 17.93 0 Td[(1)(uout)2+(uin+Amax)2. NotethatMs(t),denedin( 5 ),satises Ms(t)(jNj)]TJ /F3 11.955 Tf 17.93 0 Td[(1)2Amax. Next,wetakethesquareof( 5 )and( 5 )andthushave Ys(t+1)2Ys(t)2+Xs(t)2+(s)2)]TJ /F3 11.955 Tf 11.96 0 Td[(2Ys(t)Xs(t))]TJ /F8 11.955 Tf 11.96 0 Td[(s and Zs(t+1)2Zs(t)2+Ms(t)2+(s)2)]TJ /F3 11.955 Tf 11.96 0 Td[(2Zs(t)s)]TJ /F6 11.955 Tf 11.96 0 Td[(Ms(t). 119

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Similarly,wesumoverallthesessionsandhave XsYs(t+1)2)]TJ /F13 11.955 Tf 11.96 11.36 Td[(XsYs(t)2B2)]TJ /F3 11.955 Tf 11.96 0 Td[(2XsYs(t)Xs(t))]TJ /F8 11.955 Tf 11.95 0 Td[(s where B2=jSj(Amax)2+Xs(s)2. Also,weobtain XsZs(t+1)2)]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsZs(t)2B3)]TJ /F3 11.955 Tf 11.95 0 Td[(2XsZs(t)s)]TJ /F6 11.955 Tf 11.95 0 Td[(Ms(t) where B3=Xs(s)2+jSj(jNj)]TJ /F3 11.955 Tf 17.94 0 Td[(1)4(Amax)2. Denethesystem-wideLyapunovfunctionas L( QQQ(t))=Xn,hQn,h(t)2+XsYs(t)2+XsZs(t)2. Next,wedenetheLyapunovdrift[ 14 ]ofthesystemas =E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(L( Q(t+1)))]TJ /F6 11.955 Tf 11.96 0 Td[(L( Q(t))j Q(t).(5) Dene B=B1+B2+B3, wehave B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.04 11.36 Td[(X(m,n)2Lum,hm,n(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXhs(t)n=n0S Q(t)1A)]TJ /F3 11.955 Tf 19.26 0 Td[(2E XsYs(t)(Xs(t))]TJ /F8 11.955 Tf 11.96 0 Td[(s) Q(t)!)]TJ /F3 11.955 Tf 11.95 0 Td[(2E XsZs(t)(s)]TJ /F6 11.955 Tf 11.95 0 Td[(Ms(t)) Q(t)!. 120

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Next,wesubtractbothsidesbyJEPsUs(Xs(t))j Q(t)andhave )]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(Xs(t)) Q(t)!B)]TJ /F3 11.955 Tf 19.27 0 Td[(2Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t) Q(t)1A+2E XsXhQn0s,h(t)Xhs(t) Q(t)!)]TJ /F3 11.955 Tf 11.96 0 Td[(2E XsYs(t)Xs(t) Q(t)!+2XsYs(t)s+2E XsZs(t)Ms(t) Q(t)!)]TJ /F3 11.955 Tf 11.96 0 Td[(2XsZs(t)s)]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(Xs(t)) Q(t)!. (5) WerewritetheR.H.S.of( 5 )as R.H.S.=B+2XsYs(t)s)]TJ /F3 11.955 Tf 11.95 0 Td[(2XsZs(t)s)]TJ /F3 11.955 Tf 19.26 0 Td[(2Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t) Q(t)1A)]TJ /F6 11.955 Tf 19.26 0 Td[(E Xs2Ys(t)Xs(t))]TJ /F13 11.955 Tf 11.95 11.35 Td[(Xs2Zs(t)Ms(t))]TJ /F13 11.955 Tf 11.95 11.35 Td[(XsXh2Qn0s,h(t)Xhs(t)+JXsUs(Xs(t)) Q(t)!. WeobservethatthedynamicroutingandschedulingcomponentoftheANRAschemeisactuallymaximizing Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.35 Td[(X(m,n)2Lum,h+1m,n(t) Q(t)1A.(5) Inaddition,thejointrateadmissioncontrolandtrafcsplittingcomponentoftheANRAschemeisessentiallymaximizing E Xs2Ys(t)Xs(t))]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xs2Zs(t)Ms(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXh2Qn0s,h(t)Xhs(t)+JXsUs(Xs(t)) Q(t)!(5) withtheconstraintsof XhXhs(t)=Xs(t),8s,t.(5) 121

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Toseethis,wecandecompose( 5 )toshowthateachsessionsonlymaximizes JUs(Xs(t))+2Ys(t)Xs(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Zs(t)XhhXhs(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xh2Qn0s,h(t)Xhs(t).(5) Therefore,theproposedANRAalgorithmindeedminimizestheR.H.S.of( 5 )overallpolicies. Considerareducednetworkcapacityregion,denotedby,parameterizedby>0,as fjn,h+2g(5) whereistheoriginalnetworkcapacityregionand n,h=limT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0XsXhs(t)n=n0s.(5) DeneUastheglobaloptimumnetworkutilityachievedinthereducedcapacityregion.Apparently,wehavelim!0U!U.Inaddition,denoteXhs,(0),Xhs,(1),,Xhs,(t),astheoptimumratesequencewhichyieldsU.DeneXsastheaverageoftheoptimumratesequenceofsessions,inthereducedcapacityregion.ItisstraightforwardtoverifythatXs+isintheoriginalnetworkcapacityregion.Therefore,followingasimilaranalysisasin[ 13 15 91 ],weclaimthatthereexistsarandomizedpolicy,denotedbyR,whichgenerates E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.04 11.36 Td[(X(m,n)2Lum,h+1m,n(t)1AXhs,+(5) ifnisoneofthesourcenodesand E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t)1A(5) forothernodes.Furthermore,policyRensures E XhXhs,(t)c+! 122

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and E)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(Ms,(t)+s whereMs,(t)isgeneratedbyXhs,(t).DuetothefactthattheproposedANRAschememinimizestheR.H.S.of( 5 )overallallpolicies,includingR,wehave )]TJ /F6 11.955 Tf 11.95 0 Td[(JE XsUs(Xs(t)) Q(t)!B)]TJ /F3 11.955 Tf 11.96 0 Td[(2 Xn,hQn,h(t)+XsYs(t)+XsZs(t)!)]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(XhXhs,(t)+) Q(t)!. Wenexttaketheexpectationwithrespectto Q(t)andobtain L( Q(t+1)))]TJ /F6 11.955 Tf 11.95 0 Td[(L( Q(t)))]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(Xs(t))!B)]TJ /F3 11.955 Tf 11.95 0 Td[(2E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE XsUs(XhXhs,(t)+)!. (5) Wesumovertimeslots0,,T)]TJ /F3 11.955 Tf 11.96 0 Td[(1andhave L( Q(T)))]TJ /F6 11.955 Tf 11.96 0 Td[(L( Q(0)))]TJ /F9 7.97 Tf 11.96 14.94 Td[(T)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0JE XsUs(Xs(t))!TB)]TJ /F9 7.97 Tf 11.96 14.94 Td[(T)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0JE XsUs(XhXhs,(t)+)!(5) since E Xn,hQn,h(t)+XsYs(t)+XsZs(t)! isalwaysnonnegative.Next,wedividethebothsidesof( 5 )byTandrearrangetermstohave 1 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0JE XsUs(Xs(t))!1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0JE XsUs(XhXhs,(t)+)!)]TJ /F3 11.955 Tf 13.42 2.65 Td[(B)]TJ /F6 11.955 Tf 13.15 8.08 Td[(L( Q(0)) T wherethenonnegativityoftheLyapunovfunctionisutilized.Sinceweassumethattheinitialqueuebacklogsinthesystemareniteandthevirtualqueuesareinitiallyempty, 123

