The Research on Extending The Lifetime of Wireless Sensor Networks

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Title:
The Research on Extending The Lifetime of Wireless Sensor Networks
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english
Creator:
Yun, Youngsang
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University of Florida
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Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
Xia, Ye
Committee Members:
Dobra, Alin
Liu, Chien-Lian
Chen, Shigang
Smith, Jonathan

Subjects

Subjects / Keywords:
lifetime, linear, wireless
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
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Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
We study various problems efficient energy management for wireless sensor networks. First, we research the energy efficient deployment of wireless sensor nodes so that energy consumption rates of all nodes are equal during the lifetime of the wireless sensor network. If the sensors are deployed uniformly across the network, they experience different traffic intensities and energy depletion rates depending on their locations. Usually, the sensors near the sink tend to deplete their energy sooner; when enough of them exhaust their energy, they leave holes in the network, causing the remaining nodes to be disconnected from the sink. One of the solutions to this energy-hole problem is to deploy the sensors non-uniformly. Moreover, we describe a method for deciding the sensor deployment densities so as to equalize the energy consumption rates of all nodes. The method is general and can be applied to other objectives and constraints. Second, we propose a framework to maximize the lifetime of the wireless sensor networks by using a mobile sink when the underlying applications tolerate delayed information delivery to the sink. Within a prescribed delay tolerance level, each node does not need to send the data immediately as it becomes available. Instead, the node can store the data temporarily and transmit it when the mobile sink is at the most favorable location for achieving the longest WSN lifetime. We call the proposed framework as Delay-Tolerant Mobile Sink Wireless Sensor Network. To find the best solution within the proposed framework, we formulate optimization problems that maximize the lifetime of the WSN subject to the delay bound constraints, node energy constraints, and flow conservation constraints. We conduct extensive computational experiments on the optimization problems and find that the lifetime can be increased significantly as compared to not only the stationary sink model but also more traditional mobile sink models. We also show that the delay tolerance level does not affect the maximum lifetime of the WSN. Third, we propose an adaptive and potentially decentralized algorithm for the DT-MSM. The distributed routing algorithms are very important in developing a practical routing protocol. Distributed algorithms are generally free from the network scalability issues in several reasons. They do not need to have knowledge about the whole network configurations and they also do not require the central node to compute the routes for all nodes in the network. Lagrange multiplier method solves dual of the primal problem. Dual problem sometimes has a nice structure with which we can decompose the dual problem into several sub-problems. We use a subgradient projection method to solve the dual problem and. A sensor node in our method keeps virtual queue which is a scalar product of the Lagrange multiplier and it is used in solving sub-problems. We propose (a possibly distributed implementable) decentralized algorithms for solving sub-problems. Moreover, we analytically show the our algorithm finds a solution arbitrarily close to the optimal solution of the primal problem. It is verified through the numerical experiments.
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In the series University of Florida Digital Collections.
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Includes vita.
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Description based on online resource; title from PDF title page.
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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 Youngsang Yun.
Thesis:
Thesis (Ph.D.)--University of Florida, 2010.
Local:
Adviser: Xia, Ye.

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2.4.6 Effect Of Node Deployment Strategies

In this section, we consider the two types of node deployment strategies. One

is a uniformly distributed node deployment (or uniform distribution) and the other is a

grid-based node deployment (or grid based distribution) When a uniformly distributed

node deployment is used, the coordinate of the node / ,say (xi, yi), is determined by

uniform distribution. In other words, we suppose that x, and yi, where / = 1... n, are

identically independent random variable with uniform distribution in the interval (0, 50).

Note that our sensor field is fixed to 50x50 through this section. For the grid-based node

deployment, we are given the number of grids as a parameter, denoted by A. Then the

average number of nodes in each grid must be p = n/A. For the simplicity, we assume

p > 1. To satisfy this condition, in each grid, we need to determine the total number

of nodes and this number follows uniform distribution in [1..2p 1]. Since, there exit

average p nodes in the grid and we have A = n/p grids in the system, on average

we have n nodes in the system. This is the key to the grid-based node deployment

simulation.

Figure 2-7 shows the performances of the uniform node deployment method

and grid-based node deployment method in the both single and multiple sink system.

We observe ,in the perspective of the performance, there is no significant difference

between both deployment strategies, as the number of the nodes increases. In multiple

sinks situation, we also observe the same phenomenon as the single sinks setting.

However, the lifetime in the multiple sinks system is far better than that of the single

sink system. Finally, we conclude that the lifetime has nothing to do with the node

deployment strategy, especially for the dense sensor network.

2.4.7 Effect Of Communication Ranges

In this section, we focus on how different communication ranges affect the

performance of the sensor networks. In general, communication range determines

the neighbor set N, for the node i. If we draw a circle at node i with the radius that is








optimality condition is

KEr fi(Z) > 1, Z < Z*
(5-52)
KiEA fil(z) < 1, z > z*.
Also note that fi(z) changes only when one of the ff(z) changes. From Algorithm 5-2,
we see that, for each i, fi(z) changes only when we select the most profitable item in
the list of remaining items. Suppose item (i,, I) is selected. The new fit(z) is given by

ff(z) = (7(/) ))/e,'). Furthermore, the next time when ffi(z) changes again is when z
is incremented by Me,~)/E,.
To summarize, the procedure for searching z* is to keep track of the sequence of
points where fi(z) changes, which requires keeping track of the sequence of points
where ff/(z) changes, for each i. Consider a fixed i. Suppose (7') r'))/e,') is sorted
in decreasing order and suppose any item (i, I,j) with (r') <)) < 0 is discarded.
Starting with zo = 0, we can generate a sequence zk = Zk-1 + Me ')/E, iteratively, where
(i,j, ) used in the update to get zk is the kth item in the list. Then, ff(z) can change
only at each of the points zk. Algorithm 5-3 describes an implementation of the above
idea, as well as the solution to the subproblem 52. For each i, the array Pi[] records the
sequence of zk and f1(zk).

Algorithm 5-3 Solution for 52
for each i e Jf do
sort (i, I,j) in decreasing order of (7i) "i))/e )
discard any item (i, I,j) if ((') ~)) < 0
k = 0; zk = 0
for each of (i, I,j) in the sorted list do
zk zk + Me/) / E
Pi[k] #=z (Zk 7) Y
k k 1
end for
end for
find z* which satisfies (5-52) by searching (Pi),e
each node i applies Algorithm 5-2 with z*


108









sensor field can have an arbitrary two-dimensional shape rather than a disk and
we allow data transmission from any node to any other node.

2. The nodes in [43] generate data at the same constant rate. However, the nodes in
our models may have different event generation rates.

3. Our energy consumption model is more general: The required transmission power
of a node is a function of the transmission distance to the receiver.

The resulting problem of determining the node densities is quite different and more

complicated in our case. Furthermore, we can achieve an equal energy dissipation rate

for all nodes in the entire network.

In [30], it is assumed that the sensor nodes are deployed uniformly. The sensor field

is also decomposed into concentric rings around the sink. The assumption on routing

is that each node can only transmit data to a node in the inner adjacent ring. The main

question addressed by [30] is how to decide the widths of the rings so that all nodes in

the network exhaust their energy at the same time.

In [32], the authors provided a formal description of the problem that traffic tends to

be concentrated at the nodes close to the sink when the shortest path routing is used.

They suggested a heuristic algorithm that finds some "curved" paths to the sink and

showed that the traffic load is more balanced.

In [6, 42], density control is used as one of the means to guarantee the coverage

requirement of the sensor network rather than to provide a balanced energy dissipation

rate of the nodes. In [21], the authors studied the impact of carefully controlled

deployment of the sensor nodes and the sink on the data capacity, which is defined

as the total amount of data that can reach the sink. They proposed and analyzed several

approaches to increase the data capacity, and showed that non-uniform deployment can

outperform uniform deployment.

Many authors formulate the problem of maximizing the network lifetime as

optimization problems [12], [26], [35], [45]. In [35] and [45], the authors proposed

approximation algorithms to solve the multicommodity flow problem induced by the









problem of sensor network lifetime maximization. The authors of [17] calculated upper

bounds on the lifetime of the networks that have regular topologies, such as a regular

linear array and a regular two-dimensional circular network.

In [19], the authors proposed a dynamic node-clustering scheme known as LEACH,

where each sensor node may operate as a cluster head depending on its remaining

energy and the cluster heads change during the operation of the network. They

compared the lifetime resulting from LEACH with the MTE (minimum transmission

energy) routing in which each node uses the path that consumes the least amount of

total energy among all possible paths.

Some researchers define the lifetime of sensor networks differently from the time

until the first node dies [47], [22]. The authors of [47] introduced the a-lifetime, which is

the time until the remaining sensor nodes can still cover at least a portion of the entire

sensor field. In [22], the authors were interested in prolonging the network operation

time after the first node dies. They recursively maximize the n-th minimum lifetime of the

nodes after (n 1)-th minimum lifetime of the nodes has been maximized, for all n.

3.2 Models with Discrete Ring Structure

In this section, we show how to compute the node densities required to equalize the

energy consumption rates of all nodes in the network. We make some simplifications

on the network model to illustrate the basic ideas of the method and to make numerical

computation easier. The method can be applied to other objectives and network models

(see Section 3.4).

For this section, we assume the shape of the sensor field is a disk and the sink is

at the center of the disk (Figure 3-1). The disk is divided into concentric rings having

the same width and the final node density in each ring will be constant. We consider

the routing rule at the granularity of rings. That is, all nodes in a ring are subject to an

identical routing rule, which specifies the next-hop ring rather than the next-hop node.

The ring-based modeling approach here is typical (See [30], [43].). As a result, we












L
s. t. e 0 )x ) I 1 jeNI(i)

x i) x + y i(' i(') =0, V/ e -Vi c V
j AENI(j) jENA(i) (51
j:iLN ) jtN(i) (5-12)
I X) M = 0
S=1 j:seNIj)
yfO) = Dd,, y(L) = O, Vi E AN (5-13)

x/) > O, Vl e L, Vi e Ri, Vj e Ni(i) (5-14)

y() > O, Vi A,V/e {2,3,..., L-1} (5-15)

z > 0. (5-16)

Note that M is an upper bound of any traffic volume, a term that also includes
the buffered data. We will use the terms flow and volume interchangeably. The new
formulation has the interpretation that it minimizes the maximum energy consumption
among all nodes in a single round, normalized with respect to E,, (]L1 cjEN/(i) e(/)xN)/Ei)
while satisfying flow conservation.
Instead of tackling the problem (5-10) (5-16), we would like to consider the
following problem, where qualities in flow conservation are changed into less-than-or-equal
(<) inequalities. Later we will prove that these different formulations are, in fact,
equivalent.


min z


(5-10)









We can decompose the problem (5-45) (5-49) into the following two subproblems.
N L-1
S5: min X,(I1) i i)yi)
i=1 /=
s.t. O < y() < M, Vi e N, 1 < < L 1. (5-50)





52 : min z+ ( )_ (/ ) ) ( )
/-1 (ij)EA

s.t. O < x') < M, Vi e /, V e Vj e NI(i)
L
e ()x~()- zE, <0, Vi e
/ 1 jEN/(i)

z> 0. (5-51)

Note that we add the upper bound M to the flow variables x and y.

5.3.1 Algorithms for Subproblems

The solution for the subproblem S, is very obvious. If (,(' ) 0)) is negative, then

we assign the largest value (= M) to the variable yi). Otherwise, yi) should be 0. This

is shown in Algorithm 5-1.

Algorithm 5-1 Solution for S5
if (~'/+) T(/)) > 0 then
yP() 0
else
yi(l) M
end if


Algorithm 5-1 can be implemented in a distributed and local manner. The value

of y') can be decided locally in the sensor node i. Also, node i only needs to have the

knowledge of -j from its neighbor node set NI(i) for all / e C.


105









area (For a 50x50 sensor field, sink's coordinate is (25.0, 25.0)). However in the case

of multiple sinks, the locations of the sinks are randomly selected. For our simulation

to be more realistic, we need to know a lot of variables and constant values, and we

summarized them in the table 2-1. Note that we assume that energy required to run the

transmitter/receiver circuitry in the sensor node is negligible, thus we just set a to 0.

parameters values
communication range {3.5m 10.0m}
a 0
1.3 x 10-9J/b/m4
y {2,3,4}
Initial Energy 500J
Data Rate 500bps
Dimension of Sensor Fields {20 x 20, 30 x 30, 40 x 40, 50 x 50, 60 x 60, 70 x 70}
The number of sinks {1, 2, 3,...,60, 70, 80}
The number of nodes {200, 225, 250, 275,..., 775, 800}
Deployment methods { uniform,grid-based }
Table 2-1. System parameters used in the simulation


We try to simulate the sensor network in various environments. For example, we

want to figure out how does the lifetime change according to the different energy model,

how can the number of the sink affect the lifetime of the sensor network and is there any

noticeable relationship between the lifetime and the density of the sensor nodes, etc.

Figure 2-1 shows an example of graphs used in the simulation. In this graph,

the location of node is determined in an uniformly distributed manner. If the distance

between the node i and the node j is less than communication range, node i and node

j have a bidirectional link. In our simulation, if the whole graph does not form a single

strongly connected component, we just discard it and keep generating the new graph

until we get a strongly connected one.

Figure 2-2 shows another graph which is generated by the grid-based node

deployment. With this method, sensor field is divided into the fixed number of equal

size grids and each grid contains at least one sensor in it. The maximum number of

sensors in the grid is limited and determined by the number of nodes to be deployed









REFERENCES


[1] Ahuja, Ravindra K., Magnanti, Thomas L., and Orlin, James B. Network Flows:
Theory, Algorithms, and Applications. Prentice Hall, 1993.

[2] Akyildiz, IF, Su, W, Sankarasubramaniam, Y, and Cayirci, E. "Wireless sensor
networks: a survey." Computer networks (2002).

[3] Awerbuch, Baruch and Leighton, Tom. "A Simple Local-Control Approximation
Algorithm for Multicommodity Flow." IEEE Symposium on Theory of Computing.
1993, 459-468.

[4] "Improved Approximation Algorithms for the Multi-Commodity Flow Problem
and Local Competitive Routing in Dynamic Networks." ACM Symposium on Theory
of Computing. 1994, 487-496.

[5] Baek, SeungJun and de Veciana, Gustavo. "Spatial Energy Balancing in
Large-Scale Wireless Multihop Networks." INFOCOM 2005. 2005.

[6] Balister, Paul, Bollabas, Bela, Sarkar, Amites, and Kumar, Santosh. "Reliable
Density Estimates for Coverage and Connectivity in Thin Strips of Finite Length."
MobiHoc '07. 2007, 75-86.

[7] Basagni, Stefano, Carosi, Alessio, Melachrinoudis, Emanuel, Petrioli, Chiara, and
Wang, Z. Maria. "A New MILP Formulation and Distributed Protocols for Wireless
Sensor Networks Lifetime Maximization." IEEE International Conference on
Communications 2006. 2006, 3517-3524.

[8] "Controlled sink mobility for prolonging wireless sensor networks lifetime."
Wireless Networks 14 (2007).6: 831-858.

[9] Bazaraa, Mokhtar S., Jarvis, John J., and Sherali, Hanif D. Linear Programming
and Network Flows. Wiley-lnterscience, 2004.

[10] Boyd, Stephen and Vandenberghe, Lieven. Convex Optimization. Cambridge
University Press, 2004.

[11] Chang, J.H. and Tassiulas, L. "Routing for maximum system lifetime in wireless
ad hoc networks." 37th Annual Allertion Conf. Communication, Control, and
Computing. Monticello, IL, 1999.

[12] "Maximum Lifetime Routing in Wireless Sensor Networks." IEEE/ACM
Transactions of Networking 12 (2004): 609-619.

[13] Fleischer, Lisa K. "Approximating Fractional Multicommodity Flow Independent of
the Number of Commodities." Siam Journal of Discrete Mathematics 13 (2000):
505-520.


126









3.2.2 Deriving the Node Densities of the Rings

Our goal is to make the energy consumption rate of every node equal by controlling

the node densities in the sensor field. Since we assume that each ring has a constant

node density, we need to ask what the node density should be in each ring so that a

typical node in one ring consumes the same amount of energy per unit of time as a

typical node in any other ring.

Suppose the nodes are uniformly distributed over ring j with a density pj, 1 < j < n.

Consider an arbitrary node in ring j. Since a typical node in the sensor network both

generates traffic as a data source and relays traffic for other nodes, the node's total data

transmission rate is the sum of the rate of the locally-generated traffic and the rate of

the traffic it relays. We assume flow conservation at every sensor node: A node cannot
buffer an infinite amount of data, and, after the traffic is generated, there is no further

in-network processing that may reduce or increase the traffic volume at the node. The

data transmission rate at a node in ring j can be expressed as

G, = Sj + C, (3-3)

where Sj is the rate of the locally-generated data, and C, is the rate of the traffic to be

relayed by the node.

Let us assume that certain amount of data rate is needed to monitor a unit of area

and this rate is a constant value of K throughout the sensor field. This is the inherent

data rate needed for reporting events or conditions about a given area. There are at

least two possibilities regarding how this inherent data rate affects the actual traffic rate

generated by each nearby node. In the first, one can assume that the system has local

coordination among the nearby sensors that reduces the amount of traffic generated. In

the best case, every nearby node generates the minimum amount of traffic sufficient to

cover the area. That is, for ring j, the rate of the locally-generated traffic at a typical node









where () > 0. Now, after summing constraint (5-19) over i e f., I e C, we get

L N
( E i/) x i' + y,'-1)- yi)= 0v (5-25)
/ 1 i1 j:iEN/(i) jENI(i) (u,v)EK

After canceling common terms in LHS of (5-25), we have
L N N L
(- Y Xs M (5-26)
/I1 j:sNI(j) i=1 i=1 I=1 j:sENIj)

Recall that y() = d, and yL) = 0 for i e AV. However, since E(u,v)K -v) > 0, the RHS of

(5-25) is negative. Hence we have the following inequality.
L

/=1 j:sENIU)
L
Y Y 41) > M (5-27)
/=1 j:sENIj)

This is a contradiction to the contradiction the second part of constraint (5-19).

Therefore, for (x, y) to be feasible, all constraints in the first part of (5-19) must be

binding. And as a consequence, ,1 j:sENI(O)) = M.

Theorem 5.1. The problems as formulated in (5-10) (5-16) and (5-17) (5-23) are

equivalent.

Proof. This is a direct consequence from the Lemma 1. D

Next, we show that the removal of the second constraint in (5-19) does not

affect the optimal objective value. The modified formulation is the final one which we









higher theoretical and computational complexity. For now, we ignore this inaccuracy for

the benefit of simpler numerical computation.

3.2.3.3 Simplified uniform node selection

In this routing model, the probability that a node in ring j takes a node in ring /

as the next-hop node, Fj(i), is proportional to the number of nodes in ring /, where

(j /) < i < j. Thus, the probability can be written as follows.

pi(2i 1)
Fj(i) = ,1) (j- /)+ < i < j. (3-21)
SPk(2k 1)
k=max(j-l,1)

Since Fj(i) depends on (pj), successive substitution of the form in (3-14) are needed to

find the solution.

The following reasoning shows why this model can be viewed as a simplification of

the Uniform Node Selection scheme. Suppose the range of each node is Iw. However,

the node only selects a next-hop node in its nearby / rings, where / < /I. In this

case, the ratio of the area of Ri n Q to that of Rk n Q can be well approximated by

(2i 1)/(2k 1), for (j /) < i, k < j. (See the first two rings next to node X in Figure

3-2.)

3.3 Experimental Results

In this section, we use experimental results to show how well our method for

computing the node densities works. The radius of the sensor field is 50 and the total

number of rings is 20 (= n). The procedure of the experiments is as follows. First, for the

given routing strategy and experimental parameters, we calculate the node densities of

the rings using the equations introduced in Section 3.2. Then, in our simulation setup,

we randomly deploy the sensor nodes into the sensor field according to the calculated

densities and have each node select its next-hop neighbor according to the given routing

scheme. The density of the outmost ring pn can be tuned to control the total number of

nodes in the sensor field. In the simulation run, we measure the energy consumption









After taking the limit in T, we get (5-71).

Next, from (5-73), we have
T-1 2 -1
2e Q(k) < BT + 2() +V(q(0))- V(q(T)) z(k)
k=O k=O

< BT 2+z()+ V(q(O)). (5-75)


The above inequality is the same as

1 )B 1 V(q(O))
T Q(k)2 (c) (5-76)
T 2e 6e 2Te
k=0

After taking the limit in T, we get (5-72). O

Let (x*, y*, z*) be an optimal solution to the original problem in (5-17) (5-23). Note

that z* is also the optimal objective value. Let eo be the largest c for which the perturbed

problem (5-59) (5-62) is feasible.

Theorem 5.5. There exists a positive constant B such that for any positive 6, the

following holds.
1 B
lim sup ) z*, (5-77)
T k=o
T-1
1 B I
lim sup Q(k) < + -2(o) (5-78)
rTo T- 2c, 6C0
k-0
Proof. In (5-71), let c 0. Since by Theorem 5.3, 2(c) z* as c 0, we get

(5-77). D

By Theorem 5.5, we can take 6 small enough so that the long-time average of z(k)

is arbitrarily close to the optimum z*. But, this is at the expense of an increase in the

provable queue bound.

Now we will prove that the long-time average of x(k) and y(k) eventually satisfies

the constraint (5-19). Let ')(k) and y')(k) be (I/k) k- ')(k) and (1/k) E ) (k)

respectively.


116









of transmission power at each sensor node, so that resulting topology satisfies the

application specific constraints such as the degree of the connectivity. This is the

definition of the topology control. The proper adjustment of the transmission power also

has a good side-effect. It reduces the possibility of interference at the receiver and

leads to the increase of the throughput. However, due to immense number of sensor

nodes, topology control should be distributed. Moreover it also should be dynamic

because of frequent node failures.