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taking!0andliminfT!1yieldstheperformanceresultoftheANRAalgorithmstatedinTheorem 5.2 WenextshowthattheconstraintsoftheSNUMproblemarealsosatised.Toillustratethis,weshowthatthequeuesinthenetwork,includingrealdataqueuesandvirtualqueues,arestable.Basedon( 5 ),wesumovertimeslots0,,T)]TJ /F3 11.955 Tf 12.44 0 Td[(1andhave T)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=02E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!L( Q(0))+T)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0JE XsUs(Xs(t))!+TB.(5) DuetoXs(t)Amaxandtheassumptionsontheutilityfunction,weclaimthatUs(t)isupperboundedanddenotethemaximumutilitywithinonetimeslotasUmax,i.e., Us(t)Umax,8s,t.(5) Dividethebothsidesof( 5 )byTandwehave 1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=02E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!L( Q(0)) T+JjSjUmax+B. BytakinglimsupT!1,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!JjSjUmax+B 2. (5) Notethattheaboveanalysisholdsforanyfeasiblevalueof.Denote'asthemaximumvalueofsuchthat'isnotempty.Finally,weconcludethat limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!JjSjUmax+B 2'<1.(5) ThestabilityofthenetworkfollowsimmediatelyfromMarkov'sInequalityandthuscompletestheproof. Itisworthnotingthatasshownin( 5 ),alargevalueofJinducesalongeraveragequeuesizeinthenetwork.Therefore,atradeoffbetweenthealgorithm 124

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performanceoftheANRAschemeandtheaveragedelayexperiencedinthenetworkcanbecontrolledeffectivelybytuningthevalueofJ. 5.6CaseStudy Inthissection,wedemonstratetheeffectivenessoftheANRAalgorithmnumericallythroughasimplenetworkshowninFigure 5-2 .Westressthat,however,thisexemplifyingstudycasereproducesallthechallengingproblemsinvolvedinthecomplexcrosslayerinteractionsintimevaryingenvironments,suchasstochastictrafcarrivals,randomchannelconditionsanddynamicroutingandschedulingetc.AsshowninFigure 5-2 ,thesourcenodesinthenetworkarenodeAandBwhereasthedestinationsinknodesaredenotedbyEandF. Figure5-2. Examplenetworktopology Therearesixnodesandtwelvelinksinthenetwork.Therefore,nodeAandBeachmaintainsfourqueues,fromhop2tohop5,andnodeCandDeachmaintainsvequeues,fromhop1tohop5,inthebuffer.Ateachtimeslot,awirelesslinkisassumedtohavethreeequallypossibledatarates4,2,8and16.Thetrafcarrivalsarei.i.d.withthreeequallypossiblestates,i.e.,0,10and20.Theminimumraterequirementsofthe 4Notethattheunitofdatatransmissionsispacketperslot. 125

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adaptationalgorithm,weinvestigatethethreshold-basedrateadaptationalgorithmviaareverseengineeringapproach.WeassumethattheRTS/CTSsignalsareturnedoff.Inatimeslot,sayt,wedenotethechannelstate2ass(t)anddenotethesuccessfultransmissionprobability,giventhecurrenttransmissionrater(t)andthechannelconditions(t),as PS(s(t),r(t))=p(1)]TJ /F6 11.955 Tf 11.96 0 Td[(p)N)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1)]TJ /F6 11.955 Tf 11.95 0 Td[(e(s(t),r(t)))(6) whereedenotestheframeerrorrate(FER)andisgivenby e(s(t),r(t))=1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F6 11.955 Tf 11.96 0 Td[(Pe(s(t),r(t)))L(6) andPe(s(t),r(t))isthebiterrorrate(BER)whichisdeterminedbythecurrentdatarate,i.e.,modulationscheme,andthecurrentchannelcondition.Listheframelengthofthepacket.Similarly,wedenePF(s(t),r(t))=1)]TJ /F6 11.955 Tf 12.32 0 Td[(PS(s(t),r(t))asthetransmissionfailureprobabilityattimet.ItisworthnotingthatbothPSandPFarefunctionsofthecurrentdatarater(t)aswellastheinstantaneouschannelconditions(t),whichisrandom.Particularly,weassumethatthethreshold-basedrateadaptationalgorithmwillincreasethedataratebyanamountofifthereareuconsecutivesuccessfultransmissionsanddecreaseitbyiftherearedconsecutivefailures.Denoteu=u)]TJ /F3 11.955 Tf 11.64 0 Td[(1andd=d)]TJ /F3 11.955 Tf 12.66 0 Td[(1fornotationsuccinctness.Wedeneabinaryindicatorfunction(t)where(t)=1meansthatthetransmissionattimeslottissuccessfuland(t)=0otherwise.Mathematically,theupdatingruleofthethreshold-basedrateadaptationalgorithmcanbewrittenas 2Notethatthenumberoffeasiblechannelstatescanbepotentiallyinnite. 136

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r(t+1)=(r(t)+))]TJ /F14 7.97 Tf 11.65 -1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[())]TJ /F14 7.97 Tf 11.66 -1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0+r(t))]TJ /F9 7.97 Tf 11.65 -1.79 Td[(o.w. (6) where )]TJ /F9 7.97 Tf 6.78 -1.8 Td[(x=8>><>>:1,ifeventxistrue0,ifeventxisfalse. (6) Forexample,)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)=1=1ifthetransmissionattimetissuccessfuland)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=1=0otherwise.Thesymbolofo.w.denotestheeventthatneitheruconsecutivesuccessfulnordconsecutivefailedtransmissionshappened.Forsimplicity,weassumethatthemaximumallowabledatarateissufcientlylargeandtheminimumdatarateiszero,i.e.,nottransmitting.Dene h(t)=[r(t),r(t)]TJ /F3 11.955 Tf 11.95 0 Td[(1),,r(1),e(s(t),r(t)),,e(s(1),r(1))](6) asthehistoryvector.Inaddition,wedene Z(t+1)=Efr(t+1)jh(t)g (6) whereEistheexpectationoperator. Condition: (C.1)Thechannelstatesbetweentwoconsecutivesuccessfultransmissionsortwoconsecutivefailedtransmissionsareindependentrandomvariables. Weemphasizethattherestrictivecondition(C.1)isnotourgeneralassumptioninthiswork.If(C.1)issatised,however,thederivationofthereverseengineeringanalysiscanbepresentedinamoreconciseform,aswillbeshownshortly. 137