In WSN, a sensor node might be in one of three states: active, idle, and sleep. A

sensor node consumes significant amount of energy even it is in an idle state. In order

to save the energy, a sensor node sometimes goes into the sleep state. Thus, how to

schedule sleep-wake sequence is also another research issues. Since it is not possible

to synchronize the sleep-wake schedule for every sensor node, delay is essentially

increased. Good schedule should minimize incurred delay as possible.

In the data centric WSN, without data aggregation, the sink might acquire much

redundant data, because lots of sensor nodes would report the same event if they are

located closely. There are lots of ways to aggregate data. One of these is clustering the

sensor nodes. In each cluster, the cluster head aggregates raw data gathered by the

sensor nodes in its own cluster. Data aggregation is very useful in reducing the energy

consumption accompanied by communication as well.

Widely used routing protocol in the wired network may not be a good choice in the

perspective of efficient energy management. To ensure the longevity of each sensor

nodes, the routing path going through the node having insufficient energy should be

avoided. Lots of routing protocols taking the current energy level of the sensor node

into account have already been proposed. This kind of routing protocol is called Power-

aware routing protocol. The minimum total transmission energy paths and maximum

residual energy paths are such examples. With the help of a mature optimization theory,









In the Table 5-1, network size is represented by the number of nodes N and the

number of sink locations L. The measured data is the ratio of the computing time of

CPLEX over the computing time of Algorithm 3. As shown in the table, Algorithm 3

shows substantial enhancement in computing the solution for the sub-problem 52.

Our method requires an information exchange in the beginning of the time slots.

This information might be conveyed in the form of control message, which is not relevant

to the purpose of wireless sensor network. This overhead might not occur in the

system which runs centralized problem solver after the system acquires the topology

information of the sensor nodes at the very beginning of the operation of the system.

In our algorithm, each node should broadcast a single message containing the solution

of Algorithm 2 and exchanges the local information about the current length of the

virtual queues to its neighbor. Thus, in overall, N broadcast messages and maximum N2

unicast or point-to-point messages are needed as overhead messages. The discussion

about efficient way of broadcasting message is beyond the scope of this research. Note

that, in general, the cost of broadcast is much higher than the cost of unicast and the

message complexity of broadcast dominates the overall message complexity.


122









Theorem 5.6. For all i e ff and I c L, the following condition is satisfied.

im( ()(T) T ( )(T) +'-1)( T) () ( T)) < 0 (5-79)
j:iENI(j) jENI(i)

Proof. From the (5-57) of main algorithm, for each i E iV and I e L, we have the

following inequality.

q) (k + 1) > q')(k) g(i, /, k)

g(i, I, k) < q}')(k + 1) q}')(k) (5-80)


Since Q(k) = q(')(k) is bounded above by the second part of the Theorem

5.5 and each q(') is nonnegative, we know that q(') is also bounded above. Thus,

suppose q(i)(k) < Mq for all k.

Summing (5-80) over k = 0, 1..., T 1, we have
T-1
S-g(i, k) < q( T) q')(0) < M, (5-81)
k=0

Dividing the above inequality by T, we get
T-1
1 -g(i,, k) (5-82)
k=0

Letting T oc, we have
S T-1
lim -g(i, k) < 0 (5-83)
T--oo T
k=-0

(5-83) completes the proof. O

In fact, Theorem 5.6 states that the long-time average of (x, y) eventually satisfy the

constraints (5-19).

5.5 Experimental Results

In this section, we present the results of the numerical experiments to prove the

validity of our algorithm. First, we show that our algorithm achieves the optimal objective


117









BIOGRAPHICAL SKETCH

YoungSang Yun was born in ChangHeung, Republic of Korea, in 1970. He received

his BS and MS degrees in computer engineering at Pohang University of Science and

Technology in Korea in 1992 and 1994, respectively. He worked for the LG Electronics,

Inc., Korea from 1994 to 2003. He was involved in development of system software for

telecommunication equipment, such as ATM switches. He also managed the IP routing

protocol software development team. Since 2003, he has been conducting research with

Dr. Ye Xia in the Department of Computer and Information Science and Engineering

at the University of Florida. His research interests are wireless sensor networking and

mathematical optimization.


130




























2010 YoungSang Yun









by Ri, where Ri c Vf, and we call it the coverage of sink location /. The nodes from
outside Ri do not even attempt to communicate. The motivation is that, since the nodes

in Ri are close to location /, any coordination can be accomplished faster and with less

overhead. For instance, the announcement that the sink is at location / only needs to be
made to the nodes in R,. We assume R is given for each I. In a degenerate case, each

RI may be the same as iVf.

Thus, for each location /, there is a graph G' = (AV u {/}, A'), where A' = {(i,j) e

Ali e RI,j e Ri U {/}}. When the sink is at location /, the (downstream) neighbor set of
node / is denoted by NI(i) = {jl(i,j) e A'}.
We create an expanded graph from the graphs G', I e L. As we make clear shortly,
the lifetime maximization problem will be a network flow problem on the expanded

graph. In Figure 5-1, we show an example of the expanded graph. Some details about
its construction are as follows.

1. Start each column with G', for all / e L.

2. Relabel node / in G' as i().

3. Add a vertex s, which represents the sink.

4. For each /, replace the edge (i(,), /) with (i(l), s) and remove node I from G'.

5. For each i(), I = 1,..., L 1, add an edge (i(), i('+1)).

6. Set the supply at node i(l) to be Dd, and the demand at node s to be D Cjin di.
The cost of each vertical edge (of the form (i()0,j(0)) is assigned as follows:

c(i,j), if ij c Rj l

e,) = c(i, ), if i E RI,j = I (5-1)

oo, otherwise.

The cost of each horizontal edge (of the form of (i(1), i('+ ))) is set to be 0, because the
head and tail of this type of edge are the same physical node and real communication












its neighbor. On the other hand, node may have shorter length paths to the sink when

the number of sinks increases.

Figure 2-5C and figure 2-5D show the performance under grid-based node

deployment scheme.

Although the performance in the grid-based deployment is slightly better than the

performance in the uniform deployment in all the simulation, we can not find a significant

difference in term of the performance between both deployment strategies as the

number of the sink increases.


2.4.5 Effect Of Node Densities And The


Comparison of single sink and multiple sinks (uniform distribution)
1 4e06 -single sink ------ --
TE-single sink '
LPMi ng sink
1 2e+06 TE multiple sinks
SMTE multiple sinks ----
LP multiple sinks --0-
le+06

S800000

600000

400000 i' l

200000 -, ,
0 .-- i- iI --I-------
200 300 400 500 600 700 800
Number of nodes
A Lifetime


Number Of Sinks


Com onof rati of r man ener to iitiaeery
or single sink and multiple sinks (uniform distribution)
1 8: --EDb -i -i Eli fn E a I
S- Z X s X \ f X X / \ x / x /

07 :8,
067
0 6
05 "
04
03 -

01 MTE- rnultple sinks' ,- -
SMTE multiple sinks - '-'' ': - ?
LP E multiple sinks --
200 300 400 500 600 700 800
Number of nodes
B Ratio


Figure 2-6. Performance in different number of nodes and different number of sinks



Figure 2-6 shows the performance of the single sink network and multiple-sink

networks (In the multiple sinks case, we set the number of the sink to 4) under various

number of nodes. Obviously, the system of multiple sinks outperforms single sink

system. In both cases, the lifetimes of MTE are very poor and the lifetime of LP is

slightly superior to the that of SMTE, but its difference is negligible.

The rate of increase in lifetime for the multiple sink case is greater than the rate for

the single sink system. This fact gives a hint that putting more sinks is beneficial to the

lifetime when large number of sensors are deployed. Probably, the study of the impact of

the multiple sinks system is a good research issue.









increase. Therefore, they proposed an optimization problem for choosing a mobility

strategy that minimizes the maximum traffic load of the nodes. However, they assumed

the shortest path routing, which, in general, does not produce the best lifetime.

The problem of finding the trajectory of the mobile sink so as to optimize the lifetime

of the WSN is hard to solve due to its infinite search space when the locations for the

sink stops are not constrained. In [38], the authors studied how to find the optimal

sink stops and the schedule of visit to each of the stops. If the candidate locations for

the stops are unconstrained, this problem is also NP-hard. However, if the stops are

constrained to be selected from a finite set of known locations, the problem can be

easily formulated into linear programming. They proposed an approximation algorithm

to the unconstrained problem by properly dividing the whole sensor field into a finite

number of disjoint small areas, and then, converted the unconstrained problem into

a constrained problem. However, to obtain a good approximation ratio, the number of

small areas can potentially be very large, making the linear programming computation

time-consuming. Therefore, in this chapter, we restrict the set of potential sink stops to

be from a small number of given locations rather than from arbitrary locations.

The WSN model proposed in [31] is close to ours. The authors studied the

maximum lifetime problem of the WSN where the mobile sink can visit only small

number of locations. They showed that the lifetime can be further increased by

optimizing not only the schedule of sink visits but also routing of the traffic. However,

they did not consider applications where delayed information delivery is allowed.

4.2 Related Lifetime Maximization Problems

In this section, we discuss related lifetime maximization problems that have been

published in the literature. We will later compare their performance with our new

proposal.

First, we will describe the general assumptions about the WSN models. Let the set

of sensor nodes be denoted by iVf. For experimental convenience, we suppose they are









In (5-69), (2(e), x(e), y(e)) is an optimal solution of the c-perturbed problem defined in

(5-59) (5-62). Based on the earlier remark, y(e) is feasible to the optimization problem

in (5-55), and 2(e), x(e)) is feasible to (5-56). But, y(k) is a minimum to the optimization

problem in (5-55), and z(k), x(k)) is a minimum to (5-56). Hence, inequality (5-69)

follows.

After regrouping the terms in (5-69) and using (5-62), we have



A(k)+ z(k) IEc J
iEN
=B 2(e) 2e q) (k) (5-70)

-- B i-

In (5-70), the flow conservation constraint (5-61) is used. D

Define Q(k) = / 1 Cexv q() (k), which is the sum of the virtual queue sizes at

time slot k.

Theorem 5.4. There exists a positive constant B such that for any small positive e and 6,

the following holds.
T-1
1 B
lim sup z(k)< -+ 2(), (5-71)
ST 2
k=0
1 1
limsup Q(k) < 2(e) (5-72)
T -oo T 2e 6e
k 0
Proof. Summing the inequality in (5-66) for k = 0, 1, T 1, we have

2 T-1 T-1
V(q(T)) V(q())+ 2 z(k) < BT + T2() 2e Q(k) (5-73)
k=O k=O

After arranging the terms, we get
T-1 T-1
T-1 B T V(q(T)) 6V(q(0))
z(k) < 2(c)-- Q(k) -- +
T (-2 T 2T 2T
k=O k=o
JB 6V(q(0))
< + 2() + (5-74)
2 2T


115









that all nodes will exhaust their energy at the same time, and hence, energy holes will

not emerge.

Although the illustration of the method uses an example, the method itself is

intended to be general. It can be adapted for other lifetime-cost objectives and other

constraints, including the routing strategy. For instance, if the cost of deploying extra

sensors is not negligible, one can incorporate a cost function of the node densities as

part of the objective and compute the required deployment densities.

The method requires a model for the underlying routing strategy. We consider

several routing models, which are meant to capture the essence of the underlying

routing protocols. For tractability, the routing models are necessarily simple and may

not follow precisely the routing protocols. Some other issues are also overlooked in this

chapter, such as the algorithms and protocols for determining which nodes become

active or inactive in each region. These issues are either orthogonal to our work or left to

future studies.

Several researchers have studied the energy hole problem and the uneven traffic

distribution problem [19, 21, 23, 24, 30, 32, 43]. Among them, [43] is the most similar

to our work in its goal of obtaining a balanced energy dissipation rate everywhere

by non-uniform node deployment. The authors of [43] proved that an uneven energy

consumption rate is unavoidable if all nodes are homogeneous and are deployed

uniformly in the network. In their model, the sensor field is divided into several

concentric coronas or rings around the sink. They gave a heuristic routing scheme

that achieves an equal energy dissipation rate in all rings except the outmost one,

provided the number of nodes increases geometrically from the outmost ring inward.

However, their sensor field, energy consumption, and routing models are significantly

different from ours.

1. The nodes in [43] can send data only to the nodes in the neighboring ring.
However, even in our simpler model, nodes can send data to different inner
rings with different probabilities. We then make further generalization so that the









In both SSM and MSM, the sink collects data from each node i at the same rate at

which node i generates the data. However, in the DT-MSM, the data transmission rate at

node i during the collection time is no longer the same as the constant data generation

rate d,. When node / is not active (i.e., not covered by the current sink location), it

continues to gather data and should store the newly generated data. Hence, data

buffering is required by our framework. Within a cycle of D time units, the total stored

data at each node / is at most D d,.

For ease of presentation, we assume the sink visits all locations in L in the order of

1 2 ... -II 1 - The sink may stay at some location for zero time. With slight

abuse of terminology, we define the network lifetime T to be the number of cycles made

by the sink until the first node dies due to energy exhaustion. The actual lifetime is T D.

Once traffic is allowed to be buffered, there are different strategies on whose

traffic is buffered. Which strategy gets adopted in practice probably depends on the

application, other practical concerns, and the designer's preference. Since we do not

know these factors in advance, we next describe two strategies, or two variants of the

model: the sub-flow-based model and the queue-based model. The main purpose is to

illustrate that choices exist and they lead to different performance-complexity tradeoffs.

4.3.1 Sub-Flow-Based Model

In the sub-flow-based model, the nodes in the current coverage Ri are not allowed

to buffer the relayed traffic from other nodes; as soon as a node in Ri receives the data

from other nodes, it immediately forwards the data to its neighboring nodes. To model

this constraint at each node i, we need to differentiate the data generated by node /

itself and the data originally generated by other nodes but forwarded to node i.

Again, let x('0) be the rate assignment from node i to the node j, while the sink is

at /, for the traffic generated by node c (commodity c). Let x,) be the aggregated rate of









The following is the formulation for the lifetime maximization problem in sub-flow-based

DT-MSM.


Per-Commodity Sub-Flow-Based DT-MSM

max T


V/ e C; Vi, Vj e RI


(c,/)
s. t. x x ),
c6Ri


k:ieN (k)


jeN/(i)


Z ( X
\jeNI(i) k:ieN/(k)
ILI I
EZ C XC ()
=1 jN(i)

Wi ') = D d,,
1:iERJ

xU.O > 0, V/ ;V

w(') > 0, V/e L,Vi

z > 0, V/e

T> 0.


Ski


V/e L; Vi, Vc(4 i) e RI


V/ L; Vi e Ri


Sk:iL ) T< E,,
kieN/(k)


'i E A


i, Vc e RI; Vj e NI(i)

e RI


The above is a per-commodity-based formulation of the problem. Similar to

the case of the SSM problem (4-5) in Section 4.2.1, there is a simpler, equivalent,

aggregate-traffic formulation, using only the aggregate arc flow variables x /)

ceR, Xc f,). We can do away with constraint (4-33) in the resulting formulation.


Vi E A


(4-30)

(4-31)

(4-32)


(4-33)


(4-34)


(4-35)


(4-36)


(4-37)

(4-38)

(4-39)

(4-40)









* The two delay-tolerance formulations represent two strategies on what data to
buffer. The sub-flow-based formulation allows buffering of only self-generated
traffic; the queue-based formulation allows buffering of any traffic, which naturally
leads to the best lifetime performance among different strategies. The two models
can be considered as two "extreme cases", and various intermediate strategies
can be similarly formulated.

In the sub-flow-based formulation, the maximum required buffer size at node / is
Ddi. In the queue-based case, the maximum buffer size at a node may depend on
the total number of other nodes in the same coverage area, which can be much
larger.

The sub-flow-based formulation looks more similar to the standard multi-commodity
flow problem. It can be easier to find fast, specialized algorithms to solve this
problem.

4.3.3 Properties of Delay-Tolerant Mobile Sink Model

Both delay-tolerant models include the coverage of each sink location in the

formulation. This is motivated by practical concerns, in particular, how easy it is to

design practical protocols for coordinating the communication. When at a sink location,

it is far easier for the sink to coordinate with the nearby sensors and set up the data

collection process. Hence, a small radius of coverage is preferable from the protocol

complexity point of view. However, the radius of coverage can affect the network lifetime,

which we will explore next.

For illustration, consider the optimization problem for the sub-flow-based model.

Depending on the radius of coverage, we may obtain different instances of the

optimization problem. Thus, we can parameterize these instances according to the

radius of coverage. Let P(A/V, L, r) be the optimization problem when the radius of

coverage of the sink is r, the set of sensor nodes is AV, and the set of sink locations

is L. The value r must be large enough so that all sensor nodes can be covered by at

least one sink location and we denote this minimum radius of coverage for connectivity

by ro. Under the same configuration with Af and L, different r values only affect R, and

Ni(i). We will use the notations Ri(r) and NI(i, r) if it is necessary to specify the radius








(4-54) are identical. Hence, given the optimal solution ((x/')), (iv)), (2k), T) to problem
(4-41), we just constructed a feasible solution ((x'k)), (qc0), (2k), T) to problem (4-51)
with the same objective value T. Hence, T* > T. D

In the following theorem, we show that the maximum lifetime of the system is the
same for all values of D. Here, the maximum lifetime of the system is equal to the
product of D and the corresponding optimal objective value T*(D).
Theorem 3. Define P(D) as the lifetime optimization problem parameterized by the
value D, for some fixed network configuration. Let T*(D) and T*(D') be the optimal
objective values for the problem P(D) and P(D'), respectively. Then, T*(D) D =
T*(D') D'.

Proof. Consider the queue-based model.3 Let (x*(D), q*(D), z*(D), T*(D)) be the
optimal solution to the problem P(D), and let (x*(D'), q*(D'), z*(D'), T*(D')) be the
optimal solution to the problem P(D').
Letx = ()x*(D'), q = ()q(D'), z = z*(D'), T D (') T*(D). We want to
show that (x, q, z, T) satisfies the constraints (4-53)-(4-60). Since it is obvious that
the solution (x, q, z, T) satisfies the constraints (4-57), (4-58), (4-56), (4-59), and
(4-60), we focus here on constraints (4-53), (4-54), and (4-55) only. Since the optimal
solution (x*(D'), q*(D'), z*(D'), T*(D')) is feasible to the problem P(D'), it must satisfy
constraint (4-53). Next, let us plug ( )x, z, and (D )q into constraint (4-53) in the
places for x*(D'), z*(D'), and q*(D'), respectively. Then, we have

zi x ,(+ q(I-1) z x,(i) = ). (4-63)
k:iEN/(k) jENI(i)



3 Note that the proof can be adapted to the sub-flow-based model.









relay traffic. Hence, it is different from our multi-hop communication framework. It was

assumed that the mobile agents have plenty of energy. The movement of each mobile

agent is modeled as a random walk. It was shown that the queues in the mobile agents

and the sensor nodes are finite and the delay of the collected data is bounded. However,

the authors did not show the quantitative improvement of the network lifetime by using

mobile agents.

In [41], the authors formulated a linear programming problem of determining how

to move the mobile sink and how long to park the mobile sink at each stop along the

path of the sink so as to maximize the lifetime of the WSN. However, in their model, data

flows are not decision variables of the lifetime optimization problem. On the contrary,

in our formulations, not only the sink sojourn times at different sink stops but also the

routing scheme are decision variables. The analysis and experiments in [41] were

conducted under a simple structured network topology where the sensor nodes are

deployed in a grid-like pattern. In [7] and [8], the authors further extended the research

of [41]. The model proposed in [7] [8] includes the cost of moving the mobile sink (such

as nodal energy consumption for route establishment/release when the sink moves to

a new stop) and the sink mobility rate determined by the minimum sink sojourn time at

the sink stops. Furthermore, the model incorporates a hop-length limit when the sink

moves to next stop. This restricts the packet latency, which is related to the traveling

time of the sink between stops. The authors proposed an MILP (Mixed Integer Linear

Programming) problem formulation to obtain the optimal travel route of the sink and

the sojourn times at the sink stops for maximizing the lifetime of the system. They also

suggested a distributed heuristic algorithm to circumvent the complexity of the proposed

mathematical formulation.

The authors of [25] showed that the network lifetime can be extended significantly

if the mobile sink moves around the periphery of the WSN. They assumed that, if the

mobile sink can balance the traffic load of the nodes, the lifetime of the network can










16 16
14 14
12 SSM -- 12 SSM
S 10 MSM --..... X ... 10 MSM --..... -*
E DT-MSM ...... ..*** E DT-MSM .....-
-, 8 :-''".'".......... ............... ............. 8 ,....... ................ ................
.- 4-................................C..a
6 1 i 6
4[^ 4
2r 2
0 0
5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50
Transmission Range Tarnsmission Range
A Under minimum coverage B Under maximum coverage

Figure 4-5. Lifetimes versus transmission range, d: |IfVI = 200, I1 = 20, e = 2.0


chosen. We expect more performance gain if these candidate locations are carefully

selected.

We conduct similar experiments with the same configuration but minimum coverage.

The result is shown in Figure 4-4. Although the slope of lifetime increase of the DT-MSM

is lowered when compared to the maximum coverage case, the increase pattern is

similar. Although a larger set of sink locations increases the network lifetime, it can be

undesirable if the sink-traveling time cannot be ignored. The longer traveling time may

exceed the delay tolerance level D. Therefore, there is a tradeoff between the gain from

more sink locations and the delay or other system costs.

In Figure 4-5, we show the lifetimes of the three models under various values

for the transmission range. The transmission range determines whether a link exists

between a pair of nodes. Whether an existing link is useful or not depends on the radius

of coverage: A node cannot use a link to another node if the two nodes are not in the

common coverage area.