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First,weobtainEf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1jh(t)g=Prf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=1=1,)]TJ /F14 7.97 Tf 31.15 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=1=1,,)]TJ /F14 7.97 Tf 12.25 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1=1jh(t)g (6) Ifcondition(C.1)issatised,( 6 )canbefurtherdecomposedas Ef)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1jh(t)g=Prf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=1=1jh(t)gPrf)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1=1jh(t)gPrf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1=1jh(t)g=uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)) (6) Similarly,wehaveEf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0jh(t)g=Prf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=0=1,)]TJ /F14 7.97 Tf 31.15 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=0=1,,)]TJ /F14 7.97 Tf 12.25 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=0=1jh(t)g (6) If(C.1)isassumedtobevalid,wecanobtainEf)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.58 0 Td[(d)=0jh(t)g=dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)) (6) Fornotationsuccinctness,wewilltemporarilyassumethat(C.1)issatised.Theconditionwillberelaxedaftertheimplicitobjectivefunctionisrevealed.Therefore,we 138

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canwrite( 6 )as Z(t+1)=Efr(t+1)jh(t)g=(r(t)+)uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[()dYk=0(1)]TJ /F6 11.955 Tf 11.96 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))+r(t) 1)]TJ /F9 7.97 Tf 17.5 14.95 Td[(uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)))]TJ /F9 7.97 Tf 16.65 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))!=r(t)+ uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)))]TJ /F9 7.97 Tf 16.65 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))! (6) Letusrevisit( 6 ),whichcanberewrittenas r(t+1)=r(t)+1 (r(t)+))]TJ /F14 7.97 Tf 11.66 -1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[())]TJ /F14 7.97 Tf 11.66 -1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0+r(t))]TJ /F9 7.97 Tf 11.65 -1.79 Td[(o.w.)]TJ /F6 11.955 Tf 11.96 0 Td[(r(t)=r(t)+(t) (6) where (t)=1 n(r(t)+))]TJ /F14 7.97 Tf 11.66 -1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[())]TJ /F14 7.97 Tf 11.65 -1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.78 -1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0+r(t))]TJ /F9 7.97 Tf 11.65 -1.8 Td[(o.w.)]TJ /F6 11.955 Tf 11.96 0 Td[(r(t)o. (6) 139

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Itshouldbenotedthat Ef(t)jh(t)g=uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)))]TJ /F9 7.97 Tf 24.62 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.96 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)). (6) Therefore,weobservethat( 6 )isaformofstochasticapproximationwithrespectto( 6 ).Next,wepresentthereverseengineeringtheoremforthethreshold-basedrateadaptationalgorithm. Theorem6.1. Thethreshold-basedrateadaptationalgorithmof( 6 )isastochasticapproximationwhichsolvesanimplicitobjectivefunctionU(t),withaconstantstepsizeof,whereU(t)isintheformof U(t)=r(t)(uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)))]TJ /F9 7.97 Tf 16.65 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))o+K (6) ifcondition(C.1)issatisedandKisaconstantwithrespecttorater(t). Proof. Theorem1followsdirectlyfromthepreviousanalysis.Notethatthethreshold-basedrateadaptationalgorithmcanbewrittenas r(t+1)=r(t)+(t)(6) where(t)isthestochasticgradientandsatises Ef(t)jh(t)g=@U @r(t)jh(t)(6) HenceTheorem1holds. Ifcondition(C.1)doesnothold,wecanreplace uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)) 140

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and dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)) in( 6 )with Prf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=1,,)]TJ /F14 7.97 Tf 12.26 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.58 0 Td[(u)=1jh(t)g (6) and Prf)]TJ /F14 7.97 Tf 6.77 -1.86 Td[((t)=0,,)]TJ /F14 7.97 Tf 12.26 -1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0jh(t)g (6) respectivelyandthetheoremremainsvalid. ItisworthnotingthattheobjectivefunctionU(t)isatime-varyingfunctionwhichisdeterminedbythedatarateaswellasthechannelconditionsofthepasttimeslotswhere=max(u,d).Notethatthedataratewithinthelasttimeslotsalwaysremainsunchanged.However,thechanneluctuationsaffectthesuccessfultransmissionprobabilityPSandthusaltertheobjectivefunctionU(t). ThepartialderivativeoftheobjectiveofU(t)givenh(t),i.e., g(t)=uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)))]TJ /F9 7.97 Tf 17.31 14.94 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))(6) isalsotime-varying.Iftheprobabilitythatthelastutransmissionsareallsuccessfulisgreaterthantheprobabilitythatthelastdtransmissionsareallfailures,thestationtendstoincreasethedatarateandviceversa.Thespeedofrateincreasingordeceasingisdeterminedbythedifferenceofthesetwoprobabilities.Inotherwords,thepartialderivativein( 6 )couldbeeitherpositiveornegativewhichcorrespondstoarateupshiftoraratedownshift.Theabsolutevalueoftheinstantaneousderivativedeterminesthespeedofratechanging. Adirectcomputationofthepartialderivativein( 6 )ischallenging,ifnotimpossible,duetotheuncertaintyinducedbytheunpredictablestochasticchannel.Therefore,thethreshold-basedrateadaptationalgorithm,describedin( 6 ),utilizes 141

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Du(Dd):Theestimationvectorofu(d)wherethek-thelement,i.e.,Dku(Dkd),iscalculatedbyDku=Sku Cku(Dkd=Skd Ckd). R:Theresolutionparameterwhichisapositiveintegerandistunablebythestation. u(d):Thestepsizeparameterofu(d)andisgivenbyu=1 muR(d=1 mdR). 'u('d):Theperturbationvectorofu(d)wherethek-thelement,i.e.,'ku('kd),isazeromeanrandomvariablewhichisuniformlydistributedin[)]TJ /F14 7.97 Tf 22.49 5.26 Td[( Cku(Ckd),+ Cku(Ckd)]+1)]TJ /F5 7.97 Tf 6.59 0 Td[(1whereisasystemparameterandiscontrollablebythestation.Thenotationof[a,b]xyrepresents[max(a,y),min(b,x)]. Zu(Zd):Thecomparisonvectorofu(d)wherethek-thelement,i.e.,Zku(Zkd),isgivenbyZku=Dku+'ku(Zkd=Dkd+'kd). B:Thepredenedconvergencethreshold,e.g.,1,whichisdeterminedbythestation. J:Arunningparameterwhichrecordstheupdatedmaximumachievedthroughputduringonetimeperiod.Attimet0: Initialization: ThestationsetsPu=[p1,,pi,,pmu]wherepi=1 muforall1imu.Similarly,Pdisgivenby[p1,,pi,,pmd]wherepi=1 mdforall1imd. InitializesCu,Cd,Su,Sd,DuandDdtozeros. Randomlyselectsthevaluesofu(1)andd(1),saym,n,accordingtoPu(1)andPd(1). Transmitswithu=mandd=nuntilT(1)ends,i.e.,adataratechangeistriggered. RecordstheaveragethroughputwithinT(1)asJ.Attimetj(j1): Do: RecordstheaveragethroughputduringTjasO(j).IfO(j)>J,setsJ=O(j)andremainsJunchangedotherwise. 147

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Updatesthem-thelementinthesurplusvectorSuandthen-thelementinSdbyaddingthemeasurednormalizedthroughputofthelasttimeperiod,asSmu=Smu+O(j) JandSnd=Snd+O(j) J. Updatesthecountingvectorsbyaddingonetothem-thcounterinCuandthen-thcounterinCd,asCmu=Cmu+1andCnd=Cnd+1. Updatesthem-thelementinDuandthen-thelementinDdbyDmu=Smu CmuandDnd=Snd Cnd. ForeveryelementinZuandZd,updatesZku=Dku+'ku,k=1,,muandZkd=Dkd+'kd,k=1,,md. FindstheelementinZuwiththehighest5valueofZku,k=1,,mu,say,the~m-thelementinAu. Similarly,ndstheelementinZdwhichhasthehighestZkd,k=1,,md,say,the~n-thelementinAd. UpdatestheprobabilityvectorsofPuandPdas Pku=max)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(Pku)]TJ /F8 11.955 Tf 11.96 0 Td[(u,0ifk6=~m,k=1,,muPku=1)]TJ /F13 11.955 Tf 12.55 11.36 Td[(Xk6=~mPkuifk=~m (6) and Pkd=max)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(Pkd)]TJ /F8 11.955 Tf 11.95 0 Td[(d,0ifk6=~n,k=1,,mdPkd=1)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xk6=~nPkdifk=~n (6) whereu=1 muRandd=1 mdR. WiththeupdatedprobabilityvectorsPuandPd,newvaluesofuandd,i.e.,u(j+1)andd(j+1),areselected. StartsthetransmissionsinT(j+1)withu(j+1)andd(j+1). Until: max(Pu)Bandmax(Pd)BwhereBisthepredenedconvergencethreshold. 5Notethatatiecanbeeasilybrokenbyarandomselection. 148

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End Theproposedthresholdoptimizationalgorithmissimilartothestochasticestimatorlearningautomataproposedin[ 126 ].Thekeyfeatureofthisgenreoflearningautomataistherandomnessdeliberatelyintroducedby'kuand'kd.Notethatalthough'kuand'kdarezeromeanrandomvariables,theirvariancesaredependentonthevaluesofCkuandCkd,respectively.Specically,thevariancesapproachtozeroswiththeincreaseofthenumberoftimesthatthecorrespondingvaluesofuanddareselected.Asaconsequence,thethresholdoptimizationalgorithminclinestomorereliablestochasticestimatesandthuspossessesafasterconvergencebehaviorthanotherlearningalgorithms[ 126 ].Thevaluesthathavebeenselectedlessfrequentlystillhavethechanceofbeingconsideredasoptimal.However,themissingprobabilitydiminishestozeroalongiterations.Inthealgorithm,theresolutionparameterRcontrolsthestepsizeofprobabilityadjustmentinthealgorithm.AsmallervalueofRproducesane-grainedprobabilityadjustmentyetunavoidablyprolongstheconvergencetime.TheconvergencethresholdBdeterminesthestoppingcriteriaofthealgorithm.Therefore,atradeoffbetweenoptimalityandconvergenceratecanbeadjustedbytuningthevalueofB. Thesteadystatebehavioroftheproposedthresholdoptimizationalgorithmisprovidedinthefollowingtheorem. Theorem6.2. Theproposedthresholdoptimizationalgorithmis-optimalforanystationarychannelenvironmentwitharbitrarydistribution.Mathematically,foranyarbitrarilysmall>0and>0,thereexistsat0satisfying Prfj1)]TJ /F6 11.955 Tf 11.95 0 Td[(Pmuj1)]TJ /F8 11.955 Tf 11.96 0 Td[(8t>t0(6) and Prfj1)]TJ /F6 11.955 Tf 11.95 0 Td[(Pndj1)]TJ /F8 11.955 Tf 11.96 0 Td[(8t>t0(6) wheremandnarethestochasticoptimalvaluesofuandd,respectively. 149

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threshold-basedup/downmechanism,arecompared:(1),OP-theoptimumthroughputattainedbyassuminganoraclewhichforeseesthevariationofchannelandadaptsthedatarateoptimally.Thiscurveisattainedasaperformancecomparisonbenchmark;(2),ARF-thethreshold-basedrateadaptationalgorithmwithu=10andd=2;(3),DLA-thedynamicthresholdadjustmentalgorithmproposedin[ 93 ];(4),AARF-thethresholdadjustmentalgorithmproposedin[ 94 ];(5),TOA-thethresholdoptimizationalgorithmproposedinthiswork.Thealgorithmsarecomparedwitheachotherintermsoftheaveragesystemthroughput(inMbps).ForTOA,withoutlossofgenerality,weassumethatmu=md=10andtheresolutionparameterRis1.TheconvergencethresholdBis0.999andis1.TheperformancecurvesoftheaforementionedalgorithmsareplottedinFigure 6-1 InFigure 6-1 ,weobservethatexcepttheOPcurve,allotherrateadaptationalgorithmssufferfromperformancedegradationswithlargeDopplerspreadvalues.RecallthatalargeDopplerspreadvalueindicatesafastfadingchannelenvironmentandhencetheaveragethroughputdeterioratesduetotheincompetencyofcapturingshortchanneluctuations.Amongwhich,ARFandAARFprovideworstperformancewithaslightdifference.DLAperformsbetterduetothecapabilityofswitchingbetween Table6-1. SNRv.s.BERforIEEE802.11bdatarates SNR BPSK QPSK CCK CCK (dB) (1Mbps) (2Mbps) (5.5Mbps) (11Mbps) 1 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 5E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 8E)]TJ /F5 7.97 Tf 6.59 0 Td[(2 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(1 2 1E)]TJ /F5 7.97 Tf 6.58 0 Td[(6 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 4E)]TJ /F5 7.97 Tf 6.59 0 Td[(2 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(1 3 6E)]TJ /F5 7.97 Tf 6.58 0 Td[(8 2.1E)]TJ /F5 7.97 Tf 6.59 0 Td[(4 1.8E)]TJ /F5 7.97 Tf 6.58 0 Td[(2 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(1 4 7E)]TJ /F5 7.97 Tf 6.58 0 Td[(9 3E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 7E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 5E)]TJ /F5 7.97 Tf 6.59 0 Td[(2 5 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 2.1E)]TJ /F5 7.97 Tf 6.59 0 Td[(6 1.2E)]TJ /F5 7.97 Tf 6.58 0 Td[(3 1.3E)]TJ /F5 7.97 Tf 6.58 0 Td[(2 6 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.5E)]TJ /F5 7.97 Tf 6.59 0 Td[(7 3E)]TJ /F5 7.97 Tf 6.59 0 Td[(4 5.2E)]TJ /F5 7.97 Tf 6.58 0 Td[(3 7 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(8 6E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 2E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 8 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(9 1.3E)]TJ /F5 7.97 Tf 6.58 0 Td[(5 7E)]TJ /F5 7.97 Tf 6.59 0 Td[(4 9 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(9 2.7E)]TJ /F5 7.97 Tf 6.58 0 Td[(6 2.1E)]TJ /F5 7.97 Tf 6.58 0 Td[(4 10 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(9 5E)]TJ /F5 7.97 Tf 6.59 0 Td[(7 6E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 ... ... ... ... ... 152