Both the MSM and the DT-MSM exhibit a sharp lifetime increase when the

transmission range is small but increasing. However, as the transmission range

becomes large, the lifetime increase comes to a stop for all three models. This is

because the energy cost increases with the transmission distance, and hence, in an












Lifetime of different size of sensor fields (uniform distribution)
1 2e+06 MT
*MTE ------
le+06 LP ----

800000

600000

400000

200000

0


20


30 40 50 60
Length of the side of sensor fields
A Lifetime


70


Ratio of different size of sensor fields (uniform distribution)

09 ------
08
08 ----------- X- - -
07
S06 ...
05
04 -----
S: : : TE ----M---
03 SMTE X
0 2 .. .LP -- ---
20 30 40 50 60 70
Length of the side of sensor fields
B Ratio


Figure 2-9. Performance of different dimension of sensor fields


Since the node density is fixed and the dimension of the sensor fields varies, the

number of nodes also varies according to the area of the sensor fields and the

constant node density. We assume all sensor fields shape square, and we just change

side of the sensor fields from 20 to 70. Therefore the areas of the sensor fields are

400,900,1600, and so on. The node density used in the simulation is set to 0.16. With

these parameters, we can find the number of nodes in different sized sensor fields:

64,144,256, and so on. The other parameters are set to their default. Please refer to the

table 2-1. With the same node density, larger sensor fields means the distance between

nodes is the same without respect to the size of the sensor fields, but the paths have

more hops in the larger sensor field.

Figure 2-9A shows the lifetime of the simulation. As the dimension of the sensor

fields gets larger, the lifetime of the network gets smaller. This may be due to the longer

paths (with more hops) as well as enlarged relaying load. On the other hand, the ratio of

simulation keep increasing according to the size of sensor fields due to the decreasing

lifetime.

2.6 Summary

In this chapter, we have reviewed the past research works on the sensor networks

which basically aimed to prolong the lifetime, and we have presented the results of









multi-commodity flow problem in which all pairs of nodes can concurrently send or

receive flow. The flow is sustained through links with a certain capacities. The MCFP

tries to find the flow such that the ratio of the flow between each pair of nodes to the

demand between that pair, so called throughput, is the same for all pairs of nodes.

In addition, the solution flow should observe the flow conservation law and capacity

constraint of the link. That is, the objective of the MCFP is to find the flow maximizing

throughput in the multi-commodity network. However MCFP problem used in [35] is

little bit different from the original one in that energy constraints, which is caused by the

limited energy of the nodes, are used in the problem formulation instead of capacity

constraints of the links. The objective is to find the flow maximizing the minimum

lifetime of the system while satisfying above constraints. In [35], the authors propose an

distributed and iterative approximation algorithms for the MCFP described above: Every

node maintains the same number of queues as commodities for its links. The key of the

algorithm is to keep each queue for a link equalized at any time.

Typically, the approximation algorithms would not give us optimal solutions, but

with these algorithms we can obtain the solutions very fast and they are close enough

to the optimal solution with an acceptable error. During the past several years, lots

of approximation algorithms have been published [3, 4, 13, 15, 45]. The performance

of the approximation algorithms, in terms of time complexity, are normally expressed

as functions of several parameters including a tolerable error range.[15] solves dual

problem with an iterative method. The problem is to find the largest A such that there is

a multi-commodity flow to which Ad(i) units of commodity i is assigned. Each edge of

the graph has positive capacity (c : E R) and there are k commodities with a demand

d(i) ( i = 1, 2,..., k). Let's x(P) be the amount of flow on some path P. then primal









is

S = K/lp. (3-4)

The second possibility is that there is no local coordination among the nodes

and the nearby nodes all report the same events to the sink. In this case, the traffic

generated by any node in the network is K. Between the two extreme cases are a

large number of other possibilities, depending on the degree of local coordination

and other factors. Note that the local coordination can take the form of sleep-wake

schedules, in which only a subset of the nodes are awake at any moment. Or, the rate

of report generation at each node is made inversely proportional to the node density

nearby. In either case, one can expect that the locally generated traffic rate at a node

is proportional to K/pj, i.e., Sj = rK/pj for some constant r > 0. In this section, we

assume r = 1 for notational simplicity. In effect, we assume the first possibility (3-4)

as the local traffic generation model. But, extension to the case with a general constant

qr is trivial. For other more different models, the entire methodology in the section still

applies, although the results will differ more substantially.

The rate of the relay traffic at a node does not have a simple expression. It is

easier to write out the recursive relationship it satisfies. For each pair of rings k and j,

0 j < k, let


Fk(j) ={the probability that a node in ring k selects

a node in ring j as its next-hop neighbor}. (3-5)

Note that all nodes in the same ring have the same probability distribution. The behavior

of different routing schemes can be captured by different choices of Fk(j). Hence, we

call each particular matrix (Fk(j)) a routing model.


NkGkFkj) = PkAkGkFk(j) =Pk(2k 1)w2GkFk().









so that we could decompose the dual problem into several sub-problems. However,

in general, dual problems is not differentiable. To deal with non-differentiability, we

adopt sub-gradient method. In each iteration, it is required to solve decomposed

sub-problems. We proposed (possibly) distributed algorithms for solving them.

Moreover, we prove that our approach eventually finds the solution which is arbitrarily

close to the optimal solution of the primal problem.


125









solution within the proposed framework, we formulate optimization problems that

maximize the lifetime of the WSN subject to the delay bound constraints, node energy

constraints, and flow conservation constraints. We conduct extensive computational

experiments on the optimization problems and find that the lifetime can be increased

significantly as compared to not only the stationary sink model but also more traditional

mobile sink models. We also show that the delay tolerance level does not affect the

maximum lifetime of the WSN.

Third, we propose an adaptive and potentially decentralized algorithm for the DT-

MSM. The distributed routing algorithms are very important in developing a practical

routing protocol. Distributed algorithms are generally free from the network scalability

issues in several reasons. They do not need to have knowledge about the whole

network configurations and they also do not require the central node to compute

the routes for all nodes in the network. Lagrange multiplier method solves dual of

the primal problem. Dual problem sometimes has a nice structure with which we

can decompose the dual problem into several sub-problems. We use a subgradient

projection method to solve the dual problem and. A sensor node in our method keeps

virtual queue which is a scalar product of the Lagrange multiplier and it is used in

solving sub-problems. We propose (a possibly distributed implementable) decentralized

algorithms for solving sub-problems. Moreover, we analytically show the our algorithm

finds a solution arbitrarily close to the optimal solution of the primal problem. It is verified

through the numerical experiments.









THE RESEARCH ON EXTENDING THE LIFETIME OF WIRELESS SENSOR
NETWORKS


















By

YOUNGSANG YUN


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










12 12
S10 ..... 10
E ............ .............
8 8

SSSM -- ..... SSM -
4 -MSM ---..... 4 D MSM ....--x-- .
4 DT-MSM *.............. -o DT-MSM ...*...
2 .......... ......... ...................... ...................... 2 ) ) ~ .......... ... ...................... X
0 0
5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40
Number of Sink Location Number of Sink Location
A 100-node network B 200-node network

Figure 4-3. Lifetime against the number of sink locations; maximum coverage; e = 2.0


circular area with radius 25. We use a simple algorithm to find the minimum radius of

coverage (denoted by ro): At each sink location, we increase the radius of coverage from

0 simultaneously until the union of all coverages contains all sensor nodes. At that point,

we have reached the minimum radius of coverage required to cover all nodes. After that,

we increase the radius of coverage in 0.1 increments. The result of the experiment is

plotted in Figure 4-2. Note that, in the figure, the lifetime is normalized to the optimal

lifetime of the MSM. As shown in the figure, the lifetime of the DT-MSM increases as the

radius of coverage increases, which is consistent with Theorem 1. The increase is the

sharpest when the radius just exceeds the minimum radius required to cover all nodes.

After that, further increase of the radius has a negligible effect. Recall that, when the

mobile sink reaches one of the stops, say /, only those sensor nodes in the coverage

of I (i.e., Ri) can communicate. It is generally desirable for R to have as few nodes

as possible, since this reduces the communication and coordination complexity. The

aforementioned behavior of lifetime increase is desirable.

Next, we compare the lifetimes of models under various numbers of the sink

locations. The number of nodes is set to 100 or 200, and the path loss exponent e is

2.0. The coverage is set large enough to always cover the entire sensor field. We ran

the experiment 100 times for each configuration. The lifetimes of the MSM and DT-MSM









for receiving data. The subsequent development still applies to the general case of

7 / 0; but the expressions are more complicated.
In our energy model, the energy required to transmit a unit of data is a function of

the transmission distance. To simplify the analysis, we make the approximation that the

distance between a pair of nodes is determined by the number of rings separating them.

The energy that a node in ring j, 1 < j < n, consumes per unit of time to transfer the

portion of its data directed to the nodes in ring i, 0 < < j, is 1


P(i) = (j i)aw GF,(i).

Thus, the energy consumption rate for a node in ring j, 1 < j < n, is 2
j-1
P = P (i)
i=0-I)+
j-1
= waG, (j- i)aF(i). (3-10)


3.2.2.1 Case of density-independent routing

Our goal is to equalize the energy consumption rate of all nodes. In other words, for

all 1 < j < n, we want to make Pj = Pn. From (3-10), we get the following relationship

for all j, 1 < < n.



1 Here, the probability Fj(i) is interpreted as the portion of data at a fixed node
in ring j that is transmitted to some node in ring i. Throughout, we will use the two
interpretations (probability or proportion) interchangeably depending on convenience.
2 We use the notation (a)+ = max(a, 0).









3.2.2.2 Case of density-dependent routing . . . . .... 51
3.2.3 Models of Routing/Node Selection . . . . . . . ... 52
3.2.3.1 Uniform ring selection . . . . . . . .. .. 53
3.2.3.2 Uniform node selection . . . . . . . ... 54
3.2.3.3 Simplified uniform node selection . . . . .... 56
3.3 Experim ental Results ................... .......... 56
3.3.1 Uniform Ring Selection ......................... 57
3.3.2 Uniform Node Selection ........................ 58
3.4 General Sensor Field and Routing Models . . . . . . ..... 60
3.4.1 General Two-Dimensional Model . . . . . . . ..... 60
3.4.1.1 Node-density independent routing . . . . .... 60
3.4.1.2 Node-density dependent routing . . . . . .... 64

4 MAXIMIZING THE LIFETIME OF WIRELESS SENSOR NETWORKS WITH
MOBILE SINK IN DELAY-TOLERANT APPLICATIONS . . . . . .... 66

4.1 O verview . . . . . . . . . . . . . . . . . . 66
4.2 Related Lifetime Maximization Problems . . . . . . . ..... 70
4.2.1 Static Sink Model ....... ... ...... ... ... . .. 72
4.2.2 Mobile Sink Model ........................... 75
4.3 Lifetime Maximization in Delay Tolerant Mobile Sink Model . . . ... 77
4.3.1 Sub-Flow-Based Model ........................ 79
4.3.2 Queue-Based Model .......................... 82
4.3.3 Properties of Delay-Tolerant Mobile Sink Model . . . . ... 84
4.4 Experimental Results ................... .......... 88

5 A DECOMPOSITION TECHNIQUE FOR DT-MSM . . . . . . .... 94

5.1 O verview . . . . . . . . . . . . . . . . . . 94
5.2 SystemModel and Problem Formulation . . . . . . . . 94
5.3 Decomposition by the Lagrange Method . . . . . . .... 98
5.3.1 Algorithms for Subproblems . . . . . . . .. . .. 105
5.3.2 M ain Algorithm . . . . . . . . . . . . . . 109
5.4 Performance Analysis ............................. 110
5.5 Experim ental Results ............................. 117
5.6 Im plementation Issues ............................. 120

6 CO NC LUSIO N . . . . . . . . . . . . . . . . . . 123

R EFER ENC ES . . . . . . . . . . . . . . . . . . . 126

BIOGRAPHICAL SKETCH ................................ 130





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IwouldliketobeginbythankingProfessorYeXia,mydissertationadvisorandmentorduringpastseveralyears.Dr.YeXiahasbeenthegreatestadvisor,providingmewithwithvaluableideas,support,andhelpduringmygraduatestudies.Hiswideanddeepknowledgeandhisenthusiasmalwaysinspiresme.MyspecialthanksalsogotoProf.JonathanLiu,Prof.AlinDobra,Prof.ShigangChenandProf.ColeJ.Smithfortheircommentsandsupportduringmystudies.Lastbutnotleast,IwanttothankmyfamilyfortheirlovewhileIamstudying.Withouttheirsacrice,IwouldnotbewhatIam.IwouldliketosaythatIloveChris,Han,andmywife,KyungHee. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1Contributions .................................. 15 1.2OrganizationoftheStudy ........................... 16 2EXPERIMENTSONTHEMINIMUMTRANSMISSIONENERGYROUTING,SEQUENTIALMINIMUMTRANSMISSIONENERGYROUTING,ANDOPTIMALLIFETIMEROUTING ................................ 18 2.1Overview .................................... 18 2.2RelatedWorks ................................. 19 2.3SystemModel ................................. 22 2.3.1EnergyModel .............................. 23 2.3.2SimulationModelOfMinimumTransmissionEnergyRouting .... 24 2.3.3LinearProgrammingModel ...................... 25 2.4Experiments .................................. 26 2.4.1GraphGeneration ........................... 26 2.4.2Lifetimevs.Ratio ............................ 29 2.4.3EffectOfNodeDensities ........................ 29 2.4.4EffectOfTheNumberOfSinks .................... 32 2.4.5EffectOfNodeDensitiesAndTheNumberOfSinks ........ 34 2.4.6EffectOfNodeDeploymentStrategies ................ 35 2.4.7EffectOfCommunicationRanges ................... 35 2.5ExperimentsontheMinimumTransmissionEnergyRouting,SequentialMinimumTransmissionEnergyRouting,andOptimalLifetimeRouting .. 37 2.6Summary .................................... 38 3AMETHODFORDECIDINGNODEDENSITYINNON-UNIFORMDEPLOYMENTOFWIRELESSSENSORNETWORKS ...................... 40 3.1Overview .................................... 40 3.2ModelswithDiscreteRingStructure ..................... 44 3.2.1SensorFieldandEnergyConsumptionModels ........... 45 3.2.2DerivingtheNodeDensitiesoftheRings ............... 47 3.2.2.1Caseofdensity-independentrouting ............ 50 5

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............. 51 3.2.3ModelsofRouting/NodeSelection .................. 52 3.2.3.1Uniformringselection .................... 53 3.2.3.2Uniformnodeselection ................... 54 3.2.3.3Simplieduniformnodeselection ............. 56 3.3ExperimentalResults ............................. 56 3.3.1UniformRingSelection ......................... 57 3.3.2UniformNodeSelection ........................ 58 3.4GeneralSensorFieldandRoutingModels .................. 60 3.4.1GeneralTwo-DimensionalModel ................... 60 3.4.1.1Node-densityindependentrouting ............. 60 3.4.1.2Node-densitydependentrouting .............. 64 4MAXIMIZINGTHELIFETIMEOFWIRELESSSENSORNETWORKSWITHMOBILESINKINDELAY-TOLERANTAPPLICATIONS .............. 66 4.1Overview .................................... 66 4.2RelatedLifetimeMaximizationProblems ................... 70 4.2.1StaticSinkModel ............................ 72 4.2.2MobileSinkModel ........................... 75 4.3LifetimeMaximizationinDelayTolerantMobileSinkModel ......... 77 4.3.1Sub-Flow-BasedModel ........................ 79 4.3.2Queue-BasedModel .......................... 82 4.3.3PropertiesofDelay-TolerantMobileSinkModel ........... 84 4.4ExperimentalResults ............................. 88 5ADECOMPOSITIONTECHNIQUEFORDT-MSM ................ 94 5.1Overview .................................... 94 5.2SystemModelandProblemFormulation ................... 94 5.3DecompositionbytheLagrangeMethod .................. 98 5.3.1AlgorithmsforSubproblems ...................... 105 5.3.2MainAlgorithm ............................. 109 5.4PerformanceAnalysis ............................. 110 5.5ExperimentalResults ............................. 117 5.6ImplementationIssues ............................. 120 6CONCLUSION .................................... 123 REFERENCES ....................................... 126 BIOGRAPHICALSKETCH ................................ 130 6

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Table page 2-1Systemparametersusedinthesimulation ..................... 27 4-1Experimentalparametersandtheirvalues ..................... 89 5-1PerformancecomparisonbetweenCPLEXandAlgorithm3 ........... 121 7

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Figure page 2-1Graphexample1:nodedeploymentbytheuniformdistribution ......... 28 2-2Graphexample2:nodedeploymentbygridbasedstrategy ........... 30 2-3Lifetimevs.ratioofremainingenergytotheinitialenergy ............ 31 2-4Performanceinvariousnodedensities ....................... 32 2-5Performanceinvariousnumberofsinks ...................... 33 2-6Performanceindifferentnumberofnodesanddifferentnumberofsinks .... 34 2-7Performanceofdifferentnodedeploymentstrategies ............... 36 2-8Performanceofdifferentcommunicationranges .................. 37 2-9Performanceofdifferentdimensionofsensorelds ................ 38 3-1Sensoreldmodel .................................. 46 3-2Uniformnodeselection ............................... 54 3-3Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformringselection.=2. ........... 58 3-4Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=0. .......... 59 3-5Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=1. .......... 60 3-6Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=2. .......... 61 3-7Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=3. .......... 62 4-1Examplesofthestaticsinkmodel(SSM),mobilesinkmodel(MSM),anddelaytolerantmobilesinkmodel(DT-MSM) ....................... 77 4-2ComparisonoflifetimesofMSMandDT-MSMunderthevariousradiiofcoverage 89 4-3Lifetimeagainstthenumberofsinklocations;maximumcoverage;e=2.0 90 4-4Lifetimeagainstthenumberofsinklocations;minimumcoverage;e=3.0 91 4-5Lifetimesversustransmissionrange,d:jNj=200,jLj=20,e=2.0 92 5-1ExpandedgraphofDT-MSM ............................ 97 8

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118 5-3Lyapunovdriftofthealgorithmovertime ...................... 119 5-4Timeaverageoftotalvirtualqueuesizeovertime ................. 119 5-5Long-timeaverageofthedifferencebetweenoutowandinow ......... 120 9

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Westudyvariousproblemsefcientenergymanagementforwirelesssensornetworks.First,weresearchtheenergyefcientdeploymentofwirelesssensornodessothatenergyconsumptionratesofallnodesareequalduringthelifetimeofthewirelesssensornetwork.Ifthesensorsaredeployeduniformlyacrossthenetwork,theyexperiencedifferenttrafcintensitiesandenergydepletionratesdependingontheirlocations.Usually,thesensorsnearthesinktendtodepletetheirenergysooner;whenenoughofthemexhausttheirenergy,theyleaveholesinthenetwork,causingtheremainingnodestobedisconnectedfromthesink.Oneofthesolutionstothisenergy-holeproblemistodeploythesensorsnon-uniformly.Moreover,wedescribeamethodfordecidingthesensordeploymentdensitiessoastoequalizetheenergyconsumptionratesofallnodes.Themethodisgeneralandcanbeappliedtootherobjectivesandconstraints. Second,weproposeaframeworktomaximizethelifetimeofthewirelesssensornetworksbyusingamobilesinkwhentheunderlyingapplicationstoleratedelayedinformationdeliverytothesink.Withinaprescribeddelaytolerancelevel,eachnodedoesnotneedtosendthedataimmediatelyasitbecomesavailable.Instead,thenodecanstorethedatatemporarilyandtransmititwhenthemobilesinkisatthemostfavorablelocationforachievingthelongestWSNlifetime.WecalltheproposedframeworkasDelay-TolerantMobileSinkWirelessSensorNetwork.Tondthebest 10

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Third,weproposeanadaptiveandpotentiallydecentralizedalgorithmfortheDT-MSM.Thedistributedroutingalgorithmsareveryimportantindevelopingapracticalroutingprotocol.Distributedalgorithmsaregenerallyfreefromthenetworkscalabilityissuesinseveralreasons.Theydonotneedtohaveknowledgeaboutthewholenetworkcongurationsandtheyalsodonotrequirethecentralnodetocomputetheroutesforallnodesinthenetwork.Lagrangemultipliermethodsolvesdualoftheprimalproblem.Dualproblemsometimeshasanicestructurewithwhichwecandecomposethedualproblemintoseveralsub-problems.Weuseasubgradientprojectionmethodtosolvethedualproblemand.AsensornodeinourmethodkeepsvirtualqueuewhichisascalarproductoftheLagrangemultiplieranditisusedinsolvingsub-problems.Wepropose(apossiblydistributedimplementable)decentralizedalgorithmsforsolvingsub-problems.Moreover,weanalyticallyshowtheouralgorithmndsasolutionarbitrarilyclosetotheoptimalsolutionoftheprimalproblem.Itisveriedthroughthenumericalexperiments. 11

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ThetechnologicalbreakthroughsinMEMS(MicroElectroMechanicalSystems),DSP(DigitalSignalProcessing),integratedcircuittechnologies,microprocessorhardware,andwirelesscommunicationtechniqueshavebeenphenomenalsincelastfewdecades.Inaddition,thetechnologicaladvancethatresearchersandengineershaveachievedinad-hocnetworkingroutingandprotocol,pervasivecomputing,embeddedsystemtechnologiesmakeitpossibletomass-manufacturethelowcost,small-sizedformfactored,andversatilesensornodes.Theyintegrategeneralpurposeprocessors,widevarietiesofsensingdevices,andwirelesscommunicationdevices.AWirelessSensorNetworkstypicallyconsistsofthesecheapsensornodesdeployedintothetargetedareatobemonitored.However,tomakethecostofdeployingwirelesssensornetworktobelow,asensornodehasanon-boardbattery,aswellasalowpowerprocessorandalimitedmemoryspace.Thereforeitisnecessarytomakesensornodescollaboratewithothersensornodestoovercomethislimitation.SinceaWirelessSensorNetwork(WSN)isconstructedwithahugenumberofsensornodes,whicharedenselyandsometimesrandomlydeployedintotheregionofourinterest,thelocationofsensornodesisnotknownatthetimeofdeployment.Occasionallythesensornodesneedtobedeployedrandomlyintothehostileorhazardousterrainsothatitisnoteasyorevenimpossibletoaccessthatregion.Thus,theprotocolsoralgorithmsusedintheWSNshouldhavetheabilityoforganizingthenetworkautonomously.Furthermore,sincethenumberofdeployedsensornodesisoftentremendous,anuntetheredoperationfortheindividualsensornodeisrequired. Themissionimposedonthesensornodesisgatheringinformationaboutthesurroundingenvironment,processingsensedrawdatasuchascompressionorquantization,andtransferringprocesseddatatospeciallocationscalledthesinksforfurtherprocessing.Thesinksaretypicallymorepowerfulinprocessingpowerand 12

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Duetothelimitationofavailableenergyorconstrainedcapabilityofwirelesstransceiver,itisnotfeasibleforsensornodetocommunicatewiththesinkdirectly.Instead,asensornodeusesothersensornodesasnexthopsinordertogettothesink.Forthisreason,thecommunicationbehaviorinthewirelesssensornetworkshowsmulti-hoppattern.Thereforetheindividualwirelesssensornodeplaystherolesofasourceofthedataandarelayerofthedataforothernodes.ThisfeatureofWSNmakesitverysimilartotheMobileAd-hocNetworks(MANET)inthatcommunicationisdoneinmulti-hopfashion.However,thereisahugedifferencebetweenthesetwotypesofnetwork.WhiletheconnectivityistheultimategoaloftheMANET,thelongevityofnetworkistheprimaryobjectiveoftheWSN. Thefunctionsofsensorsareverydiverse:seismic,magnetic,thermal,acoustic,visual,andradioactive.Asthevarietyofthefunctionsofsensor,thepotentialapplicationsoftheWSNisalsoboundless.Accordingto[ 2 ],theapplicationsofWSNcanbecategorizedintothemilitaryapplications,environmentalapplications,healthcareapplicationsandhomeapplications.Butthisclassicationcanbebroadenintomorecategories,suchasapplicationsinspaceexplorationordisasterrelief. Sincesensornodesreplyontheembeddedbatterypower,sometimesreplenishmentofbatteryisverycostly.Whenconsideringthenumberofsensornodesdeployeditdoesnotseempossible.ThereforeclevermanagementofenergyreservoirisrequiredtoextendthelifetimeorimprovethethroughputoftheWSN.Therehavebeenseveralchallengingresearchissuesontheefcientenergymanagement. Typicallysensornodeisabletoalteritstransmissionpowerandasaconsequence,itcanchangereachabletransmissiondistance.Bychangingthetransmissionpower,onecanmakeasensornodehavethedifferentsetofneighbornodesattimes.Intheperspectiveofnetwork,differenttopologiescanbeobtainedbytheadjustment 13

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InWSN,asensornodemightbeinoneofthreestates:active,idle,andsleep.Asensornodeconsumessignicantamountofenergyevenitisinanidlestate.Inordertosavetheenergy,asensornodesometimesgoesintothesleepstate.Thus,howtoschedulesleep-wakesequenceisalsoanotherresearchissues.Sinceitisnotpossibletosynchronizethesleep-wakescheduleforeverysensornode,delayisessentiallyincreased.Goodscheduleshouldminimizeincurreddelayaspossible. InthedatacentricWSN,withoutdataaggregation,thesinkmightacquiremuchredundantdata,becauselotsofsensornodeswouldreportthesameeventiftheyarelocatedclosely.Therearelotsofwaystoaggregatedata.Oneoftheseisclusteringthesensornodes.Ineachcluster,theclusterheadaggregatesrawdatagatheredbythesensornodesinitsowncluster.Dataaggregationisveryusefulinreducingtheenergyconsumptionaccompaniedbycommunicationaswell. Widelyusedroutingprotocolinthewirednetworkmaynotbeagoodchoiceintheperspectiveofefcientenergymanagement.Toensurethelongevityofeachsensornodes,theroutingpathgoingthroughthenodehavinginsufcientenergyshouldbeavoided.Lotsofroutingprotocolstakingthecurrentenergylevelofthesensornodeintoaccounthavealreadybeenproposed.ThiskindofroutingprotocoliscalledPower-awareroutingprotocol.Theminimumtotaltransmissionenergypathsandmaximumresidualenergypathsaresuchexamples.Withthehelpofamatureoptimizationtheory, 14

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IfthemobilitycanbeappliedtothecomponentofWSN,thebetterperformancemightbeexpected.Themobilesinkseemsapreferentialchoiceratherthanthemobilesensornodes.Whenthesinkhasacapabilityofmoving,overallenergyconsumptionofparticularsensornodescanbemitigatedascontrastedwiththenetworkwherethesinkdoesnotmovebecauseaheavyrelayingburdenalsotendstofollowthesink'smovement. 3 ,chapter 4 ,andchapter 5 .First,westudythedensitycontrolthatmakeseverysensornodeexperiencethesamerateofenergyconsumption.ThesameenergyconsumptionforallsensornodeisakeyfactorinmaximizingthelifetimeoftheWSN.Ourdensitycontrolnecessitatesanon-uniformnodedeploymentwherethedensityatapointisdeterminedaccordingtotheroutingprotocolusedanditsdistancefromthesinknode.Weproposeaverygeneralroutingmodelthatcapturesvariousexistingroutingprotocolstrategies.Wealsoproposeaniterativemethodthateventuallyndsdensitiesoftheringwherethesensornodesarealmostsimilarlydistantfromthesink. Second,weproposenewframeworkthatexploitsthemobilesinks.Theapplicationweareinterestedincantoleratesomeextentofdelayofdeliveryofinformationcollectionofsenseddata.WeformulatetheLinearProgrammingproblemthatreectsfeaturesofourframework.Throughthesimulation,wealsoshowitsperformanceisbetterthanothermodels:StaticSinkModelandGeneralMobileSinkModel. Third,wedevisetheadaptiveandpotentiallydistributedalgorithmfortheframeworkwepropose.Ouralgorithmcanworkonlywiththeinformationaboutthecurrentqueuesize,receivingandtransmittingtrafcatthesensornode.Weshowthatouralgorithmndasolutionwhichisarbitrarilyclosetotheoptimalsolution. 15

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2 ,asapreliminarywork,WecomparethelifetimesofseveralroutingprotocolswhichmightbeusedintheWSN.ThesimplestonemightbetheMinimumTransmissionEnergy(MTE)routinginwhicheverysensornodesgreedilychoosethepathalongwhichtotalenergyconsumptionintransmissionisminimal.Ontheotherhand,Wecanusethemaximallifetimeroutinginwhichroutingpathsarecalculatedbysolvingtheoptimizationproblemwiththeglobalviewofthenetworktopology.IntheMTE,aftertherstnodediesduetoenergyexhaustion,alargenumberofnodesstillpossessplentyofenergy.Thus,ratherthantopausetheoperationofthewholesystem,thenetworkcancontinueworkingaslongastheconnectivitytothesinkispreservedfromallremainingsensornodes.IntheSequentialMinimumTransmissionEnergyprotocol(SMTE),alivesensornodeskeeproutingbasedonMTEafterthefailedsensornodeisremovedfromthenetworktopologyuntilthesinkisisolatedfromthenetwork.Thesethreeroutingprotocolsarecomparedthroughextensiveexperimentswithvarioussimulationsettings. Inchapter 3 ,Wediscussnon-uniformnodedeploymentstrategies.Whennodesareuniformlydistributedintheregion,theenergyholeproblemnearthesinkmaynotbeanavoidablephenomenon.Therationaleofnon-uniformdeploymentistoputmorenodestotheplacewhererelayingburdenisexcessive.Theareanearbythesinkissuchaplace.Inthischapter,Wesolvetheproblemofhowmanynodesshouldbedeployedinaspecicareaofthesensornetwork.Thesending(oroutgoing)trafcfromanodeiscomposedofrelayingtrafcandself-generatedtrafc.Weestablishtherecurrencerelationsshowinghowtheoutgoingtrafcofnodesisrelatedaccordingtothedistancefromthesink,whenageographicalroutingisused.Inaddition,Wementionthewaytoobtainthesolutionthatsatisfyingthesystemofrecurrencerelation. Inchapter 4 ,WestudyhowthemobilitywouldimprovethelifetimeoftheWSN.Especiallythemobilityofthesink,orthemobilesink,isconsideredinthischapter.To 16

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Inchapter 5 ,WeextendourworkontheDT-MSMproposedinthechapter 4 .AlthoughtheDT-MSMshowssubstantialimprovementoftheperformance,itrequiressolutionfromtheLinearProgrammingproblem.However,thereisnowaytosolvetheLinearProgrammingproblemsinthedistributedmannerandevenWeneedaverypowerfulcomputingnodewhichrunsLPsolver.Tobeapracticalroutingmethod,itisnecessarytobeimplementableinadecentralizedmanner.Inthischapter,Weproposeapartiallydistributedalgorithmimplementingourframework.TheproposedalgorithmdoesnotrequireanLPsolver. Inchapter 6 ,Weconcludethisdissertation. 17

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Sensornodestypicallyoperatewithalimitedon-boardbatteryandit'sveryhardorevenimpossibletoreplacethebatteryorreplenishtheenergyofthenodes.Thus,thelongevityofwirelesssensornetworkshasbeenconsideredasaprimarygoalinwirelesssensornetworkresearchesandmanyworkstoprolongthelifetimeofwirelesssensornetworkshavebeenpublished[ 12 ][ 11 ][ 19 ][ 35 ][ 45 ].Ontheotherhand,someresearchersfocusonhowtoapplytheproposedalgorithmstotherealworld,forinstance,distributedimplementationofthelinearprogrammingapproaches(referto[ 26 ][ 35 ])oraprroximatedalgorithms(referto[ 35 ][ 15 ][ 3 ][ 45 ]). Ingeneral,therearemanyfactorsthatdenesthelifetimeofthesensornetwork,forexample,thesizeofthesensoreld,thenumberofsensornodes,thenumberofthesink,etc.Inpractice,knowinghowtheseparametersaffectthelifetimeofthenetworks.However,investigatingthisproblemtakingalltheseintoaccountissodifculttask.Asapreliminarystepofthisresearch,wedoextensivesimulationstogetsomeintuitionontherelationshipsthatmaybeexistbetweenthelifetimeandthesefactors. 18

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19 ],theauthorsproposetheLEACH(Low-EnergyAdaptiveClusteringHierarchy)tominimizeglobalenergyconsumptionbydistributingtherelayingburdentoallthenodes.Sensornodesaredividedintoseveralclustersandineachclusterthereisaclusterheadwhichisresponsibleforrelayingtheallthetrafcgeneratedinthecorrespondingclustertothesink.Inaddition,theroleofclusterheadisnotxedtothespecicnode,thatis,theclusterheadisassignedtothehighestenergynodeinaadaptivemanner.whichimprovesthelifetimeofsensornetworksignicantly. In[ 12 ],theauthorsdenethesensornetworklifetimeasthetimeuntilthenodedrainsoutitsenergyatthersttime.Theyalsoshowsthatminimumenergyrouteswhichminimizesenergyconsumptionsaspossiblewhilerelayingthetrafcateachnodearenotgoodintheperspectiveofthelifetimeofthesensornetworksduetounevenenergydepletionbehavior.Thatis,theminimumenergyroutescausesthefastenergyconsumptionrateofsomenodeswhichareonthefrequentlyusedroutes.Theauthorsformulatetheroutingproblemasanoptimizationproblemwheretheobjectiveistondtheowsthatmaximizethesensornetworklifetime. FormulationofmaximizingthelifetimeasaLinearProgramcanbefoundinmanyliteratures.In[ 26 ],theauthorsproposetwodistributedalgorithmsforthesameproblemas[ 12 ].Thekeyideaoftheiralgorithmsistosolvedualprobleminsteadofsolvingtheprimalproblemdirectly.Bychangingtheobjectivefunctionsofdualproblemslightly(Inthiscase,theobjectfunctionisnomorelinearfunction),Wecanmaketheseparablenonlinearprogramproblem,andthisnicestructureoftheproblemmakeeasytodeviseadistributedalgorithmforsolvingmaximumlifetimeproblem.Oneoftheiralgorithmsisapartiallydistributedone,theothercanbeimplementedinfullydistributedmanner.However,bothalgorithmsuseawell-knownsub-gradientalgorithm.[ 35 ]alsodenesthemaximumlifetimeroutingproblemasavariantofmaximumconcurrentowproblem(MCFP),asortoflinearprogramproblem.TheoriginalMCFPdenedin[ 37 ]isa 19

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35 ]islittlebitdifferentfromtheoriginaloneinthatenergyconstraints,whichiscausedbythelimitedenergyofthenodes,areusedintheproblemformulationinsteadofcapacityconstraintsofthelinks.Theobjectiveistondtheowmaximizingtheminimumlifetimeofthesystemwhilesatisfyingaboveconstraints.In[ 35 ],theauthorsproposeandistributedanditerativeapproximationalgorithmsfortheMCFPdescribedabove:Everynodemaintainsthesamenumberofqueuesascommoditiesforitslinks.Thekeyofthealgorithmistokeepeachqueueforalinkequalizedatanytime. Typically,theapproximationalgorithmswouldnotgiveusoptimalsolutions,butwiththesealgorithmswecanobtainthesolutionsveryfastandtheyarecloseenoughtotheoptimalsolutionwithanacceptableerror.Duringthepastseveralyears,lotsofapproximationalgorithmshavebeenpublished[ 3 4 13 15 45 ].Theperformanceoftheapproximationalgorithms,intermsoftimecomplexity,arenormallyexpressedasfunctionsofseveralparametersincludingatolerableerrorrange.[ 15 ]solvesdualproblemwithaniterativemethod.Theproblemistondthelargestsuchthatthereisamulti-commodityowtowhichd(i)unitsofcommodityiisassigned.Eachedgeofthegraphhaspositivecapacity(c:E!R)andtherearekcommoditieswithademandd(i)(i=1,2,...,k).Let'sx(P)betheamountofowonsomepathP.thenprimal 20

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Now,thedualproblemislike:minXec(e)l(e) Notethatl(e),zjaredualvariableassociatedtotheconstraintsfortheedgesandnodesintheprimalproblem,respectively.Inthedevelopmentoftheapproximationalgorithm,thedualproblemcanbeconsideredasanassignmentoflengths(l:E!R)totheedgessuchthatD(l)=(l)isminimized,where(l)=kj=1mincostj(l)andD(l)=el(e)c(e).mincostj(l)istheminimumcostofd(j)unitsofcommoditytoowfromsourceofjtosinkofj.Notethatl(e)meansthecostofoneunitofcommoditytoowalongedgeeandactuallyisthedualvariableofthedualproblem.Theauthorsalsoshowthatiftheshortestpathalgorithmcanbeusedasasubroutineineachiterationinsteadofndingtheminimumcostowforasinglecommodity,convergencetimeofthealgorithmcanbeimproved.[ 3 4 ]isverysimilarto[ 35 ]inthattheyareiterativealgorithmstryingtobalancingthequeuelengthforasinglelink(minimizingthe 21

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13 ]isalmostidenticaltothatof[ 15 ],butbychangingtheterminationcondition,runningtimeofthe[ 13 ]isnowindependentofthenumberofcommodities.[ 45 ]alsosolvestheMCFPbasedonthealgorithmof[ 15 ],buttheauthorsusesaggregationtree,whichisanaggregatedstructureofunicastroutesfromallsourcestothecommonsink.Therunningtimetondaggregationtreeforacommonsinkisnotthatmuchlargerthanthatofndingtheshortestpathbetweenanindividualsourcetothecommonsink.Thus,insteadofcalculatingtheshortestpathforeverysources,[ 45 ]triestondtheaggregationtreeforthecommonsink. Inotherhands,[ 17 ]solvesgivenLinearProgramforasimpleandregulartopologies,forexamplelineararraytopology,withananalyticalmanner.However,sincetheiranalysisisrestrictedtotheregulartopology,itmakesnosensetoapplytheiranalysisinrealworld. Someresearchersstudyhowthemulti-pathroutingcancontributetoreducetheimbalanceofenergyburdenofthesensornodes.Forexample,[ 5 ]explainstheoptimizingtrade-offsbetweentheenergycostofspreadingtrafcandtheimprovedspatialbalanceofenergyburden.Theauthorsproposethemulti-pathroutingbasedonthenodeproximity.Theiralgorithm,infact,isanheuristicapproachtopreventenergyholearoundthesinknodefromoccurringataratherearlytime. 22

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whiletheenergyconsumedbyanodeiinreceivingaunitofdatafromanodej,denotedaserjiisgivenby, wheredijistheEuclideandistancebetweennodeiandnodej,andisaconsumedenergytorunthetransmitterorreceivercircuitryinthesensor,andistherequiredenergytorunthetransmitteramplier[ 19 ].Althoughshouldbedeterminedfromvariousenvironmentalfactors,typicallyitisaconstantbetween2and4.Forexample,infreespacepropagationmodel,isconsideredas2,sothatenergyconsumptionforthetransmissionofasingleunitofdataisproportionaltothesquareofthedistance[ 33 ]. Asfortheinitialsettingofnetworks,weassumethateverysourcehasaconnectivitytothesink,thatis,thereareatleastonepathtothesinkfromeverysources. 23

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45 ].Referto 2.3.1 formoredetails.Infact,MTEisanotheroptimizationproblemandwecanformulateitasaLinearProgram:minXi2NXj2Nietijxij (2)TXj2NietijxijE(i)8i2N Ifthedenedinthe 2.3.1 equalsto1,thenMTEisequivalenttoMinimumHopRouting.Duetothetriangularinequality,MTEprefersthelongeredgestogettotheneighborclosetothesink.Iftheisgreaterthan1,MTEprefersthepathswhichconsistofmanyshorteredgestothepathsoffewlongeredges. However,thelifetimeoftheMTEstrategycanperformarbitrarilybadly,becausethisdoesnotconsidertheresidualenergyoftheintermediatenodesalongthepaths,thuscancauseanunbalancedenergyconsumptiondistribution:thenodesontheminimumenergypathtothesinkaredrainedtheirenergyveryfast,sothatentirenetworkcanbepartitioned[ 11 ],[ 12 ].[ 30 ]and[ 39 ]arguethatbythetimethesensorsclosetothesinkisdraineditsenergy,sensorsfartherawayfromthesinkstillhaveplentifulenergy.From 24

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Nowlet'schangethedenitionofthenetworklifetimefromthetimeuntiltherstnodediedtothetimeuntilnetworkisdisconnected.Thismeansthatwecanconsiderthatthenetworkisstilloperationalatthetimetherstnodedied,unlessremainingnetworkispartitioned.Hence,MTEneedstobemodiedandextendedasfollows:Foragivennetworkinwhichtheedgecostisthetransmissioncost,everysourcenodesneedtondtheshortestpathtothesinkandthispathisusedasatheminimumtransmissioncostpathforanode.Whenacertainnodehasdraineditsenergy,bydeletingthisnodeandalltheedgesincidenttothisnode,wemaygetareducednetwork.Wecancheckwhetherthereducedgraphisstronglyconnectedcomponentornot.Ifthegraphisstillastronglyconnectedcomponentandthatmeanstheeverysourcenodesmusthaveaconnectivitytothesink.Thus,MTEcanbeappliedonthereducedgraphagain.Wecanapplythisprocessoverandoveruntiltheresultingnetworkisnolongerstronglyconnectedcomponent.WecallsucharepeatedMTEasaSequentialMTE(SMTE).NotethatabovedenitionofthelifetimeofthenetworkintheSMTEisequivalenttothetimeuntilnetworkispartitioned.Infact,wedonotconsiderthesituationsuchthatpartsofthesensingareabecomeuncoveredduetotheoutageofthesensornodes. j2Nieijtxij(2) ThenetworklifetimeTunderowxisthetimeuntiltherstnodeusedupitsenergy.Thatis, 25

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(2)TXj2NieijxijE(i)8i2N Notethatthisistheproblemthatmaximizingtheminimumlifetimeofsourcenodes.Theowconservationinthenetworkisrepresentedas( 2 ),andinequalities( 2 )meantheenergyconstraintsoneachnode.BychangingthevariableTto1=z,wecanobtainaminimizationproblembutequivalenttotheabovemaximizationproblem.(P)minz (2)Xj2NieijxijzE(i)8i2N Notethatinthisproblem,wemustdeterminex=fxijgandz. 2.4.1GraphGeneration 26

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2-1 .Notethatweassumethatenergyrequiredtorunthetransmitter/receivercircuitryinthesensornodeisnegligible,thuswejustsetto0. parameters values communicationrange Systemparametersusedinthesimulation Wetrytosimulatethesensornetworkinvariousenvironments.Forexample,wewanttogureouthowdoesthelifetimechangeaccordingtothedifferentenergymodel,howcanthenumberofthesinkaffectthelifetimeofthesensornetworkandisthereanynoticeablerelationshipbetweenthelifetimeandthedensityofthesensornodes,etc. Figure 2-1 showsanexampleofgraphsusedinthesimulation.Inthisgraph,thelocationofnodeisdeterminedinanuniformlydistributedmanner.Ifthedistancebetweenthenodeiandthenodejislessthancommunicationrange,nodeiandnodejhaveabidirectionallink.Inoursimulation,ifthewholegraphdoesnotformasinglestronglyconnectedcomponent,wejustdiscarditandkeepgeneratingthenewgraphuntilwegetastronglyconnectedone. Figure 2-2 showsanothergraphwhichisgeneratedbythegrid-basednodedeployment.Withthismethod,sensoreldisdividedintothexednumberofequalsizegridsandeachgridcontainsatleastonesensorinit.Themaximumnumberofsensorsinthegridislimitedanddeterminedbythenumberofnodestobedeployed 27

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Graphexample1:nodedeploymentbytheuniformdistribution 28