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Figure6-2. Thetrajectoriesofuanddinachievingthestochasticoptimumvalues i.e.,PuandPd,aredemonstratedinFigure 6-3 andFigure 6-4 ,respectively.Notethat,inFigure 6-3 ,thesixthcurvewhichrepresentsthevalueofP6usoonexcelsothersandapproachesto1.Correspondingly,thevalueofuconvergesto6rapidlyasdepictedinFigure 6-2 .InFigure 6-4 ,thesecondcurveapproachestounitygraduallywhileothersdiminishtozeros.Asaconsequence,inFigure 6-2 ,thevalueofdconvergestothestochasticvalue,i.e.,2.Inthissamplerunofsimulation,thestochasticoptimumvaluesofuandd,whichmaximizestheexpectedthroughputofthesystem,isgivenbyu=6andd=2.Theproposedthresholdoptimizationalgorithmndsthesevalueseffectivelyandefciently,whileprovidingsuperiorperformancethanothernon-adaptivethreshold-basedrateadaptationalgorithms,asdemonstratedinFigure 6-1 Figure6-3. EvolutionoftheprobabilityvectorPu 154

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interestingandneedsfurtherinvestigation.Duetothecompetitivenatureofchannelaccess,astochasticgameformulationmaybeutilized. 156

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Therestofthischapterisorganizedasfollows.TherelatedworkisreviewedinSection 7.2 andSection 7.3 providesthesystemmodelofourwork.ThestochastictrafcengineeringproblemwithconvexityisinvestigatedinSection 7.4 .InSection 7.5 ,weextendourframeworktothenon-convexstochastictrafcengineeringproblem.PerformanceevaluationisprovidedinSection 7.6 ,followedbyconcludingremarksinSection 7.7 7.2RelatedWork Traditionaltrafcengineering(TE)algorithmsareproposedasthesolutiontothetrafcmanagementofthenetworkinacost-efcientmanner.DifferentfromthetraditionalQoSrouting,thetrafcengineeringsolutionnotonlyguaranteesacertainQoSlevelforeachow,butalsooptimizesaglobalperformancemetricoverthewholenetwork,bysplittingtheingresstrafcoptimallyamongseveralavailablepaths.Themulti-pathroutingisusuallysupportedbytheMulti-ProtocolLabelSwitching(MPLS)techniqueswheretheexplicitroutingpathforapacketispredeterminedratherthanbeingcomputedinahop-by-hopfashion.Forapairofsourceanddestinationnodes,thesetofavailablepaths,a.k.a.,labelswitchedpaths(LSP),areestablishedandmanagedbysignallingprotocolssuchasRSVP-TE[ 141 ]andCR-LDP[ 142 ]ormanuallyconguration.Thetraditionaltrafcengineeringsolutionevolvestothestochastictrafcengineering(STE)solutionwhenuncertaintyexistsinthenetwork,e.g.,therandomreturnsoftheprimaryusersinourscenario.TEsolutionsrequireconsistentroutechangeswhichareunfavorableinthatthenetworkwillbeoverwhelmedbytheoscillationsinducedbytheunpredictablebehaviorsoftheprimaryusers.Inlightofthisstabilityconcern,STEsolutionalternativelypursuesanoptimummulti-pathroutingstrategysuchthattheexpectedutilityofthenetworkismaximized.Thestochastictrafcengineeringwithuncertaintiesarediscussedintheliteraturesuchasin[ 143 ]and[ 144 ].However,thepreviousworksusuallyassumeaprobabilitydistributionoftheuncertainty,whileinourscenario,thebehaviorsoftheprimaryusersarecompletelyunpredictable 160

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7.4StochasticTrafcEngineeringwithConvexity 7.4.1Formulation Inthestandardnetworkutilitymaximizationframework,eachuserhasautilityfunctionUl(xl),whichreectsthedegreeofsatisfactionofuserlbytransmittingatarateofxl,e.g.,Ul(xl)=log(xl).Inthissection,weassumetheutilityfunctionstobeconcaveanddifferentiable.Thenon-convexutilityfunctionsareconsideredinSection 7.5 .Notethatthefairnessissuecanbeembodiedintheutilityfunctions[ 4 ].Forexample,intheseminalpaper[ 113 ],thelog-utilityfunctionsareadoptedtoachievetheproportionalfairnessamongdifferentows. Deneafeasiblestochastictrafcengineeringsolutionasr=[r1,,rjLj]whererl,[r1l,,rjPljl].WecanformulatethestochastictrafcengineeringproblemasP1: maxr0Xl2LUl(Xk2Plrkl)s.t.Xk2Plrkldl8l2L (7)Xs2SsXk2Plrkl0@Xe2Pkllse(fe,cse)1Abl8l2L (7)feXs2Sscse8e2E (7)fe=Xl2LXk2PlHle,krkl8e2E (7)cse=Wse1 Tlog2(1+Kse)8e2E (7) wheree2Pklrepresentsthelinksalongthek-thpathofuserl.ThevariableinP1isthevectorofr.Therstsetofconstraintsreectthattheoveralldataratesofallpathscannotexceedthetrafcdemanddl.Thesecondsetofconstraintsindicatethatfor 167

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eachuserl,theexpectedcosthastobenomorethanapredenedconstraintbl.Thethirdsetofconstraintsrepresentthattheaggregatedowonlinkecannotexceedtheaveragelinkcapacity.Apparently,iftheunderlyingprobabilitydistributionofeachstatesisknownasapriori,P1isadeterministicconvexoptimizationproblemandthuseasytosolve.However,inpractice,theaccuratemeasurementofprobabilitydistributionisanon-trivialtask.In[ 20 ],weutilizedastochasticapproximationbasedapproachtocircumventthedifcultyofestimatingtheprobabilitydistribution.Inthefollowing,wewillextendthistechniqueanddevelopatailoreddistributedalgorithmtoaddresstheissuesoftime-varyinglinkcapacitiesaswellastheuser-specicQoSrequirements,whichareofparticularinterestinmulti-hopcognitivewirelessmeshnetworks. First,denetheLagrangianfunctionofP1asL(r,,,v)=Xl2LUl(Xk2Plrkl)+Xl2Ll(dl)]TJ /F13 11.955 Tf 12.24 11.35 Td[(Xk2Plrkl)+Xl2Lvl0@bl)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xs2SsXk2Plrkl(Xe2Pkllse(fe,cse))1A)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xe2Ee(fe)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xs2Sscse)=Xl2L(Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.24 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F6 11.955 Tf 9.3 0 Td[(vlXs2SsXk2Plrkl(Xe2Pkllse(fe,cse))9=;)]TJ /F13 11.955 Tf 11.96 11.35 Td[(Xe2Ee(fe)]TJ /F13 11.955 Tf 11.95 11.35 Td[(Xs2Sscse)=Xs2Ss(Xl2L Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.25 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F13 11.955 Tf 11.58 11.35 Td[(Xk2Plrkl(Xe2Pkl(vllse(fe,cse)+e))1A+Xe2Eecse) 168

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DeneMs(,,v)=supr0(Xl2L Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.24 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F13 11.955 Tf 11.58 11.35 Td[(Xk2Plrkl(Xe2Pkl(vllse(fe,cse)+e))1A+Xe2Eecse) (7) Let~rbetheoptimumsolutionof( 7 ).Wewilldiscusshowtoobtain~rshortly.ThedualfunctionofP1isobtainedby g(,,v)=Xs2SsMs(,,v).(7) Thus,thedualproblemofP1isgivenbyP2: min,,v0g(,,v).(7) 7.4.2DistributedAlgorithmicSolutionwiththeStochasticPrimal-DualApproach Inthissubsection,weproposeadistributedalgorithmicsolutionofP1,orequivalentlyP2,basedonthestochasticprimal-dualmethod.Inordertoreachthestochasticoptimumsolution,thedualvariables,andvareupdatedaccordingtothefollowingdynamics l(n+1)=[l(n))]TJ /F8 11.955 Tf 11.95 0 Td[(l(n)l(n)]+8l2L (7) e(n+1)=[e(n))]TJ /F8 11.955 Tf 11.95 0 Td[(e(n)e(n)]+8e2E (7) vl(n+1)=[vl(n))]TJ /F8 11.955 Tf 11.95 0 Td[(b(n)l(n)]+8l2L (7) where[x]+denotesmax(0,x)andnistheiterationnumber.l(n),e(n)andb(n)arethecurrentstepsizeswhilel(n),e(n)andl(n)arerandomvariables.Moreprecisely,theyarenamedthestochasticsubgradientofthedualfunctiong(,)andthefollowing 169

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requirementsneedtobesatisedEfl(n)j(1),,(n)g=@lg(,,v)8l2L (7)Efe(n)j(1),,(n)g=@eg(,,v)8e2E (7)Efl(n)jv(1),,v(n)g=@vlg(,,v)8l2L (7) whereE(.)istheexpectationoperatorand(1),,(n),(1),,(n)andv(1),,v(n)denotethesequencesofsolutionsgeneratedby( 7 ),( 7 )and( 7 ),respectively.ByDanskin'sTheorem[ 162 ],wecanobtainthesubgradientsasl(n)=dl)]TJ /F13 11.955 Tf 12.25 11.35 Td[(Xk2Pl~rkl(n)8l2L (7)e(n)=cse(n))]TJ /F3 11.955 Tf 12.05 2.66 Td[(~fe(n)8e2E (7)l(n)=bl)]TJ /F13 11.955 Tf 12.24 11.35 Td[(Xk2Pl~rkl(n)Xe2Pkllse(~fe(n),cse(n))8l2L (7) where~rklistheoptimumsolutionof( 7 ).Notethatcse(n)denotestheinstantaneouschannelcapacityonlinkeatiterationn. WenextshowhowtocalculateMs(,,v)in( 7 ),i.e.,ndingtheoptimumsolution,denotedby~r,whichmaximizesXl2L Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.24 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F13 11.955 Tf 11.58 11.35 Td[(Xk2Plrkl(Xe2Pkl(vllse(fe,cse)+e))1A+Xe2Eecse (7) Notethatwhenupdatingtheprimalvariable,i.e.,r,thelinkcostsaredeterministicwhichareobtainedviathefeedbacksignal,e.g.,ACKmessages.Therefore,byutilizingthesamestochasticsubgradientapproach,wehave rkl(n+1)=rkl(n)+r(n)(n)dl0 (7) 170

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where (n)=@Ul @Pk2Plrkl(n))]TJ /F8 11.955 Tf 11.95 0 Td[(l)]TJ /F13 11.955 Tf 13.55 11.36 Td[(Xe2Pkl(e+vlle(fe,cse)) (7) isthestochasticsubgradientmeasuredattimen. Theorem7.1. TheproposedalgorithmconvergestotheglobaloptimumofP1withprobabilityone,ifthefollowingconstraintsofstepsizesaresatised:(1)(n)>0,(2)P1n=0(n)=1,and(3)P1n=0((n))2<1,8l2Lande2E,whererepresentse,l,bandrgenerally. Proof. First,letusrevisittheupdatingequationsof( 7 ),( 7 )and( 7 ).Notethatinthestochasticsubgradientapproach,themeasuredvaluesofl(n),e(n)andl(n)areconsideredastheinstantaneousobservationoftherealgradients,denotedby l(n), e(n)and l(n),respectively.Weconsidertherelationshipofl(n)and l(n)forinstance.Theobservationvalue,i.e.,l(n),canberewrittenas l(n)=l(n))]TJ /F25 11.955 Tf 11.95 0 Td[(E(l(n))+E(l(n)))]TJ ET q .478 w 305.43 -335.28 m 329.96 -335.28 l S Q BT /F8 11.955 Tf 305.43 -345.92 Td[(l(n)+ l(n)= l(n)+E(l(n)))]TJ ET q .478 w 248.61 -362.17 m 273.13 -362.17 l S Q BT /F8 11.955 Tf 248.61 -372.82 Td[(l(n)+l(n))]TJ /F25 11.955 Tf 11.95 0 Td[(E(l(n))= l(n)+]l(n)+[l(n) (7) whereEistheexpectationoperatorand ]l(n)=E(l(n)))]TJ ET q .478 w 275.28 -460.8 m 299.8 -460.8 l S Q BT /F8 11.955 Tf 275.28 -471.44 Td[(l(n) (7) [l(n)=l(n))]TJ /F25 11.955 Tf 11.95 0 Td[(E(l(n)). (7) Notethat]l(n)isthedifferencebetweentheexpectationoftheobservationsandtherealgradient.Hence,itisthebiasedestimationerrorterm.Next,weexaminethat E([l(n)j\l(n)]TJ /F3 11.955 Tf 11.95 0 Td[(1),,[l(0))=08n.(7) 171

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Therefore,theseriesof[l(n)isamartingaledifferencesequence[ 163 ].Therelationshipof( 7 )indicatesthattheobservationvalueistherealgradientdisturbedbyabiasedestimationerroraswellasamartingaledifferencenoise.Wenextinvestigatetheconvergenceconditionsofthestochasticprimal-dualapproach.For]l(n),thefollowingrequirement 1Xn=0e(n)jE(l(n)))]TJ ET q .478 w 261.26 -122.19 m 285.78 -122.19 l S Q BT /F8 11.955 Tf 261.26 -132.83 Td[(l(n)j<1(7) issatisedduetothestationaryassumption.Similarly, E([l(n)2)=E)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((l(n))]TJ /F25 11.955 Tf 11.96 0 Td[(E(l(n)))2(7) isboundedaswell.Thesimilaranalysiscanbeextendedtoe(n),l(n)and(n)in( 7 ),( 7 )and( 7 )straightforwardly.Therefore,thestandardconditionsaresatisedandtheconvergenceresultofTheorem1follows[ 8 ]. Itisworthnotingthattheaforementioneddistributedalgorithmenjoysthemeritofdistributedimplementationfromanengineeringperspective.Withthecurrentvaluesofdualvariables,eachsourcenodeS(l)optimizes( 7 )accordingto( 7 )and( 7 ).Theinformationneedediseitherlocallyattainableoracquirablebythefeedbackalongthepaths.Forexample,thechannelstatesoftheintermediatenodesalongpathscanbepiggybackedbytheend-to-endacknowledgementmessagesfromthedestinationnode,i.e.,thegatewaynodeinourscenario.Thesourcenodeupdatesthelandvlaccordingto( 7 )and( 7 )wheretheneededinformationiscalculatedby( 7 )and( 7 ),respectively.Foreachlinke,thecurrentstatusof( 7 )ismeasured.Next,thevalueofeisupdatedfollowing( 7 ).Theiterationcontinuesuntilanequilibriumpointisreached.Notethatourframeworkcanincorporatethewirelesslossynetworkscenariosbyreplacingtheowratewiththeeffectiveowrateintheleaky-pipeowmodel[ 164 ]. 172