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Inthischapter,wemainlyfocusontwotypesofmetricsasperformancemeasures.Oneisaboutthenetworklongevity,andtheotherisabouttheenergyefciencyofthealgorithm.Ofcourse,thenetworklongevitycanbemeasuredbythelifetime,andtheenergyefciencycanberepresentedbytheratioofthetotalremainingenergytothetotalinitialenergyaftersimulationisnished.Whenwesayratiothroughoutthischapter,itfollowsabovedenitionoftheratio. 2-3 showshowtheratiochangesduringthelifetimeofsensornetwork.ThisistheresultoftheSMTEsimulationandweassumethatthecommunicationrangeforanodeissufcientlylargesothateverynodecanreachthesinkdirectly.Inthisgure,Wecanobservethattheratioisdroppedsharplywhenthelifetimeexceedsacertainpoint.Wesuspectlotsofnodesinthesensornetworkdiedduetoenergyexhaustionafterthelifetimeexceedsthethresholdpoint,thusthedistancetotraverseinasinglehopmightincrease.Themorenodesdie,thefasterenergyisused.Infact,wemightnotgetthesameresultforallthenodesasthegure 2-3 ,forexampleforthenodenearthesink,thissuddendroppingoftheratiomightcomeatearliertime.Ontheotherhand,forthenodefartherfromthesink,thisdroppinghappensatthelatertime,butitisverysteep. 2-4A showstheperformanceofthesimulationundervariousnumberofnodes.Inthisgraph,MTE,LP,andSMTEmeanstheMinimumTransmissionEnergysimulation,LinearProgramsolution,andSequentialMTE,respectively.Inourexperiments,thenumberofnodes 29

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Graphexample2:nodedeploymentbygridbasedstrategy 30

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Lifetimevs.ratioofremainingenergytotheinitialenergy areincreasedby25from200to800andwerun100timesforeachconguration.Weassumethatthereisonlyonesinkattheverycenterofthesensoreldandwesettheoftheenergymodelto4.Allthreesimulationsshowthatlifetimeincreasewithsteadypacesasthenumberofnodesincrease.NotethatthelifetimesoftheSMTEandLParealmostthesame.WeobservethedecreaseintheratiofortheLPcase,asthenodedensityincreases.However,incasesofMTEandSMTE,theratioseemstobeconstantwithoutregardtotheincreaseofnodedensity.Hence,weconcludethatfortheMTE,theincreaseofthenodedensitymaynotcontributetothelifetimethatmuch.Asthenodedensityincreases,averagedistancebetweennodesgetshorter,thuseachnodemaygetenergysavingduetotheshorterdistancehop.However,increasednumberofnodesmightalsoputtherelayingstressonthenodesandthisstressmightbestrongertothenodesclosertothesinkthanthenodesfartherfromthesink.Thisrelayingburdencutsdownthebenetoftheshortenedaveragedistancebetweennodes.FortheLPcases,thedecreasedratiomayaccountfortheincreasedlifetimeofthenetwork.SMTEisvery 31

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Performanceinvariousnodedensities schemethatmaymakefulladvantageoftheincreaseofnodedensity.ThelifetimeforSMTEismuchlongerthatMTEanditsenergyefciencyisbetterthanLP. 32

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2-5A andgure 2-5B .Asweseeingure 2-5A ,thenumberofsinksbecomeslarger,SMTE Performanceinvariousnumberofsinks outperformstheLP.Ratiosofallthreeroutingcasestendstobeconstantorslightlyincreasingafteracertainnumberofsinks.Interestingly,wendthatitismoreeffectivetoincreasethenumberofthesinksratherthantoincreasethenodedensityforthenetworklifetime.Notethat,whenthenodedensityincreasesnodehasshorterlinkswith 33

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Figure 2-5C andgure 2-5D showtheperformanceundergrid-basednodedeploymentscheme. Althoughtheperformanceinthegrid-baseddeploymentisslightlybetterthantheperformanceintheuniformdeploymentinallthesimulation,wecannotndasignicantdifferenceintermoftheperformancebetweenbothdeploymentstrategiesasthenumberofthesinkincreases. Performanceindifferentnumberofnodesanddifferentnumberofsinks Figure 2-6 showstheperformanceofthesinglesinknetworkandmultiple-sinknetworks(Inthemultiplesinkscase,wesetthenumberofthesinkto4)undervariousnumberofnodes.Obviously,thesystemofmultiplesinksoutperformssinglesinksystem.Inbothcases,thelifetimesofMTEareverypoorandthelifetimeofLPisslightlysuperiortothethatofSMTE,butitsdifferenceisnegligible. Therateofincreaseinlifetimeforthemultiplesinkcaseisgreaterthantherateforthesinglesinksystem.Thisfactgivesahintthatputtingmoresinksisbenecialtothelifetimewhenlargenumberofsensorsaredeployed.Probably,thestudyoftheimpactofthemultiplesinkssystemisagoodresearchissue. 34

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Figure 2-7 showstheperformancesoftheuniformnodedeploymentmethodandgrid-basednodedeploymentmethodinthebothsingleandmultiplesinksystem.Weobserve,intheperspectiveoftheperformance,thereisnosignicantdifferencebetweenbothdeploymentstrategies,asthenumberofthenodesincreases.Inmultiplesinkssituation,wealsoobservethesamephenomenonasthesinglesinkssetting.However,thelifetimeinthemultiplesinkssystemisfarbetterthanthatofthesinglesinksystem.Finally,weconcludethatthelifetimehasnothingtodowiththenodedeploymentstrategy,especiallyforthedensesensornetwork. 35

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Performanceofdifferentnodedeploymentstrategies equaltothecommunicationrange,theneighborsetNiofnodeisisthesetofallnodesresidewithinthiscircle.Thus,largercommunicationrangemeansthebiggerneighborset.Inaddition,neighborsetswithdifferentcommunicationrangeshaveaninclusionproperty,thatis,theneighborsetwithalargercommunicationrangemustincludetheneighborsetwithasmallercommunicationrange. Figure 2-8 showsthelifetimesandratiosofthesimulationsofvariouscommunicationranges.Inthesesimulations,thenumberofnodesisxedto400andthecommunicationrangevariesfrom3.5metersto10meters.Ineachsimulation,weusethesamenetworktopologyforthefairness.NotethatthelifetimesandratiosremainconstantforthecaseofMTEwithoutregardtothechangesofthecommunicationrange.IntheMTErouting, 36

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Performanceofdifferentcommunicationranges eachnodepreferstheshortestcosthopamongitsneighborset.Duetotheinclusionproperty,thisshortestcosthopneighborisincludedinthebiggerneighborsetwithalargercommunicationrange.Thus,anenlargedneighborsetduetotheincreasedcommunicationrangedoesnotinuencetheselectionofthenexthopduringrouting.However,tooshortcommunicationrangeperhapsdestroysthenetworkconnectivity.Inconclusion,aslongastheconnectivityofthenetworkisguaranteed,thecommunicationrangedoesnothaveaneffectontheperformanceofthenetworkwhoseroutingschemeisMTE. Inthissection,westudytheeffectsofthedifferentsizeofsensoreldsontheperformanceofthesensornetworks.Inthissimulation,wedonotxthenumberofnodestoaparticularvalue.Instead,wesetthenodedensitytoaconstantvalue. 37

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Performanceofdifferentdimensionofsensorelds Sincethenodedensityisxedandthedimensionofthesensoreldsvaries,thenumberofnodesalsovariesaccordingtotheareaofthesensoreldsandtheconstantnodedensity.Weassumeallsensoreldsshapesquare,andwejustchangesideofthesensoreldsfrom20to70.Thereforetheareasofthesensoreldsare400,900,1600,andsoon.Thenodedensityusedinthesimulationissetto0.16.Withtheseparameters,wecanndthenumberofnodesindifferentsizedsensorelds:64,144,256,andsoon.Theotherparametersaresettotheirdefault.Pleaserefertothetable 2-1 .Withthesamenodedensity,largersensoreldsmeansthedistancebetweennodesisthesamewithoutrespecttothesizeofthesensorelds,butthepathshavemorehopsinthelargersensoreld. Figure 2-9A showsthelifetimeofthesimulation.Asthedimensionofthesensoreldsgetslarger,thelifetimeofthenetworkgetssmaller.Thismaybeduetothelongerpaths(withmorehops)aswellasenlargedrelayingload.Ontheotherhand,theratioofsimulationkeepincreasingaccordingtothesizeofsensoreldsduetothedecreasinglifetime. 38

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Inourobservation,wefoundthatthemosteffectivewaytoincreasethelifetimeofthenetworksistoputmoresinknodesintothesensornetworks.Thestudyontheperformanceofthesensornetworkswithmultiplesinksmightbeoneofourfutureresearch. 39

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Thelifetimeofasensornetworkhasseveraldenitionsintheliterature.Oneofthemostpopulardenitionsistheintervalfromthetimewhenthesystemstartsoperationuntilthetimewhentherstnodeexhaustsitsenergy.Thisisequaltotheshortestlifetimeofthenodes.Muchworkhasbeenpublishedonhowtoprolongthelifetimeofsensornetworks[ 12 ][ 11 ][ 17 ][ 19 ][ 35 ][ 45 ].Someproposedenergyefcientroutingstrategies,whereasothersproposedefcientwaysofclusteringthenodestosaveenergy. Regardlessoftheenergy-savingstrategiesused,sensornetworksoftenexperienceunbalancedtrafcdistributionbecausethemulti-hoptrafcpatternistypicallymany-to-one[ 24 27 30 32 34 ].Sensornodesinthenetworkactasdataoriginatorsanddatarelayers.Thetrafctransmittedbyeachnodetypicallyincludesbothself-generatedandrelayedtrafc.Sincetheentirenetworktrafcowstowardthesink,thenodesclosertothesinktendtoexperiencemoretrafc.Asaresult,theirenergyconsumptionratestendtobehigherthanthosenodesthatarefarawayfromthesink,assumingthetransmissiondistanceisthesame.Thiscausesthenodesclosertothesinktodie 40

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23 24 43 ].In[ 30 ],theauthorsclaimedthatwhenthenodesonehopawayfromthesinkuseupalltheirenergy,theremainingnodeshaveusedonly7%oftheirenergyonaverage.Ithasbeenshownanalyticallyin[ 23 24 ]thattheenergyholeproblemexistsinvarioussensornetworks. Oncloserexamination,theenergyholeismostoftenobservedinnetworkswherethesensornodesarehomogeneousanduniformlydeployed,andtheyreporteventsgeneratedataconstantratetothesink.Ifweallownon-uniformdeploymentofthesensornodes,bycarefullyincreasingthenumberofnodesaroundthesink,wecanpreventthesensornodesnearthesinkfromdepletingtheirenergyfasterthanothers,andhence,resolvetheenergyholeproblem.Addingmorenodesinthesensoreldalsohasotherbenets,suchasbetterconnectivityandhigherreliability.Ontheotherhand,addingmorenodesmeansahighercost.Hence,thissolutionmakessenseinsituationswhereinexpensivesensorscanbemass-producedorhavingalongernetworklifetimeoutweighsthecostoftheextrasensors.Recentadvancesinmicro-electro-mechanicalandintegrationtechnologiesmaketherstsituationmoreandmorelikelytooccur. Thischaptercontributestotheresearchareathatseekstoextendnetworklifetimebydeployingthesensorsnon-uniformlyandbycarefullycontrollingthenodedensitiesindifferentpartsofthesensoreld.Themainquestiontobeaddressedishowmanynodesperunitarea(i.e.,thenodedensity)shouldbedeployedateachlocationinordertoachieveaprescribedlifetime-costobjective.Themainresultofthechapterisamathematicalmethodforcomputingthenodedensities. Thischapterillustratesthemethodusingaparticularexample,infact,aparticularobjective.Weshowhowtoderivethelocation-dependentnodedensitiesthatequalizetheenergydissipationrateofthesensornodesthroughoutthenetwork.Theresultis 41

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Althoughtheillustrationofthemethodusesanexample,themethoditselfisintendedtobegeneral.Itcanbeadaptedforotherlifetime-costobjectivesandotherconstraints,includingtheroutingstrategy.Forinstance,ifthecostofdeployingextrasensorsisnotnegligible,onecanincorporateacostfunctionofthenodedensitiesaspartoftheobjectiveandcomputetherequireddeploymentdensities. Themethodrequiresamodelfortheunderlyingroutingstrategy.Weconsiderseveralroutingmodels,whicharemeanttocapturetheessenceoftheunderlyingroutingprotocols.Fortractability,theroutingmodelsarenecessarilysimpleandmaynotfollowpreciselytheroutingprotocols.Someotherissuesarealsooverlookedinthischapter,suchasthealgorithmsandprotocolsfordeterminingwhichnodesbecomeactiveorinactiveineachregion.Theseissuesareeitherorthogonaltoourworkorlefttofuturestudies. Severalresearchershavestudiedtheenergyholeproblemandtheuneventrafcdistributionproblem[ 19 21 23 24 30 32 43 ].Amongthem,[ 43 ]isthemostsimilartoourworkinitsgoalofobtainingabalancedenergydissipationrateeverywherebynon-uniformnodedeployment.Theauthorsof[ 43 ]provedthatanunevenenergyconsumptionrateisunavoidableifallnodesarehomogeneousandaredeployeduniformlyinthenetwork.Intheirmodel,thesensoreldisdividedintoseveralconcentriccoronasorringsaroundthesink.Theygaveaheuristicroutingschemethatachievesanequalenergydissipationrateinallringsexcepttheoutmostone,providedthenumberofnodesincreasesgeometricallyfromtheoutmostringinward.However,theirsensoreld,energyconsumption,androutingmodelsaresignicantlydifferentfromours. 1. Thenodesin[ 43 ]cansenddataonlytothenodesintheneighboringring.However,eveninoursimplermodel,nodescansenddatatodifferentinnerringswithdifferentprobabilities.Wethenmakefurthergeneralizationsothatthe 42

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2. Thenodesin[ 43 ]generatedataatthesameconstantrate.However,thenodesinourmodelsmayhavedifferenteventgenerationrates. 3. Ourenergyconsumptionmodelismoregeneral:Therequiredtransmissionpowerofanodeisafunctionofthetransmissiondistancetothereceiver. Theresultingproblemofdeterminingthenodedensitiesisquitedifferentandmorecomplicatedinourcase.Furthermore,wecanachieveanequalenergydissipationrateforallnodesintheentirenetwork. In[ 30 ],itisassumedthatthesensornodesaredeployeduniformly.Thesensoreldisalsodecomposedintoconcentricringsaroundthesink.Theassumptiononroutingisthateachnodecanonlytransmitdatatoanodeintheinneradjacentring.Themainquestionaddressedby[ 30 ]ishowtodecidethewidthsoftheringssothatallnodesinthenetworkexhausttheirenergyatthesametime. In[ 32 ],theauthorsprovidedaformaldescriptionoftheproblemthattrafctendstobeconcentratedatthenodesclosetothesinkwhentheshortestpathroutingisused.Theysuggestedaheuristicalgorithmthatndssomecurvedpathstothesinkandshowedthatthetrafcloadismorebalanced. In[ 6 42 ],densitycontrolisusedasoneofthemeanstoguaranteethecoveragerequirementofthesensornetworkratherthantoprovideabalancedenergydissipationrateofthenodes.In[ 21 ],theauthorsstudiedtheimpactofcarefullycontrolleddeploymentofthesensornodesandthesinkonthedatacapacity,whichisdenedasthetotalamountofdatathatcanreachthesink.Theyproposedandanalyzedseveralapproachestoincreasethedatacapacity,andshowedthatnon-uniformdeploymentcanoutperformuniformdeployment. Manyauthorsformulatetheproblemofmaximizingthenetworklifetimeasoptimizationproblems[ 12 ],[ 26 ],[ 35 ],[ 45 ].In[ 35 ]and[ 45 ],theauthorsproposedapproximationalgorithmstosolvethemulticommodityowprobleminducedbythe 43

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17 ]calculatedupperboundsonthelifetimeofthenetworksthathaveregulartopologies,suchasaregularlineararrayandaregulartwo-dimensionalcircularnetwork. In[ 19 ],theauthorsproposedadynamicnode-clusteringschemeknownasLEACH,whereeachsensornodemayoperateasaclusterheaddependingonitsremainingenergyandtheclusterheadschangeduringtheoperationofthenetwork.TheycomparedthelifetimeresultingfromLEACHwiththeMTE(minimumtransmissionenergy)routinginwhicheachnodeusesthepaththatconsumestheleastamountoftotalenergyamongallpossiblepaths. Someresearchersdenethelifetimeofsensornetworksdifferentlyfromthetimeuntiltherstnodedies[ 47 ],[ 22 ].Theauthorsof[ 47 ]introducedthe-lifetime,whichisthetimeuntiltheremainingsensornodescanstillcoveratleastportionoftheentiresensoreld.In[ 22 ],theauthorswereinterestedinprolongingthenetworkoperationtimeaftertherstnodedies.Theyrecursivelymaximizethen-thminimumlifetimeofthenodesafter(n1)-thminimumlifetimeofthenodeshasbeenmaximized,foralln. 3.4 ). Forthissection,weassumetheshapeofthesensoreldisadiskandthesinkisatthecenterofthedisk(Figure 3-1 ).Thediskisdividedintoconcentricringshavingthesamewidthandthenalnodedensityineachringwillbeconstant.Weconsidertheroutingruleatthegranularityofrings.Thatis,allnodesinaringaresubjecttoanidenticalroutingrule,whichspeciesthenext-hopringratherthanthenext-hopnode.Thering-basedmodelingapproachhereistypical(See[ 30 ],[ 43 ].).Asaresult,we 44

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30 ],[ 43 ](SeethediscussioninSection 3.1 .),andisalreadyusefulforanumberofpracticalcases.LaterinSection 3.4 ,wewillmakegeneralizationthatallowssensoreldsofarbitrarytwo-dimensionalshapes,routingbetweentwoarbitrarylocationsintheeld,andnodedensityasafunctionofthepreciselocation. 3-1 withthesinkatthecenter.Thecommunicationcapabilityofthenodesislimitedsothatmulti-hoproutingisnecessarytodeliverthedatatothesink.Weintroducesomedenitionsandnotations. Theenergyconsumption/dissipationmodelofthesensornodesaffectsthenaldensityofeachring.Weadopttheenergydissipationmodelof[ 19 ].Therequiredtransmissionenergytosendoneunitofdatatoanodeatadistancedawayfromthesenderisgivenby whereistherequiredenergytooperatethetransceivercircuitry,isaparametercharacterizingthetransceiverampliersenergyconsumption,andistheso-calledpathlossexponent.Normally,isbetween2to6dependingontheoperating 45

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Sensoreldmodel environment.Someamountofenergyisalsorequiredtoreceiveaunitofdata.Theenergyconsumedatareceiverisasfollows. whereisthesameasin( 3 ).Normally,thereceivingenergyrequirementdoesnotdependonthedistancefromthetransmitter.Foreaseofpresentation,intheanalysistobeshownlater,wesometimesignorethereceivingenergy. Themaximumtransmissionrangeofasensornodeisalsoanimportantparameter.Inthissection,weassumethisrangeislrings,1ln.Thatis,asensornodecantransmitdatauptolringsawaywithoutrelaying.Wecallthismaximumrangethemaximumjump.Ifthemaximumjumpisequalton,theneverysensornodeisabletosenddatatothesinkdirectly,whichistheassumptionof[ 30 ]. Theroutingstrategyhasmajorimpactonthenaldensities.Wewillconsiderseveralroutingstrategieslater. 46

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Supposethenodesareuniformlydistributedoverringjwithadensityj,1jn.Consideranarbitrarynodeinringj.Sinceatypicalnodeinthesensornetworkbothgeneratestrafcasadatasourceandrelaystrafcforothernodes,thenode'stotaldatatransmissionrateisthesumoftherateofthelocally-generatedtrafcandtherateofthetrafcitrelays.Weassumeowconservationateverysensornode:Anodecannotbufferaninniteamountofdata,and,afterthetrafcisgenerated,thereisnofurtherin-networkprocessingthatmayreduceorincreasethetrafcvolumeatthenode.Thedatatransmissionrateatanodeinringjcanbeexpressedas whereSjistherateofthelocally-generateddata,andCjistherateofthetrafctoberelayedbythenode. LetusassumethatcertainamountofdatarateisneededtomonitoraunitofareaandthisrateisaconstantvalueofKthroughoutthesensoreld.Thisistheinherentdatarateneededforreportingeventsorconditionsaboutagivenarea.Thereareatleasttwopossibilitiesregardinghowthisinherentdatarateaffectstheactualtrafcrategeneratedbyeachnearbynode.Intherst,onecanassumethatthesystemhaslocalcoordinationamongthenearbysensorsthatreducestheamountoftrafcgenerated.Inthebestcase,everynearbynodegeneratestheminimumamountoftrafcsufcienttocoverthearea.Thatis,forringj,therateofthelocally-generatedtrafcatatypicalnode 47

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Thesecondpossibilityisthatthereisnolocalcoordinationamongthenodesandthenearbynodesallreportthesameeventstothesink.Inthiscase,thetrafcgeneratedbyanynodeinthenetworkisK.Betweenthetwoextremecasesarealargenumberofotherpossibilities,dependingonthedegreeoflocalcoordinationandotherfactors.Notethatthelocalcoordinationcantaketheformofsleep-wakeschedules,inwhichonlyasubsetofthenodesareawakeatanymoment.Or,therateofreportgenerationateachnodeismadeinverselyproportionaltothenodedensitynearby.Ineithercase,onecanexpectthatthelocallygeneratedtrafcrateatanodeisproportionaltoK=j,i.e.,Sj=K=jforsomeconstant>0.Inthissection,weassume=1fornotationalsimplicity.Ineffect,weassumetherstpossibility( 3 )asthelocaltrafcgenerationmodel.But,extensiontothecasewithageneralconstantistrivial.Forothermoredifferentmodels,theentiremethodologyinthesectionstillapplies,althoughtheresultswilldiffermoresubstantially. Therateoftherelaytrafcatanodedoesnothaveasimpleexpression.Itiseasiertowriteouttherecursiverelationshipitsatises.Foreachpairofringskandj,0j
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Recallthat,inmin(n,j+l),ringnistheoutmostringandlisthemaximumjump. Nowwecangetthetotaldatatransmissionrateforanodeinringj,1j
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Inourenergymodel,theenergyrequiredtotransmitaunitofdataisafunctionofthetransmissiondistance.Tosimplifytheanalysis,wemaketheapproximationthatthedistancebetweenapairofnodesisdeterminedbythenumberofringsseparatingthem.Theenergythatanodeinringj,1jn,consumesperunitoftimetotransfertheportionofitsdatadirectedtothenodesinringi,0i
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NotethatGnisgivenby( 3 ),whichdependsonlyonn.Throughout,ncanbeconsideredasagivenparameter.Hence,ifFk(j)hasnodependencyonanyk,onecancomputeGjusing( 3 )forallj.TheresultingGjisparameterizedbyn.Afterthat,onecancomputethedensitiesjiterativelyusing( 3 )foralljfromn1downto1. 3 )dependsontheunknownk.Onecanstillstartwith( 3 )andeliminateallGj'sfrom( 3 )byusing( 3 )and( 3 ).First,notethat Then,( 3 )canbere-writtenasfollows. (2j1)Fk(j)k. Theaboveexpressiondoesnotimplythat,ifnisgiven,thenonecancomputethedensitiesjforallotherj.ThisisbecauseFk(j)maydependonthedensitiesincomplicatedways.Thesetofequationsin( 3 ),for1j
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3 )doessuggestadifferentkindofiterativemethodtocomputeeachj,whichwillbecalledsuccessivesubstitution.Suppose,weinitializetheiterationatsomeconstant(0)jforall1j0foralli,1in.Itcanbeobservedthatif(j)n1j=1isasolutionto( 3 )givenn,then(j)n1j=1isasolutionto( 3 )givenn.Inthiscase,onecandecidethenodedensitiesasfollows:Chooseanarbitrarypositivevalueforn;computealljfor1j
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3 ),( 3 )and( 3 )(or,equivalently,( 3 ))tocomputetherequirednodedensities.Next,wewillconsidersomesimpleroutingschemesasexamplesandlatershownumericalresultsaboutthem.Manymoreroutingschemescanbemodeledsimilarly. Notethat,inthiscase,Fj(i)isindependentonthenodedensities.From( 3 ),Gjbecomes 3 ),thenodedensitiescanbecomputediterativelyfromn1to1bythefollowingexpression.j=nmin(n,l) min(j,l)Pj1i=(jl)+(ji) (2j1)1 min(j,l)k. 53

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Figure3-2. Uniformnodeselection Figure 3-2 illustratestheunderlyinggeometryforUniformNodeSelection.NodeXcanchooseanynodeintheshadedpartinthegureasthenext-hopnode.LettheregionwithinthecommunicationrangeofnodeXbedenotedbyQ.AsshowninFigure 3-2 ,nodeXcanchooseanext-hopnodeintheintersectionofQandtheinnerringsthatXcanreach,i.e,[j1k=(jl)+(Rk\Q).Whenthenodesinringkareuniformlydistributed,thenumberofnodesinRk\Qisk(Rk\Q),wherethenotation(S)representstheareaofaregionS.Thenumberofpossiblenext-hopnodesfornodeXisPj1k=(jl)+k(Rk\Q).Hence,theprobabilityFj(i)is,for(jl)+i
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2p TheareaofR(jl)+\Qisgivenby TheprobabilityFj(i)canbeobtainedbypluggingequations( 3 )and( 3 )into( 3 ).ByapplyingtheexpressionsforFj(i)to( 3 ),wederiveasetofequationsinjonly,fordifferentj.From( 3 ),notethatFj(i)dependsonvariousi.Hence,theresultingequationscannotbesolvedbyderivingeachifori=n1downto1iterativelyusing( 3 ).But,theymaybesolvedbysuccessivesubstitution,whichistoiterate((t)j)overtasin( 3 ). Notealsothat,previously,wehaveassumedthatthedistancebetweennodeXanditsnext-hopnodeisdeterminedbytheringsinwhichthetwonodeslie.Thisassumptionisnotaccurateinthecurrentmodel.Thedistancetothenext-hopnodedependsonwherethenext-hopnodeliesinitsring.Wewilllaterdescribeanextendedformulationthatincorporatestheaccuratedistancesbetweennodes,butwilldosoinamuchmoregeneralsettingwithrespecttootheraspectsaswell(Section 3.4 ).Thepricetopayis 55

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SinceFj(i)dependson(j),successivesubstitutionoftheformin( 3 )areneededtondthesolution. ThefollowingreasoningshowswhythismodelcanbeviewedasasimplicationoftheUniformNodeSelectionscheme.Supposetherangeofeachnodeislrw.However,thenodeonlyselectsanext-hopnodeinitsnearbylrings,wherellr.Inthiscase,theratiooftheareaofRi\QtothatofRk\Qcanbewellapproximatedby(2i1)=(2k1),for(jl)+
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3-3 ,wherethepathlossexponent,,is2.Wehaveconductedextensiveexperimentsforothervaluesof;buttheresultsareomittedforbrevity.InFigure 3-3 ,weshowboththeaverageper-nodeenergyconsumptionrateandthecalculatednodedensityineachoftherings.Severalobservationscanbemade.First,theaverageper-nodeenergyconsumptionratesoftheringsarenearlyidentical.Thisdemonstratesthatourmodelingapproachandanalyticalmethodarehighlyaccurate,andthatcorrectnodedensitiescanbederivedfromtheresultingmathematicalexpressions.Second,theshapeofthedensityfunction,asafunctionoftheringindex,issomewhatsurprisinginsomecases.Thefunctionsarenotevenmonotonicinthecaseofl=10orl=20. Inthecasesofl=1orl=2,thedensityfunctionismonotonicandincreasesveryfastastheringgetsclosertothesink.Itiseasytoexplainthecaseofl=1.Sincethemaximumjumpsizeis1,allthetrafcofanodemustowthroughtheadjacentringontheinside.Therefore,thetrafcloadbecomesheavierastheringgetsclosertothesink.Itisnecessarytodeploymorenodesintheringsclosertothesinksoastobalancetheenergydissipationratesacrosstherings.Asitapproachesthesink,theareaoftheringdecreaseswhilethenumberofnodesintheringincreases.Hence,thedensityincreasesfast. Forlargervaluesofthemaximumjump,e.g.,l=10,itisnotnecessarilytruethathighernodedensitiesarerequiredforringsclosertothesink.Thisismoreduetotheboundaryeffect.Inthiscase,eachnodecandirectlytransmititstrafctomultipleinsiderings.However,longertransmissiondistancerequiresmoreenergy.Anodeinoneofthelinner-mostrings(Ri,1il)hasfewerthanlringsleftontheinside. 57

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Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformringselection.=2. Hence,itsmaximumtransmissiondistanceislessthanlringsaway.Asaresult,ittendstoconsumelessenergyonaveragethananodeinaringfurtheroutside,sayRjforj>l.Theprecisesituationiscomplicated,dependingontheparametersoftheenergyconsumptionmodelandtheroutingprobabilities. 3 ).Theresultsforthecaseof=0aregiveninFigure 3-4 .Theaverageper-nodeenergyconsumptionratesinallringsarenearlyidenticalineachofthefourplots,whichcorrespondtol=1,2,10and20,respectively.Whenthepathlossexponent,,is0,ittakesaconstantamountofenergyforanodeto 58

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Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=0. Theresultsfor=1,2and3areshowninFigure 3-5 3-6 and 3-7 ,respectively.Inallplots,thecurvefortheaverageper-nodeenergyconsumptionrateisat.Thismeansthat,ifwedeploythenodesaccordingtothecomputeddensities,wecanachieveanevenenergydissipationrateinallrings.Theseresultsindicatethatourmodelingapproach,analyticalmethodandnumericalsolutionsareallaccurateorsound.Observethecurvesforthenodedensities,whichcanbequiteoscillatoryorirregular.Weseethatitishardtopredictthedeploymentdensitieswithoutprecisecomputation. 59

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Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=1. 3.4.1.1Node-densityindependentrouting 60

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Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=2. sink.OnceatransmissionreachesinsideB(),itisreceivedbythesink.WeletAbethesensoreldwithB()removed,i.e.,A=AnB().WewishtondthenodedensityinA. Considerapointy2A.Letg(y)bethetotaltrafcrateofanodeaty.Letc(y)betherateofthetrafctoberelayedbyanodeaty.Weagainassumethattherateofthelocallygeneratedtrafcateachpointisinverselyproportionaltothenodedensityatthatpoint.Then,thisrateatanodeisK=(y)forsomeconstantK>0.Wehave, 61

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Nodedensitiesandaverageper-nodeenergyconsumptionratesforvariousmaximumjumpsizes,l,underuniformnodeselection.=3. Letf(y,x)betheprobabilitydensityfunctionforanodeatytochooseanodeatlocationxasthenext-hopnode,x2A.ItsatisesRAf(y,x)dx=1foreveryy.3Notethatanodeatyingeneralmaynotbeabletoreachdirectlyeverywhereinthewholesensoreld.IftheregionthatitcanreachdirectlyisdenotedbyA(y),whereA(y)A,wecanassumethatf(y,x)isnon-zeroonlyonA(y),andRA(y)f(y,x)dx=1. 62

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Notethatc(y)dependsonthenodedensityfunction.But,wesuppressitinthenotationfornow.Then,wehave Afterrearrangement, g(y),y2A.(3) LetP(y)betheexpectedenergyconsumptionforanodeaty2A. LetybeanarbitrarypointontheboundaryofA,whosenodedensityisassumedtobeaknownparameter.OurobjectiveistohaveP(y)=P(y)forally2A,where Thisgivesthefollowing. Intheabove,weassumefisindependentonthenodedensityfunction.(y)isaconstant(parameter).Then,g(y)canbedeterminedforally2A.Then,( 3 )isalinearintegralequationwiththeunknownfunction.ItisknownasaFredholmequationofthesecondkind[ 28 ],whichhasthefollowinggeneralform. In( 3 ),isaknownconstant,(x)isanunknownfunctionandf(x)isaknownfunction,f(x)6=0,where,f:U!RforsomeURm.Thefunctionk(x,s)iscalleda 63

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3 ),VisasubsetofU.Forourcaseof( 3 ),=1,theunknownfunctionis,theknownfunctionisK=g,andthekernelisk(z,y)=g(z) IthasbeenshownbythetheoremsknownasFredholmAlternativesthatthesolutionto( 3 )nearlyalwaysexistsandisnearlyalwaysunique. Then,wecanwrite, Ifitisrequiredthattheper-nodeenergyconsumptionrateisequaleverywhere,theoutgoingtrafcfromanodeatysatisesthefollowing,whichalsodependson. Thefunctionsatisesthefollowingfunctionalequation. g(y,),y2A.(3) 64

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3 )isnotatypicallinearintegralequationandthemathematicaltheoryontheexistenceanduniquenessofthesolutionisunknownatthispoint.However,theearliermodelwiththeringstructureandenergy-dependentroutinginSection 3.2.2.2 isaspecialdiscreteanalogofthisand,there,ourcomputationexperiencehasshownthatsuccessivesubstitutionalwaysndsasolution.Therearegoodreasonstobelievethatasolutionto( 3 )oftenexistsandcanbefoundbysuccessivesubstitution. 65

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AlthoughthelifetimeofaWSNcanbedenedinmanyways,weadoptthewidelyuseddenition,whichisthetimeuntiltherstnodeexhaustsitsenergy.MuchworkhasbeendoneduringrecentyearstoincreasethelifetimeofaWSN.Amongthem,inspiteofthedifcultiesinrealization,takingadvantageofmobilityintheWSNhasattractedmuchinterestfromresearchers[ 7 14 16 25 31 36 38 40 41 ].WecantakethemobilesinkasanexampleofmobilityinaWSN.CommunicationinaWSNoftenhasthemany-to-onepropertyinthatdatafromalargenumberofsensornodesneedstobeconcentratedtooneorafewsinks.Sincemulti-hoproutingisgenerallyneededfordistantsensornodestosenddatatothesink1,thenodesnearthesinkcanbeburdenedwithrelayingalargeamountoftrafcfromothernodes.Thisphenomenonissometimescalledthecrowdedcentereffect[ 32 ]ortheenergyholeproblem[ 23 24 44 ].Itresultsinearlyenergydepletionatthenodesnearthesink,potentially 66

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ThischapterproposesaframeworktomaximizethelifetimeofaWSNbytakingadvantageofsinkmobility.Comparedwithothermobile-sinkproposals,themainnoveltyisthatweconsiderthecasewheretheunderlyingapplicationstoleratedelayedinformationdeliverytothesink.Oneoftheapplicationexamplesisbattleeldsurveillance,wheresensornodesaredeployedtomonitorthemovementofenemyvehiclesortroops.Amobilesinkattachedtoanunmannedaerialvehicleiesoverthemonitoredregionregularlytoharvestthecollectedintelligence.Toavoidbeinginterceptedordetectedbyenemyforces,themobilesinkneedstooperateinonlyafewsafelocationswithinalimitedoperationtime.Anotherexampleishabitatmonitoringwhereamobilerobotisusedtocollectinformationfromthesensornodesintheeld.Ifmuchofthehabitatareaisnotbeaccessiblebytherobotorifitisdesirabletominimizedisturbancetothetargetedanimalspecies,themobilerobotwilltracepredeterminedpathsandstopbyasetofpre-arrangedlocationsregularlyfordatacollection. Inourproposal,withinaprescribeddelaytolerancelevel,eachnodedoesnotneedtosendthedataimmediatelyasitbecomesavailable.Instead,thenodecanstorethedatatemporarilyandtransmititwhenthemobilesinkisatthelocationmostfavorableforachievingthelongestnetworklifetime.Tondthebestsolutionwithintheproposedframework,weformulateoptimizationproblemsthatmaximizethelifetimeoftheWSNsubjecttothedelayboundconstraints,nodeenergyconstraintsandowconservationconstraints.Anotheroneofourcontributionsisthatwecompareourproposalwithseveralotherlifetime-maximizationproposalsandquantifytheperformancedifferencesamongthem.Ourcomputationalexperimentshaveshownthatourproposalincreasesthelifetimesignicantlywhencomparedtonotonlythestationarysinkmodelbutalsomoretraditionalmobilesinkmodelswithoutdelaytolerance. 67

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Beingoneoftheearlypapersonextendingthenetworklifetimewithmobilityanddelaytolerance,thechapterfocusesonformulatingseveralsimpleandtypicallifetime-maximizationproblemsandevaluatingthelifetimeimprovement.Therecanbemanyvariantsoftheproblemformulation,someofwhichcanbeverydifcult,ofteninvolvingNP-hardcombinatorialsub-problems.Thedegreeoflifetimeimprovementdemonstratedbythischaptercanjustifyfurtherworkonmoredifcultproblems. Wenowbrieyreviewthemostrelevantworkonhowtoexploitmobilitytoincreasethenetworklifetime.In[ 36 ],theauthorsintroducedmobileagents,whichmovearoundandcollectdatafromnearbysensornodesonbehalfoftheimmobilesink.Whenthemobileagentsmovetothevicinityofthesink,theyforwardthecollecteddatatothesink.Inthatframework,communicationoccursonlyfromthesensornodestothemobileagentsorfromthemobileagentstothesinkviaasinglehop;thesensornodesdonot 68

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In[ 41 ],theauthorsformulatedalinearprogrammingproblemofdetermininghowtomovethemobilesinkandhowlongtoparkthemobilesinkateachstopalongthepathofthesinksoastomaximizethelifetimeoftheWSN.However,intheirmodel,dataowsarenotdecisionvariablesofthelifetimeoptimizationproblem.Onthecontrary,inourformulations,notonlythesinksojourntimesatdifferentsinkstopsbutalsotheroutingschemearedecisionvariables.Theanalysisandexperimentsin[ 41 ]wereconductedunderasimplestructurednetworktopologywherethesensornodesaredeployedinagrid-likepattern.In[ 7 ]and[ 8 ],theauthorsfurtherextendedtheresearchof[ 41 ].Themodelproposedin[ 7 ][ 8 ]includesthecostofmovingthemobilesink(suchasnodalenergyconsumptionforrouteestablishment/releasewhenthesinkmovestoanewstop)andthesinkmobilityratedeterminedbytheminimumsinksojourntimeatthesinkstops.Furthermore,themodelincorporatesahop-lengthlimitwhenthesinkmovestonextstop.Thisrestrictsthepacketlatency,whichisrelatedtothetravelingtimeofthesinkbetweenstops.TheauthorsproposedanMILP(MixedIntegerLinearProgramming)problemformulationtoobtaintheoptimaltravelrouteofthesinkandthesojourntimesatthesinkstopsformaximizingthelifetimeofthesystem.Theyalsosuggestedadistributedheuristicalgorithmtocircumventthecomplexityoftheproposedmathematicalformulation. Theauthorsof[ 25 ]showedthatthenetworklifetimecanbeextendedsignicantlyifthemobilesinkmovesaroundtheperipheryoftheWSN.Theyassumedthat,ifthemobilesinkcanbalancethetrafcloadofthenodes,thelifetimeofthenetworkcan 69

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TheproblemofndingthetrajectoryofthemobilesinksoastooptimizethelifetimeoftheWSNishardtosolveduetoitsinnitesearchspacewhenthelocationsforthesinkstopsarenotconstrained.In[ 38 ],theauthorsstudiedhowtondtheoptimalsinkstopsandthescheduleofvisittoeachofthestops.Ifthecandidatelocationsforthestopsareunconstrained,thisproblemisalsoNP-hard.However,ifthestopsareconstrainedtobeselectedfromanitesetofknownlocations,theproblemcanbeeasilyformulatedintolinearprogramming.Theyproposedanapproximationalgorithmtotheunconstrainedproblembyproperlydividingthewholesensoreldintoanitenumberofdisjointsmallareas,andthen,convertedtheunconstrainedproblemintoaconstrainedproblem.However,toobtainagoodapproximationratio,thenumberofsmallareascanpotentiallybeverylarge,makingthelinearprogrammingcomputationtime-consuming.Therefore,inthischapter,werestrictthesetofpotentialsinkstopstobefromasmallnumberofgivenlocationsratherthanfromarbitrarylocations. TheWSNmodelproposedin[ 31 ]isclosetoours.TheauthorsstudiedthemaximumlifetimeproblemoftheWSNwherethemobilesinkcanvisitonlysmallnumberoflocations.Theyshowedthatthelifetimecanbefurtherincreasedbyoptimizingnotonlythescheduleofsinkvisitsbutalsoroutingofthetrafc.However,theydidnotconsiderapplicationswheredelayedinformationdeliveryisallowed. First,wewilldescribethegeneralassumptionsabouttheWSNmodels.LetthesetofsensornodesbedenotedbyN.Forexperimentalconvenience,wesupposetheyare 70

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19 ],theenergyrequiredperunitoftimetotransmitdataattherateofxijfromnodeitojcanbedeterminedasfollows. whereCtijistherequiredenergyfortransmittingoneunitofdatafromnodeitojanditcanbemodeledasfollows[ 33 ]. whered(i,j)istheEuclideandistancebetweennodeiandj,andarenonnegativeconstants,andeisthepathlossexponent.Typically,eisintherangeof2to6,dependingontheenvironment.Here,theenergycostperunitofdatadoesnotdependonthelinkrate,andthisisvalidforthelowrateregime.Hence,weneedtoassumethatthetrafcratexijissufcientlysmallcomparedtothecapacityofthewirelesslink. Theenergyconsumedatnodeiperunitoftimeforreceivingdatafromnodekisgivenby[ 19 ] whereisagivenconstant.Hencethetotalenergyconsumptionperunittimeatnodeiis Weassumethateachsensornodehasthesametransmissionrange.Letldenotethesink.Inthispaper,wetaketheconventionthatthesinkisaspecialnodedifferentfromthesensornodesandl=2N.Therequiredenergyfortransmittingoneunitof 71

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4 )withjreplacedbyl.Wedenethe(downstream)neighborsofnodeiasN(i)=fj2N[flgjd(i,j)dg,whenthetransmissionrangeisd.Notethattheneighborsmayincludethesink. ThepaperdoesnotconsiderMAC-layercontention.ItisassumedthatcontentionisresolvedbysomeMAC-layerprotocol.TheoperationoftheMAC-layerprotocoldeterminesthelinkrates,whichareassumedtobelargeenoughsothattheydonotimposeaconstraintonthedatarates.Futureworkmaytrytorelaxtheseassumptions.Conversely,ifthedataratesaresmall,thenevensimpleMAC-layerprotocolswillbeabletodelivertherequiredlinkrates.Inotherwords,itcanbeeasytodesignoneofsuchprotocols. 1 15 ].Letxcijbetherateassignmentfromnodeitothenodejforthetrafcgeneratedbynodec(commodityc).Theproblemofmaximizingthelifetimeinthismodelisformulatedasfollows[ 11 12 ]. 72

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Theconstraint( 4 )istheowconservationconstraint,whichstatesthat,atanodei,thesumofalloutgoingowsforacommoditycisequaltothesumofallincomingowsforthecommodityc.Ifi=c,theincomingowsshouldincludetheowsgeneratedatnodeiitself,ordi.Theinequality( 4 )istheenergyconstraintanditmeansthatthetotalenergyconsumedbyanodeduringthelifetime(Z)cannotexceedtheinitialenergyofthenode.Withthisformulation,theroutingisdynamicandallowsmultipathcommunications.Thereisnoassumptiononxed-pathrouting,suchastheshortestpathrouting.Theaboveoptimizationproblemcanbeeasilyconvertedintoalinearprogramming(LP)problem. Theparticularformulationaboveisequivalenttothefollowingformulation,wheretheowsofthecommoditiesareaggregatedintoasinglearcow.Thenewformulationhasmuchreducedcomplexityandisusefulforndingnumericalsolutionsquickly. 73