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wherewql,kistheq-thelementofwl,k,q=0,,Q. Next,weformulateanidenticalinterestgamewheretheplayersarethejLjsourcenodesandthecommonobjectivefunctionistheoverallnetworkutility,i.e.,thesummationoftheutilityfunctions.Inaddition,foreachsourcenode,ateamoflearningautomata[ 167 ]isconstructed.Ateachtimestep,everysourcenodepicksthedataratesonitsownpathsaccordingtotheprobabilityvectors,whicharedeterminedbytheweightingvectors.Basedonthefeedbacksignal,whichwillbedenedshortly,eachsourcenodeadjuststheweightingvectorsandtheiterationcontinues.Theexecutedalgorithmoneverysourcenode,sayl,isprovidedasfollows.Algorithm: Repeat: Foreverypath,sayk,randomlyselectsatransmissionraterjl,kfromR,accordingtothecurrentprobabilityvectorpl,k(n)wherendenotesthecurrenttimeslot. Afterreceivingthefeedbacksignal(n)fromthegatewaynode,ifthecostconstraintissatised,theweightingvectorwl,kisupdatedaswql,k(n+1)="wql,k(n)+(n)(n)(1)]TJ /F6 11.955 Tf 28.6 8.09 Td[(ewql,k PQq=0ewql,k)+p (n)&ql,k(n)#L0forq=j;wql,k(n+1)=hwql,k(n)+p (n)&ql,k(n)iL0forq6=j; (7) Otherwise,theweightingvectorremainsthesame. Theprobabilityvectorpl,kisthenupdated,followingequation( 7 ). Until: max(pl,k(n+1))>BwhereBisapredenedconvergencethreshold. Inthealgorithm,(n)isthelearningparameterofthealgorithmsatisfying0<(n)<1.Lisasufcientlylargeyetnitenumberwhichkeepstheweightingvectorbounded.Thesequenceof&ql,k(n)isasetofi.i.d.randomvariableswithzeromeanandavarianceof2(n).Theglobalfeedbacksignal(n)iscalculatedbythegatewaynode 174

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andsentbacktoallsourcenodes,as (n)=Pl2L(Ul(Pk2Plrkl)) J(7) whereJisanumbertonormalizethefeedbacksignal.Forexample,wecansetJtothemaximumvalueofoverallutilitytillnandupdatethisvalueonthey.Therefore,thevalueof(n)lieswithin[0,1].Ulisthenon-convexutilityfunctionforthel-thuser.Notethattheutilityfunctionsofallusersareassumedtobetrulyacquiredbythegatewaynode[ 168 ].Inpractice,thevalueof(n)canbecirculatedefcientlybyestablishedmulticastalgorithmssuchas[ 169 ].Basedonthefeedback,thelearningautomatateamadjuststheweightingvectorinadecentralizedfashion.Inaddition,notethatBisthepredenedconvergencethreshold,e.g.,B=0.999,whichprovidesatradeoffbetweentheperformanceofthealgorithmanditsconvergencespeed. Beforeanalyzingthesteadystatebehavioroftheproposedalgorithm,werstdiscussthefollowingconcepts. Denotethemaximumnetworkutilityoftheoriginaltrafcengineeringproblem,i.e.,P1,asO.Next,wedenethenaloutcomeoftheproposedalgorithmasO0.Wesaythatthealgorithmprovidesan-accuratesolution,ifforanyarbitrarilysmall>0,thereexistsaQ0suchthat jO)]TJ /F6 11.955 Tf 11.95 0 Td[(O0j<8Q>Q0(7) Apotentialgame[ 67 ]isdenedasagamewherethereexistsapotentialfunctionVsuchthat V(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))]TJ /F6 11.955 Tf 11.96 0 Td[(V(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(l)=Ul(a0,a)]TJ /F9 7.97 Tf 6.58 0 Td[(l))]TJ /F6 11.955 Tf 11.95 0 Td[(Ul(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(l)8l,a0,a00(7) whereUlistheutilityfunctionforplayerlanda0,a00aretwoarbitrarystrategiesinitsstrategyspace.Thenotationofa)]TJ /F9 7.97 Tf 6.59 0 Td[(ldenotesthevectorofchoicesmadebyallplayersotherthanl. 175

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Aweightedpotentialgame[ 67 ]isdenedasagamewherethereexistsapotentialfunctionVsuchthat (V(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))]TJ /F6 11.955 Tf 11.96 0 Td[(V(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))hi=Ul(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))]TJ /F6 11.955 Tf 11.96 0 Td[(Ul(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l) (7) foralll,a0,a00wherehi>0. Accordingtothedenitions,itisapparentthattheformulatedidenticalinterestgameisaspecialcaseofweightedpotentialgames.Inthefollowingtheorem,wewillprovidetheconvergencebehaviorofamoregeneralsettingforweightedpotentialgames,andhencetheresultappliestoourspecicscenarionaturally. Theorem7.2. ForanN-personweightedpotentialgamewhereeachpersonrepresentsateamoflearningautomata,theproposedalgorithmcanconvergetotheglobaloptimumsolutionasymptotically,whichisan-accuratesolutiontotheoriginalstochastictrafcengineeringproblem,forsufcientsmallvalueofand. Proof. Theprooffollowssimilarlinesasin[ 167 ].However,weextendtheresulttoamoregeneralsettingwheretheunderlyingN-personstochasticgameisaweightedpotentialgame.Therefore,theproofin[ 167 ]canbeviewedasaspecialcase.DeneVasthepotentialfunctionofthegame.Notethattheselectedrateisdeterminedbytheprobabilityvectorwhichisgenerateduniquelybytheweightingvector.Therefore,wecanviewtheweightingvectorasthevariableinthiscaseandtheobjectivefunctionisgivenbyz=E(Vjw).Intheupdatingprocedureof( 7 ),signalisreplacedbyV.Werstverifythat @z @wql,k=@E(Vjw) @wql,k=@Pqpql,kE(Vjw,rql,k) @wql,k=Xqewql,k PQq=0ewql,k(1)]TJ /F6 11.955 Tf 28.6 8.09 Td[(ewql,k PQq=0ewql,k)E(Vjw(n),rql,k). 176

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NotethatE(V(1)]TJ /F6 11.955 Tf 28.59 8.09 Td[(ewql,k PQq=0ewql,k)jw(n))=Xqpql,k(1)]TJ /F6 11.955 Tf 28.59 8.09 Td[(ewql,k PQq=0ewql,k)E(Vjw(n),rql,k)=Xqewql,k PQq=0ewql,k(1)]TJ /F6 11.955 Tf 28.6 8.08 Td[(ewql,k PQq=0ewql,k)E(Vjw(n),rql,k)=@z @wql,k. Next,itisstraightforwardtoverifythatthestandardconditionsin[ 170 ](Chap.6,Theorem7)aresatised.Weomitthevericationssincetheyarethesimilarproceduresasin[ 167 ].Thus,weconcludethattheabovedynamicweaklyconvergestothefollowingSDE[ 170 171 ] dw=rz+dW(7) forasufcientlysmall!0whereisthestandarddeviationofthei.i.d.randomvariables&ql,kandWisastandardWienerProcess.NotethattheSDE( 7 )fallsintothecategoryofLangevinequation[ 172 ]whichiswell-knownthattheprobabilitymeasureconcentratesontheglobalmaximumsolutionofzforasufcientlysmall[ 167 172 ].Therefore,weconcludethatintheweightedpotentialgamescenario,theproposedalgorithmwillconvergetotheglobaloptimumoftheobjectivefunction,forthequantizeddataratesetting.Theassociationofthe-accuratesolutiontotheoriginalstochastictrafcengineeringproblemofP1followstheresultof[ 173 ]. NotethatTheorem2establishestheconvergencebehavioroftheaforementionedalgorithmwithnoadditionalrequirementfortheproblemstructure.Incontrast,weproposeastochasticprimal-dualapproachinSection 7.4 ,whichrequirestheunderlyingproblemtobeastochasticconvexprogramming.Thestochasticprimal-dualapproachcannotbeappliedefcientlyotherwise.However,theaforementionedlearning-basedalgorithmissuitableforalmosteveryaspectsuchasstochastic 177

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paths,weselectthefollowingavailablepathsforedgerouters,assummarizedinTable 7-1 Figure7-2. Exampleofcognitivewirelessmeshnetwork Table7-1. Availablepathsforedgerouters A P1A: fA!X!GWg P2A: fA!X!Y!GWg P3A: fA!X!Y!Z!GWg B P1B: fB!X!GWg P2B: fB!Y!GWg P3B: fB!Y!X!GWg P4B: fB!Y!Z!GWg P5B: fB!Z!GWg C P1C: fC!Z!GWg P2C: fC!Z!Y!GWg P3C: fC!Z!Y!X!GWg Thereareveprimaryusersinthearea,denotedby1,2,3,4and5whereeachonehasaprimarybandof10MHz.ThecommonISMbandisassumedtobe10MHz.Thereturnprobabilityoftheprimaryusersisgivenas$=[0.2,0.3,0.4,0.3,0.3].Thetransmittingpowerofeachnodeisxedas100mWandthenoisepowerisassumedtobe3mW.Weconsideramodelwherethereceivedpowerisinverselyproportionaltothesquareofthedistance.Notethatthetransmittingpowerisuniformlyspreadonallavailablebands.Inaddition,weexplicitlyspecifytheaffectingprimaryusersforaparticularnode.Weusefi,j,k,gtorepresentthataparticularnodeisaffectedbyprimaryuseri,j,k,.Forexample,nodeXislabeledwithf1,2gwhichindicatesthat 179 PAGE 180 thetransmissionofnodeXwilldevastatethetransmissionsofprimaryuser1and2ifthecorrespondingprimarybandisutilized.Notethatthecentralnode,namely,Y,ismostseverelyaffectedbyallprimaryusers.Intuitively,toachieveanexpectedoptimumsolution,thestochastictrafcengineeringalgorithmsareinclinedtosteerthetrafcawayfromY.Wewilldemonstratethisdetoureffectnext. Werstconsiderthecognitivewirelessmeshnetworkwithconvexity,e.g.,Ul(xl)=logxltoachieveaproportionalfairnessamongtheows[ 161 ].Thelinkcostisassumedtobeintheformoflse(fe,cse)=1 cse)]TJ /F9 7.97 Tf 6.58 0 Td[(fe,whichreectsthedelayexperiencedforaunitowonlinkeundertheM=M=1assumption[ 76 ].Notethatiffece,thecostis+1.Wesetthetrafcdemandofalledgeroutersasdl=30Mbpswhilethecostbudgetisbl=5.Thestepsizesarechosenas=1=nwherenisthecurrentiterationstep.Figure 7-3A illustratesthetrajectoriesoftheratevariablesandFigure 7-3B showstheconvergenceofthenetworkoverallutilityaswellastheindividualutilityfunctions4.Weobservethatwhiletheratevariablesconvergeastheiterationsgo,theoverallobjective,i.e.,thesumoftheindividualutilities,approachestotheglobaloptimumindicatedbythedashedline,whichisattainedbycalculatingthesteadystatedistributionfollowingthereturnprobability$. Inaddition,Table 7-2 providestherateoneachpathafterconvergenceforasamplerunofthealgorithm.Forcomparison,weprovidetheconvergencerateswhennodeYisswitchedfromthemostaffectednodetotheleastaffectednode,i.e.,nodeYcanutilizeallveprimarybandsallthetime,inTable 7-3 .FromTable 7-2 and 7-3 ,itisinterestingtonotethat,intherstscenario,eachuserallocatesarelativelysmallamountofowonthepathswhichtraversethroughnodeY.Onthecontrary,whennodeYislessaffected,alltheowsallocatenoticeablylargerdataratesonpathsthattraversethrough 4NotethatFigure 7-3B alsoreectstheevolutionofthethroughputofeachedgerouterlogarithmically. 180

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Table7-2. ConvergencerateswhenYisaffectedbyallveprimaryusers A P1A: 15.3035 P2A: 3.2458 P3A: 2.7362 B P1B: 7.3725 P2B: 1.9584 P3B: 2.1752 P4B: 2.6458 P5B: 5.3489 C P1C: 17.3051 P2C: 2.3792 P3C: 2.5675 Table7-3. ConvergencerateswhenYisnotaffectedbyanyoftheprimaryusers A P1A: 9.6413 P2A: 11.1335 P3A: 9.1861 B P1B: 5.0101 P2B: 6.2625 P3B: 4.9711 P4B: 4.9921 P5B: 8.7878 C P1C: 15.3118 P2C: 9.1042 P3C: 5.6121 Wenextconsideracognitivewirelessmeshnetworkwithnon-convexutilityfunctions.Specically,weconsidertheutilityfunctionas Ul(xl)=8>><>>:1,ifxl2Mbps0,ifxl<2Mbps.(7) whileothersettingsarethesameasinthepreviousscenario.Additionally,weutilizediminishingvaluesofandas=1=nand=1=n,wherenistheiterationstep5.Withoutlossofgenerality,wesetL=100andthequantizationlevelQ=20.The 5Byutilizingdiminishingparameters,atradeoffbetweentheperformanceandtheconvergencespeedcanbeachievedbytuningthedecreasingspeed[ 78 ]. 182

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