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Here,xijistheaggregateowrateofallcommoditiesfromnodeitonodej,i.e.,Pcxcij=xij.Theequivalenceoftheproblems,( 4 )and( 4 ),canbearguedasfollows.Clearly,wecanalwaysconstructafeasiblesolutiontoproblem( 4 )fromanyfeasiblesolutiontoproblem( 4 )bylettingxij=Pcxcij.Conversely,givenafeasiblesolutionfxijgtoproblem( 4 ),onecanapplytheowdecompositionalgorithm[ 1 ]tothearcowsfxijgandobtainpathowsforthecommodities2.Thepathowsinturngivetheper-commodityarcowsfxcijgfeasibletoproblem( 4 ). 4 )is 4 ),theowsoneverycyclemustbezero.Hence,wecanrestrictourselftothesetoffeasibleowsthatcanbedecomposedintopathowsonly. 74

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4 )isnotnecessarilyusefulifonewishestoincorporatemoreconstraints.Butitisusefulinthispaperbecauseitiseasiertocompute. 38 41 ]. Aspreviousauthors[ 38 ],throughoutthepaper,wemaketheassumptionthatthetravelingtimeofthesinkbetweenlocationsisnegligible.Thisway,theresultingproblemformulationsaresimpleenoughforustoobtainprecisenumericalsolutionsforevaluationpurpose.Theassumptionisappropriatewhenthetravelingtimeismuchsmallerthanthetimespentbythesinktocollectdataineachlocation. Inthismodel,theorderofvisittothestopshasnoeffectonthenetworklifetimeandcanbearbitrary.Thesinksojourntimeatalocationl2Lisdenotedbyzl;itisthetimethatthesinkspendsatltocollectdatafromthesensornodes.TheoverallnetworklifetimeisZ=Pl2Lzl.Whenthesinkisatstopl,wedenotethe(downstream)neighborsofnodeias Tondtheoptimalnetworklifetime,weneedtoconsidertheroutingofthetrafcaswellasthedurationofthesink'ssojourntimeateachstop(alsosee[ 16 31 38 41 ]). SimilartothecaseofthestaticsinkmodelinSection 4.2.1 ,thereisaper-commodity-basedformulationofthelifetime-maximizationproblem,andthereisanequivalent,simpler,aggregate-trafc-basedformulation.Forbrevity,wewillonlypresentthelatter.However,were-iteratethat,ifadditionalconstraintsarepresent,theper-commodity-based 75

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Constraint( 4 )denotestheowconservationforallnodeswhenthesinkisatl.Constraint( 4 )saysthatthetotalenergyconsumedatthenodeicannotexceedtheinitialenergyEi.Bymultiplying( 4 )withzlandsubstitutingx(l)ijzlwithanewvariabley(l)ij,wecanreplace( 4 )withthefollowingnewconstraint. Similarly,constraint( 4 )canbechangedinto Withtheconstraints( 4 ),( 4 ),( 4 ),andthenon-negativityconstraintsfory(l)ij,theaboveoptimizationproblemisconvertedintoanLPproblem.Here,y(l)ijisinterpretedasthetotaltrafcvolumethatnodeisendstonodejwhilethesinkstaysatl. 76

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Examplesofthestaticsinkmodel(SSM),mobilesinkmodel(MSM),anddelaytolerantmobilesinkmodel(DT-MSM) LetDbethemaximumtolerabledelay,orthedelaytolerancelevel.Weassumethatthesinknishesoneroundofvisittoallthestops(wherethesinkstaysforapositivedurationtocollectdata)inDtimeunits,andthen,repeatswithanotherroundagainandagain.NotethattwoconsecutivevisitstothesamestoptakesatimeD. Let'stakeanexampletoshowhowourframeworkcanoutperformotherones.Considerthetwo-nodeexampleshowninFigure 4-1 .N1andN2aretwosensornodesandL1andL2arethecandidatestopsofthemobilesink.Supposeweignorethereceivingenergyrequirementandsupposethetransmissionenergyperunitofdataisequaltothesquareofthedistancebetweenthesenderandthereceiver.BothnodesN1andN2generatedataat1bpsandhave100unitsofenergyinitially.IfthesinkislocatedatOintheSSM,bothnodesspend4unitsofenergyforsendingabitofdata.Itisobviousthattheoptimallifetimeis25seconds.IntheMSMwithsinklocations 77

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UnliketheMSMorSSM,thesinkintheDT-MSMcancollectdatafromonlyasubsetofthesetofallsensornodes,N,ateachstop.LetRlbethesubsetofNsuchthatonlynodesinRlcanparticipateinthecommunicationtowardthesinkwhenthesinkisatl2L.WecallRlthecoverageofthesinklocationl.NotethattheunionofRloverl2Lmustbethesetofallsensornodes,N.Inotherwords,anysensornodeshouldbecoveredbyatleastonesinklocation.WhenthenodeiisinRl,nodeiissaidtobeactiveatl2L.AlthoughwecanconstructRlinmanywaysdependingontheapplicationofinterest,inthispaper,averysimplemethodofconstructingRlisconsidered.Fixapositivenumberr.Wecallrtheradiusofcoverageofthesink.Foreachl2L,ifd(i,l)r,wherei2N,theni2Rl.Here,theradiusofcoverageofthesink(r)shouldbelargeenoughsothateverysensornodebelongstoatleastoneRl.Notethattheminimumrdependsonthelocationsofthesinkstops. 78

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Foreaseofpresentation,weassumethesinkvisitsalllocationsinLintheorderof1!2!jLj!1.Thesinkmaystayatsomelocationforzerotime.Withslightabuseofterminology,wedenethenetworklifetimeTtobethenumberofcyclesmadebythesinkuntiltherstnodediesduetoenergyexhaustion.TheactuallifetimeisTD. Oncetrafcisallowedtobebuffered,therearedifferentstrategiesonwhosetrafcisbuffered.Whichstrategygetsadoptedinpracticeprobablydependsontheapplication,otherpracticalconcerns,andthedesigner'spreference.Sincewedonotknowthesefactorsinadvance,wenextdescribetwostrategies,ortwovariantsofthemodel:thesub-ow-basedmodelandthequeue-basedmodel.Themainpurposeistoillustratethatchoicesexistandtheyleadtodifferentperformance-complexitytradeoffs. Again,letx(c,l)ijbetherateassignmentfromnodeitothenodej,whilethesinkisatl,forthetrafcgeneratedbynodec(commodityc).Letx(l)ijbetheaggregatedrateof 79

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Sinceatnodei2N,thecommodityorsub-owofothernodesc2Rl,c6=imustbeforwardedassoonasithasbeenreceived,wemusthave Here,wedeneNl(i)=Rl\N(i,l),whereN(i,l)isasgivenin( 4 ).Theowconservationatnodeicanbeexpressedasfollows,whichisthesameasintheMSMexceptthattheamountsoftrafcoriginatedfromnodeiitself,(w(l)i;l2L,i2Rl),arenowdecisionvariables.zl0@Xj2Nl(i)x(l)ijXk:i2Nl(k)x(l)ki1A=w(l)i. Thedatabufferedduringtheprevioussink-movementcyclemustbeclearedinthecurrentcycle.Thisrequirementcanbewrittenas 80

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Theaboveisaper-commodity-basedformulationoftheproblem.SimilartothecaseoftheSSMproblem( 4 )inSection 4.2.1 ,thereisasimpler,equivalent,aggregate-trafcformulation,usingonlytheaggregatearcowvariablesx(l)ij=Pc2Rlx(c,l)ij.Wecandoawaywithconstraint( 4 )intheresultingformulation. 81

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Theequivalenceofthetwoformulationscanbearguedasfollows.First,itisclearthatthefeasibilitysetoftheper-commodity-basedformulationisinsidethefeasibilitysetoftheaggregate-trafc-basedformulation.Conversely,givenafeasiblesolutionfx(l)ij,w(l)i,zlgtothelatterformulation,wecantreatw(l)i=zlasthesupplyateachnodeiwhenthesinkisatstopl.Foreachl,wecandecomposethearcowsfx(l)ijgintopathowsforallthecommodities,fx(c,l)ijg(again,withoutlossofgenerality,weassumethedecompositiondoesnotleadtopositivecycleows),satisfying( 4 ),( 4 )and( 4 ). Hence,fx(c,l)ij,w(l)i,zlgisafeasiblesolutiontotheper-commodity-basedformulation. 82

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Theenergyconstraintscanbeexpressedinthesamewayasinthesub-owbasedMSM.Fromtheabovediscussion,wehavethefollowingoptimizationproblemformaximizingthelifetime. TheproblemshownabovecanbeconvertedintoanLPproblembysubstitutingy(l)ijforzlx(l)ijandintroducingthenewvariableu=1=T.Thislinearizationmethodcanalsobeappliedtothesub-owbasedMSM.

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Forillustration,considertheoptimizationproblemforthesub-ow-basedmodel.Dependingontheradiusofcoverage,wemayobtaindifferentinstancesoftheoptimizationproblem.Thus,wecanparameterizetheseinstancesaccordingtotheradiusofcoverage.LetP(N,L,r)betheoptimizationproblemwhentheradiusofcoverageofthesinkisr,thesetofsensornodesisN,andthesetofsinklocationsisL.Thevaluermustbelargeenoughsothatallsensornodescanbecoveredbyatleastonesinklocationandwedenotethisminimumradiusofcoverageforconnectivitybyr0.UnderthesamecongurationwithNandL,differentrvaluesonlyaffectRlandNl(i).WewillusethenotationsRl(r)andNl(i,r)ifitisnecessarytospecifytheradius 84

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Proof. Supposethat(^x,^w,^z,^T)isafeasiblesolutiontotheproblemP(N,L,r1).Now,considerequation( 4 )fortheoptimizationproblemP(N,L,r2).For8l2L,8i2Rl(r2), WehavethefollowingbyseparatingtheneighborsetsintoA,A,B,andB. Fixl2L.Supposei2Rl(r1).Weextendthevector^xsothat^x(l)ij=0whenj2Aand^x(l)ki=0whenk2B.Then,forsuchlandi,theextendedvector(^x,^w,^z,^T)satises( 4 )sincetheoriginalvectorsatises( 4 )fori2Rl(r1). Nextsupposei2Rl(r2)nRl(r1).Then,wecanextendthevector^xfurtherbysetting^x(l)ij=0forallj2Nl(i,r2)and^x(l)ki=0forallksuchthati2Nl(k,r2).Furthermore,weextend^wbysetting^w(l)i=0.Aftersuchextension,(^x,^w,^z,^T)satises( 4 )fori2Rl(r2)nRl(r1)(sincealltermsarezero). 85

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4 ),wecanapplyasimilarprocedure.Hence,wecanconcludethatanyfeasiblesolutiontotheproblemP(N,L,r1),aftersuitableextension,isalsoafeasiblesolutiontotheproblemP(N,L,r2). Next,thequeue-basedmodelislessconstrainingthanthesub-ow-basedmodel;thisresultsinlifetimegainsintheformermodel.Thefollowingtheoremformalizesthefactthatthequeue-basedmodelalwaysoutperformsthesub-ow-basedmodel. 4 ),andTbetheoptimalobjectivevaluetoproblem( 4 )withthesameconguration(N,L)andthesameradiusofcoverager.Then^TT. Proof. 4 ).Wewillprovethistheorembyconstructingafeasiblesolutiontoproblem( 4 )with((^x(l)ij),(^w(l)i),(^Zk),^T)andshowingthatunderthisfeasiblesolution,theobjectivevalueofproblem( 4 )is^T. Wenowdeneavectorwasfollows.Foreachl2L,weletw(l)i=^w(l)iifi2Rl,andw(l)i=0otherwise.Then,weletq(0)i=Ddi,andq(l)i=q(l1)iw(l)iforalli2N.Wehavethefollowingsequenceofassignmentsfortheq()i. 4 )andtheconstructionofw.Hence,( 4 )issatised.Sincethecongurationandradiusofcoveragerforproblem( 4 )arethesameasthoseforproblem( 4 ),Nl(i),i2N,l2Larethesameforbothproblems.Becauseofthisandby( 4 )andw(l)i=q(l1)iq(l)i,( 4 )issatised.Theenergyconstraints( 4 )and 86

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4 )areidentical.Hence,giventheoptimalsolution((^x(l)ij),(^w(l)i),(^zk),^T)toproblem( 4 ),wejustconstructedafeasiblesolution((^x(l)ij),(q(l)i),(^zk),^T)toproblem( 4 )withthesameobjectivevalue^T.Hence,T^T. Inthefollowingtheorem,weshowthatthemaximumlifetimeofthesystemisthesameforallvaluesofD.Here,themaximumlifetimeofthesystemisequaltotheproductofDandthecorrespondingoptimalobjectivevalueT(D). Proof. Letx=(D D0)x(D0),q=(D D0)q(D0),z=z(D0),T=(D0 4 )-( 4 ).Sinceitisobviousthatthesolution(x,q,z,T)satisestheconstraints( 4 ),( 4 ),( 4 ),( 4 ),and( 4 ),wefocushereonconstraints( 4 ),( 4 ),and( 4 )only.Sincetheoptimalsolution(x(D0),q(D0),z(D0),T(D0))isfeasibletotheproblemP(D0),itmustsatisfyconstraint( 4 ).Next,letusplug(D0 4 )intheplacesforx(D0),z(D0),andq(D0),respectively.Then,wehave 87

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D0)Tintheplacesforx(D0),z(D0)andT(D0)onthelefthandsideofconstraint( 4 ),wehave D0.(4) AftercancelingDandD0,itiseasytoseethatthenewsolution(x,q,z,T)satisestheenergyconstraintoftheproblemP(D). Fromtheconstraint( 4 )fortheproblemP(D0),wehaveq(0)i(D0)=D0di.Sinceq=(D D0)q(D0),q(0)i(D0)=q(0)i(D0 Fromaboveargument,wehaveshownthatnewsolution(x,q,z,T)isfeasibletotheproblemP(D).Hence,wehaveT(D)T=D0 Usingasimilarargument,wecanalsoconcludethatT(D)DT(D0)D0.Hence,T(D)DmustequaltoT(D0)D0 88

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Experimentalparametersandtheirvalues #ofsensornodes 500J Datagenerationrate(di) 500bps Figure4-2. ComparisonoflifetimesofMSMandDT-MSMunderthevariousradiiofcoverage Wehaveexperimentedwithdifferentparametersextensively,suchasthenumberofnodes,thenumberofpossiblesinklocationsandtheparametersfortheenergyconsumptionmodel.Onlyasmallsubsetoftheresultsarereportedhereforbrevity.InTable 4-1 ,weprovidethesystemparametersandtheirvaluesforthereportedexperimentsinthispaper.Weadoptthedataforthelastfourparametersfrom[ 18 ].Inallexperiments,weuseGLPKforsolvingthelinearprogrammingproblem. First,wewouldliketomentiontheimpactoftheradiusofcoverageofthesinkontheperformanceoftheDT-MSM.Forthisexperiment,thepositionsfor100nodesand20mobile-sinklocationsarerandomlygenerated(jNj=100,jLj=20)ina 89

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B200-nodenetwork Lifetimeagainstthenumberofsinklocations;maximumcoverage;e=2.0 4-2 .Notethat,inthegure,thelifetimeisnormalizedtotheoptimallifetimeoftheMSM.Asshowninthegure,thelifetimeoftheDT-MSMincreasesastheradiusofcoverageincreases,whichisconsistentwithTheorem 1 .Theincreaseisthesharpestwhentheradiusjustexceedstheminimumradiusrequiredtocoverallnodes.Afterthat,furtherincreaseoftheradiushasanegligibleeffect.Recallthat,whenthemobilesinkreachesoneofthestops,sayl,onlythosesensornodesinthecoverageofl(i.e.,Rl)cancommunicate.ItisgenerallydesirableforRltohaveasfewnodesaspossible,sincethisreducesthecommunicationandcoordinationcomplexity.Theaforementionedbehavioroflifetimeincreaseisdesirable. Next,wecomparethelifetimesofmodelsundervariousnumbersofthesinklocations.Thenumberofnodesissetto100or200,andthepathlossexponenteis2.0.Thecoverageissetlargeenoughtoalwayscovertheentiresensoreld.Werantheexperiment100timesforeachconguration.ThelifetimesoftheMSMandDT-MSM 90

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B200-nodenetwork Lifetimeagainstthenumberofsinklocations;minimumcoverage;e=3.0 4-3 ,thelifetimeoftheMSMisabout100%200%greaterthanthatoftheSSM.However,theDT-MSMis200%1000%betterthantheSSM.Moreover,thecurvesalllooklinear;theperformancegapcangrowevenlargerwithmoresinklocations. Interestingly,thelifetimeoftheMSMincreasesveryslowlywiththenumberofsinklocations.Asexplainedin[ 38 ],intheoptimalsolution,onlyafewlocationsfromthesetofsinklocationsarechosenasthetruestopsforthesink.However,inDT-MSM,therateoflifetimeincreaseissubstantialasjLjincreases.ThisisbecauseeachnodecanhavebetterandbettersinklocationasjLjincreases,anditisnotforcedtoparticipateinthecommunicationwhenthecurrentcoverageisnotthemostfavorableforenergysaving,evenifthenodemaybelongtothatcoverage.ThisisnotpossibleintheMSMbecausenomatterwherethesinkstops,everynodemustparticipateinthecommunication. Wewishtomakethefollowingremarks.First,ourformulationsandreportedexperimentsallusetheoptimalroutingwithrespecttomaximizingthesystemlifetime.Theroutingstrategyisimportantforincreasingthesystemlifetime.Forinstance,basedonourexperiences,whentheshortestpathroutingisusedinthestaticsinkmodel(resultsnotshown),thelifetimeperformanceisquiteinferiortothecaseofoptimalrouting.Second,inourmodel,thelocationsofthesinkstopcandidatesarerandomly 91

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BUndermaximumcoverage Lifetimesversustransmissionrange,d:jNj=200,jLj=20,e=2.0 Weconductsimilarexperimentswiththesamecongurationbutminimumcoverage.TheresultisshowninFigure 4-4 .AlthoughtheslopeoflifetimeincreaseoftheDT-MSMisloweredwhencomparedtothemaximumcoveragecase,theincreasepatternissimilar.Althoughalargersetofsinklocationsincreasesthenetworklifetime,itcanbeundesirableifthesink-travelingtimecannotbeignored.ThelongertravelingtimemayexceedthedelaytolerancelevelD.Therefore,thereisatradeoffbetweenthegainfrommoresinklocationsandthedelayorothersystemcosts. InFigure 4-5 ,weshowthelifetimesofthethreemodelsundervariousvaluesforthetransmissionrange.Thetransmissionrangedetermineswhetheralinkexistsbetweenapairofnodes.Whetheranexistinglinkisusefulornotdependsontheradiusofcoverage:Anodecannotusealinktoanothernodeifthetwonodesarenotinthecommoncoveragearea. BoththeMSMandtheDT-MSMexhibitasharplifetimeincreasewhenthetransmissionrangeissmallbutincreasing.However,asthetransmissionrangebecomeslarge,thelifetimeincreasecomestoastopforallthreemodels.Thisisbecausetheenergycostincreaseswiththetransmissiondistance,andhence,inan 92

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93

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LetNidenotethesetof(downstream)neighbornodesofnodei,i.e.,N(i)=fjj(i,j)2Ag.Letc:A!R+beagivencostfunctionontheedgeset.Thecostc(i,j)istherequiredenergytosendaunitofdatafromnodeitojanditisusuallyafunctionofthedistancebetweeniandj.LetLbethesetofthesinklocationsindexedfrom1throughL. 94

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DT-MSMwasintroducedin[ 46 ].Itissuitableforanapplicationthatcantolerateacertainamountofdelay.InDT-MSM,eachnodecanpostponethetransmissionofdatauntilthesinkisatthelocationmostfavorableforextendingthesystemlifetime.However,thereisusuallyamaximumdelaythattheapplicationcantolerate.ThismaximumdelaytoleranceisdenotedbyD.Thesinkmustcompleteoneofitstoursfromnode1toLandbackto1withinDtimeunitsandthenrepeatsthesametourinthenextround. SinceeachtourtakesDtimeunits,theproblemofmaximizingthesystemlifetimeistomaximizethenumberoftours,whichisdenotedasT.TheactuallifetimeisTD.Thedecisionvariablesarehowmuchtimethesinkstaysateachlocationl2Lwithineachtour,denotedbytl,andwhattherateofdatatransmissionfromnodeitojwillbe,denotedbya(l)ij.Notethatinanoptimalsolution,themobilesinkdoesnotnecessarilyvisitallthesinklocations.Inthatcase,westillletthesinkvisitsuchanode;butthetimeofstayis0. Itturnsout,intheproblemformulation,tlanda(l)ijalwaysshowuptogetherintheformtla(l)ij.Wecandenex(l)ij=tla(l)ijtoreplacetla(l)ij.Clearly,x(l)ijcanbeinterpretedasthetrafcvolumeonthelink(i,j)whenthesinkisatl.Wewilltaketheviewoftrafcvolumeinthefollowingdiscussion. Unlikethenon-delay-tolerantmobilesinkmodel,where,regardlesswherethesinkis,everysensornodeimusttransmitallnewlygenerateddataatthedata-generationratedi,asensornodeinDT-MSMcantemporarilydelaydatatransmissionbystoringthedatainthelocalbuffer.Withdelaytoleranceanddatabuffering,thereisanotherexibilityofsimilarnaturethatonecantakeadvantageof.Whenasinkisatlocationl,itmayberequiredtocollectdataonlyfromnearbysensors.Thissetofsensorsisdenoted 95

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Thus,foreachlocationl,thereisagraphGl=(N[flg,Al),whereAl=f(i,j)2Aji2Rl,j2Rl[flgg.Whenthesinkisatlocationl,the(downstream)neighborsetofnodeiisdenotedbyNl(i)=fjj(i,j)2Alg. WecreateanexpandedgraphfromthegraphsGl,l2L.Aswemakeclearshortly,thelifetimemaximizationproblemwillbeanetworkowproblemontheexpandedgraph.InFigure 5-1 ,weshowanexampleoftheexpandedgraph.Somedetailsaboutitsconstructionareasfollows. 1. StarteachcolumnwithGl,foralll2L. 2. RelabelnodeiinGlasi(l). 3. Addavertexs,whichrepresentsthesink. 4. Foreachl,replacetheedge(i(l),l)with(i(l),s)andremovenodelfromGl. 5. Foreachi(l),l=1,...,L1,addanedge(i(l),i(l+1)). 6. Setthesupplyatnodei(1)tobeDdiandthedemandatnodestobeDPi2Ndi. Thecostofeachverticaledge(oftheform(i(l),j(l)))isassignedasfollows: Thecostofeachhorizontaledge(oftheformof(i(l),i(l+1)))issettobe0,becausetheheadandtailofthistypeofedgearethesamephysicalnodeandrealcommunication 96

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ExpandedgraphofDT-MSM doesnotoccur.Letx(l)ijandy(l)ibethetrafcvolumeonedge(i(l),j(l))and(i(l),i(l+1)),respectively. Theowconservationlawatthesensornodesisasfollows. Theinterpretationisthefollowing.AtthebeginningofacycleoflengthDtimeunits,nodeihasaccumulatedDdiamountofdata,whichwasgeneratedinthepreviouscycle.Thisamountofdatamustbedeliveredtothesinkbytheendofthecurrentcycle.x(l)ijistheamountofdatasentonedge(i,j)whenthesinkisatlocationl;y(l)iistheamount 97

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Inaddition,atthesink(nodes),allarrivaltrafcmustbedrained.Thus,wehave Theproblemweaddressinthispaperistomaximizethenumberofrounds(orcycles),T,madebythemobilesinkwhilemaintainingtheowconservation( 5 )and( 5 ),subjecttotheenergyconstraintsatthesensornodes.Moreprecisely,theproblemcanbewrittenasfollows.maxT 52 ),( 53 ) Constraint( 5 )meansthatthetotalenergyexpenditureatanodeduringTroundsshouldbelessthanorequaltothenode'sinitialenergyendowment.Theaboveproblemcanbeeasilytransformedintoalinearprogrammingproblem,whichwillbeshownnext. 5.2 .Thefollowingistheequivalentlinearproblem,whichisobtainedfromthemaximizationproblemof( 5 )-( 5 )byreplacing1=Twithz.Forconvenience,wealsodeneM=PNi=1Ddi,y(0)i=Ddiandy(L)i=0foralli2N. 98

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NotethatMisanupperboundofanytrafcvolume,atermthatalsoincludesthebuffereddata.Wewillusethetermsowandvolumeinterchangeably.Thenewformulationhastheinterpretationthatitminimizesthemaximumenergyconsumptionamongallnodesinasingleround,normalizedwithrespecttoEi,(PLl=1Pj2Nl(i)e(l)ijx(l)ij=Ei)whilesatisfyingowconservation. Insteadoftacklingtheproblem( 5 )-( 5 ),wewouldliketoconsiderthefollowingproblem,whereequalitiesinowconservationsarechangedintoless-than-or-equal()inequalities.Laterwewillprovethatthesedifferentformulationsare,infact,equivalent. 99

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Weuseanordered-pairnotation(i,l)todenotetheindexoftheowconstraint( 5 )ofthenode(exceptthesink)iwhenthesinkisatl.Foragiven(x,y),iftheowconstraint(i,l)satisesequality,wesaythatconstraint(i,l)isbinding.(Someauthorsmayusethetermtightoractiveinstead.)Thatis,thebindingconstraint(i,l)impliesthatPj:i2Nl(i)x(l)jiPj2Nl(i)x(l)ij+y(l1)iy(l)i=0. 5 )ifonlyifallconstraint( 5 )for(x,y)arebinding. Proof. 5 )isfeasiblesolution.TOprovetheoppositedirectionofthelemma,let'ssuppose(x,y)isfeasibleandsomeconstraintof( 5 )arenotbinding.LetKbethesetofanorderedpair(i,l),suchthatPj:i2Nl(i)x(l)jiPj2Nl(i)x(l)ij+y(l1)iy(l)i<0. Wemaywriteaunbindingconstraint(i,l)asfollows. 100

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5 )overi2N,l2L,weget AftercancelingcommontermsinLHSof( 5 ),wehave Recallthaty(0)i=diandy(L)i=0fori2N.However,sinceP(u,v)2K(v)u>0,theRHSof( 5 )isnegative.Hencewehavethefollowinginequality.LXl=1Xj:s2Nl(j)x(l)js+M<0,LXl=1Xj:s2Nl(j)x(l)js>M Thisisacontradictiontothecontradictionthesecondpartofconstraint( 5 ). Therefore,for(x,y)tobefeasible,allconstraintsintherstpartof( 5 )mustbebinding.Andasaconsequence,PLl=1Pj:s2Nl(j)x(l)js=M. 5 )-( 5 )and( 5 )-( 5 )areequivalent. Proof. 1 Next,weshowthattheremovalofthesecondconstraintin( 5 )doesnotaffecttheoptimalobjectivevalue.Themodiedformulationisthenalonewhichwe 101

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Letbe^zandzbetheoptimalvalueoftheformulation( 5 )-( 5 )andformulation( 5 )-( 5 )respectively. 5 )-( 5 )andformulation( 5 )-( 5 )respectively.Then,^z=z. Proof. 5 )-( 5 )isbiggerthanthefeasiblesetof( 5 )-( 5 ). SinceTheorem 5.1 saysthattheformulations( 5 )-( 5 )and( 5 )-( 5 )areequivalent,wecomparetheoptimalobjectivevalueof( 5 )-( 5 )tothatof( 5 )-( 5 ).Let(z,x,y)beoneofoptimalsolutionoftheformulation( 5 )-( 5 ).Theinequalityconstraint( 5 )canbetransformedintoequalityconstraintbyaddingslackvariables.Xj:i2Nl(j)x(l)ijXj2Nl(i)x(l)ji+y(l)iy(l1)i0 102

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5 )-( 5 )withconstraint( 5 )replacedbytheaboveequalityconstraintwithslackvariables. Supposethat^z0.Theaboveassignmentsatisesthefollowingproperty. Ifweconstructedgeowfromthenewpathowassignment,f,theedgeowassignmentsatisestheowconservationconstraintas( 5 ). Theenergyconstraint( 5 )canbealsorepresentedasapathowasfollows. Putting^f(p)=f(p)+^f(p)(sli=P^f(p))intotheenergyconstraint,weget 103

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Hence,ournewpathow(orequivalentedgeow)canhaveanobjectivevalue,sayz,suchthatz<^z,whichisthecontradictiontoourpremise. Therefore,^z=z. Now,weturnourattentiontoderivingthealgorithmbyLagrangianrelaxation.Let(l)ibetheLagrangemultipliersassociatedwiththeconstraintsin( 5 ).TheLagrangianfunctionof( 5 )is where=((l)i),overi2N,l2L. Aftergroupingthetermsbasedontheprimalvariablesxandy,weget Sincethelasttermof( 5 )isaconstantforagiven,wecanignoreitwhenderivingthedualfunction.TheLagrangiandualfunction()isnowgivenby()=minL(z,x,y,) 104

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5 )-( 5 )intothefollowingtwosubproblems.S1:minNXi=1L1Xl=1((l+1)i(l)i)y(l)is.t.0y(l)iM,8i2N,1lL1. NotethatweaddtheupperboundMtotheowvariablesxandy. 5-1 5-1 canbeimplementedinadistributedandlocalmanner.Thevalueofy(l)icanbedecidedlocallyinthesensornodei.Also,nodeionlyneedstohavetheknowledgeofjfromitsneighbornodesetNl(i)foralll2L. 105

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Thelastequalitystatesthatthemaximizationproblemforndingf(z)canbefurtherdecomposedintosmallermaximizationproblemsinwhicheachnodeitriestondfi(z).Theproblemtondfi(z)ateachnodeicorrespondstothefractionalknapsackproblem,whichhasapolynomialtimegreedyalgorithm[ 20 ]. SupposethereareNknapsacksandknapsackihasaweightcapacityofzEi.Forknapsacki,wecanpackitems,denotedby(i,l,j)suchthatj2fjj(i,j)2Al,8l2Lg.Weassumethateachitemcanbeinnitelydivisible.Supposethereisareward((l)i(l)j)whenwepackaunitofitem(i,l,j).Also,considere(l)ijastheweightofoneunitofitem(i,l,j).Again,recallthatthemaximumavailableamountofanitemislimitedbyM.Theprotofanitem(i,l,j)isdenedastherewardperunitweightofthatitem,or((l)i(l)j)=e(l)ij.Thefractionalknapsackproblemistoselecttheitemstopacksubjecttothecapacityconstraintoftheknapsacksuchthatthetotalrewardismaximized. Thesolutionisstraightforward.Wegreedilypackthemostprotableitemamongtheremainingonesuntilthatitemisexhaustedortheknapsackcapacityisreached.Thisoperationisrepeateduntilallprotable(thatis,withpositiveprot)itemsarepackedortheknapsackisfull.DetailsarelistedinAlgorithm 5-2 .Insomecases,theknapsack 106

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Notethatthisalgorithmcanbeimplementedinadistributedandlocalmanner.Anodeionlyrequirestheknowledgeof(l)jofeachneighborj2Nl(i),foralll. if((l)i(l)j<0)then Considertherightderivativesofthesepiecewiselinearfunctions.Notethatthe(right)derivativeofthefunctionf(z)canbewrittenasf0(z)=Pi2Nfi0(z)1.The 107

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Alsonotethatf0(z)changesonlywhenoneofthefi0(z)changes.FromAlgorithm 5-2 ,weseethat,foreachi,fi0(z)changesonlywhenweselectthemostprotableiteminthelistofremainingitems.Supposeitem(i,j,l)isselected.Thenewfi0(z)isgivenbyfi0(z)=((l)i(l)j)=e(l)ij.Furthermore,thenexttimewhenfi0(z)changesagainiswhenzisincrementedbyMe(l)ij=Ei. Tosummarize,theprocedureforsearchingzistokeeptrackofthesequenceofpointswheref0(z)changes,whichrequireskeepingtrackofthesequenceofpointswherefi0(z)changes,foreachi.Consideraxedi.Suppose((l)i(l)j)=e(l)ijissortedindecreasingorderandsupposeanyitem(i,l,j)with((l)i(l)j)0isdiscarded.Startingwithz0=0,wecangenerateasequencezk=zk1+Me(l)ij=Eiiteratively,where(i,j,l)usedintheupdatetogetzkisthekthiteminthelist.Then,fi0(z)canchangeonlyateachofthepointszk.Algorithm 5-3 describesanimplementationoftheaboveidea,aswellasthesolutiontothesubproblemS2.Foreachi,thearrayPi[]recordsthesequenceofzkandfi0(zk). endfor 5 )bysearching(Pi)i2N 5-2 withz

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5-3 isambiguousonhowtomakethealgorithmdistributedand(partially)local.Therearedifferentpossibilities.Inoneversion,eachnodeiexecutestheouterfor-loopinparallel,andforthat,itrequiresonlylocalinformation.Aftertheone-dimensionalarrayPi[]iscomputed,nodeicanbroadcastthisarraytoallothernodes.Afteranodecollectsthecompletetwo-dimensionalarray,itcancomputezbyitself.AnotherpossibilityisthateachnodeisendsthearrayPi[]tothesink;thesinkcomputeszandsendsitbacktoeverynode,whichgoesontoexecuteAlgorithm 5-2 5 )isDual:max()s.t0 Considerthesubgradientprojectionmethodtosolveproblem( 5 ).Theupdateofateachiterationisgivenbythefollowingequations.(l)i(k+1)=[(l)i(k)(Xj2Nl(i)x(l)ij(k)+y(l)i(k)Xj:i2Nl(j)x(l)ji(k)y(l1)i(k))]+,8i2N,8l2L where[b]+=maxf0,bgand(>0)isasufcientlysmallnumber. Ouralgorithmismotivatedbythesubgradientalgorithm,butnotexactlyidentical.Thestandardconvergenceresultsofthesubgradientalgorithmdonotapply.InSection 5.4 ,wewilluseadifferentanalyticalframeworktoprovetheoptimalityofthealgorithm. Letq(l)i(k)=(l)i(k).Wehavethefollowingalgorithm.Fortechnicalreasons,theupperboundoftheowvariablesismodiedfromMtoM(),M+NL,whereisasmallpositivevalue. 109

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Notethat( 5 )and( 5 )aresolvedbyAlgorithm 5-1 and 5-3 ,respectively,withsuitablemodicationofthenotations.Wecanconsiderq(l)iasavirtualqueueatnodei(l)inFigure 5-1 ,and( 5 )canbeunderstoodasthequeuedynamic.Thatis,thequeuelengthofthenodeattimeslotk+1isequaltothequeuelengthattimeslotkplusthenewarrivals(Pj:i2Nl(j)x(l)ji+y(l1)i)andminusthetotalservice(Pj2Nl(i)x(l)ij+y(l)i).Sinceq(k+1)s=0,allowreachingthesinkshouldbedrainedout. 29 ]. 110

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Theusualnon-negativityconstraintsofthevariablesarestillrequired.Intheaboveproblem,weinjectextrasupplyintheamountateachnodei(l),i2N,l2L.ThedemandatthedestinationnodesisnowM+NL,sothatthereexistsafeasibleow. 5 ),and(z,x)isfeasibleto( 5 ). Inthefollowinglemma,wediscussthepropertiesoftheoptimalobjectivevaluefunctionofthe-perturbedproblem.Forsimplicityofdiscussion,weconsiderthestandardlinearprogrammingproblem:(P)mincTxs.t.Ax=dx0. 111

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Letfandf()betheoptimalobjectivevaluesforproblem(P)and(P()),respectively. Proof. 9 ],weknowthatf()=maxf(d+d)Tj2Dg.Forany,theoptimalobjectivevaluecanbeobtainedatoneoftheextremepointsofD.LetbethesetofextremepointsofD,whichisaniteset.Hence,f()=maxf(d+d)Tj2g.Foreach2,wehave 112

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10 ]. 5 )-( 5 )andzbetheoptimalvaluefortheunperturbedproblem.Then,z()!zas!0. Proof. 2 Next,wewanttoproveouralgorithmconvergestotheoptimalobjectivevalueinthetimeaveragesense.LetusdeneaLyapunovfunctionofthequeuesbyV(q)=Pi2NPl2L(q(l)i)2.Let(k),V(q(k+1))V(q(k)). Proof. 5 )andarrangingit,weget(q(l)i(k+1))2(q(l)i(k))2g2(i,l,k)2q(l)i(k)g(i,l,k), 113

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whereB,LN3M2.( 5 )canbeobtainedbyregroupingthetermsbasedonvariablesxandy.Notethatthethirdterminthelastequalityexcludeslinkstothesink.Adding2qs(k)(PlP(j,s)2Alx(l)js(k))=0to( 5 ),wehave Notethatthethirdtermnowincludesthelinkstothesink.Wealsousedthefacty(0)i(k)=Ddi. Adding(2=)z(k)tobothsidesofinequality( 5 ),weget(k)+2 114

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5 ),(^z(),^x(),^y())isanoptimalsolutionofthe-perturbedproblemdenedin( 5 )-( 5 ).Basedontheearlierremark,^y()isfeasibletotheoptimizationproblemin( 5 ),and^z(),^x())isfeasibleto( 5 ).But,y(k)isaminimumtotheoptimizationproblemin( 5 ),andz(k),x(k))isaminimumto( 5 ).Hence,inequality( 5 )follows. Afterregroupingthetermsin( 5 )andusing( 5 ),wehave In( 5 ),theowconservationconstraint( 5 )isused. DeneQ(k)=PLl=1Pi2Nq(l)i(k),whichisthesumofthevirtualqueuesizesattimeslotk. 5 )fork=0,1,,T1,wehave Afterarrangingtheterms,weget1 2T+V(q(0)) 2TB 2T. 115

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5 ). Next,from( 5 ),wehave2T1Xk=0Q(k)BT+2T Theaboveinequalityisthesameas 2T.(5) AftertakingthelimitinT,weget( 5 ). Let(x,y,z)beanoptimalsolutiontotheoriginalproblemin( 5 )-( 5 ).Notethatzisalsotheoptimalobjectivevalue.Letobethelargestforwhichtheperturbedproblem( 5 )-( 5 )isfeasible. 5 ),let!0.SincebyTheorem 5.3 ,^z()!zas!0,weget( 5 ). ByTheorem 5.5 ,wecantakesmallenoughsothatthelong-timeaverageofz(k)isarbitrarilyclosetotheoptimumz.But,thisisattheexpenseofanincreaseintheprovablequeuebound. Nowwewillprovethatthelong-timeaverageofx(k)andy(k)eventuallysatisestheconstraint( 5 ).Letx(l)ij(k)andy(l)i(k)be(1=k)Pk1k=0x(l)ij(k)and(1=k)Pk1k=0y(l)i(k)respectively. 116

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5 )ofmainalgorithm,foreachi2Nandl2L,wehavethefollowinginequality.q(l)i(k+1)q(l)i(k)g(i,l,k),g(i,l,k)q(l)i(k+1)q(l)i(k) SinceQ(k)=PLl=1PNi=1q(l)i(k)isboundedabovebythesecondpartoftheTheorem 5.5 andeachq(l)iisnonnegative,weknowthatq(l)iisalsoboundedabove.Thus,supposeq(l)i(k)Mqforallk. Summing( 5 )overk=0,1,...,T1,wehave DividingtheaboveinequalitybyT,weget LettingT!1,wehave ( 5 )completestheproof. Infact,Theorem 5.6 statesthatthelong-timeaverageof(x,y)eventuallysatisfytheconstraints( 5 ). 117

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Convergencetotheoptimalvalue,z 5 )-( 5 ).Next,weshowhowtheLyapunovdriftandthequeuesizeevolve. Forthesimulationexperiments,werandomlyplace50sensornodesinacircularregionwitharadius25m.Wealsogenerate6sinklocationsinthesameregionforthemobilesinktovisit.Thecostoftransmissionbetweentwonodesdependsonthedistancebetweenthem. where=0.0013pJ=bit=m2[ 19 ].Thedatagenerationrateofnodeiisrandomlyselectedfrom[0,500]bpsandeachnodehas500Jofinitialenergy. Transmissioncanonlyoccurwithinalimitedrange,whichisdenedtobe7.5minoursetting.Inthesubgradientprojectionmethod,weuseaconstantstepsize,whichis=108.Inallsimulations,theperturbationparameteris=108. Figure 5-2 showstheconvergenceresultofouralgorithmtotheprimaloptimalvalue.Asareference,theoptimalsolutionoftheprimalproblem( 5 )-( 5 )isobtainedbytheCPLEXlinearprogrammingsolver.Thecurvelabeledasz(k)isthetimeaveragevalueofz(k)atiterationk.Figure 5-2 veriestherstpartofourmaintheorem,Theorem 5.5 118

PAGE 119

Lyapunovdriftofthealgorithmovertime Figure5-4. Timeaverageoftotalvirtualqueuesizeovertime WealsomeasuretheLyapunovdrift,(k)=V(q(k+1))V(q(k)),ateveryiteration.AsexpectedbyLemma 3 ,wecanobservethatthedriftisboundedfromabove. Figure 5-4 showsthetimeaveragedvalueofthetotalqueuesize,XlXiq(l)i.BythesecondpartofTheorem 5.5 ,thisvalueisboundedfromabove,whichisveriedhere. Figure 5-5 showthelong-timeaveragevalueofg(i,l,k)=Pjx(l)ji(k)Pjx(l)ij(k)+y(l1)i(k)y(l)i(k)forafewselectediandl.Asshowninthegure,itcanbeobservedthatlong-timeaveragevalueofg(i,l,k)convergesto0.Itmeansthatthelong-time 119

PAGE 120

Long-timeaverageofthedifferencebetweenoutowandinow averagesolutionofsequencefx(k),y(k)ggeneratedbyouralgorithmeventuallyfallintothefeasibleset. Ingeneral,thedistributedalgorithmshaveseveralbenetsoverthecentralizedalgorithms.First,thereisnosinglepointoffailure.Inthecentralizedimplementation,thespecialnodecomputingthewayofcontrollingthesystemmightbesuchapoint.Therefore,ifsuchacomputingnodefails,thewholenetworkstopstowork.Second,thedistributedalgorithmtypicallypossessesthescaleability.Asthenetworksizebecomeslarger,thegrowthrateincomputationtimeofthecomputingnodemightbemuchhigherthanoneinthedistributedalgorithm.Third,mostdistributedalgorithmsarealsoadaptiveanddynamicalgorithmsasouralgorithm.Asaconsequence,itiseasy 120

PAGE 121

PerformancecomparisonbetweenCPLEXandAlgorithm3 networksize(#ofsensornodes,#ofsinklocations) ratioofcomputingtime (50,5) 30.333 (100,5) 25.644 (200,5) 23.110 (50,10) 34.474 (100,10) 24.714 (200,10) 21.701 forthedistributedalgorithmtocopewiththechangeofthenetwork.However,whenthenetworkchanges,thecomputingnodemustredothecomputation.Inotherwords,thedistributedsystemensuresacertaindegreeoffaulttoleranceinnature.Forthosereasons,distributedalgorithmsometimesispreferableasanetworkcontrolalgorithmforthelargescalenetworks. However,thesebenetscomeatthecost.Inourdistributedalgorithm,everynoderunsthreealgorithmspresentedintheprevioussections.Eachnoderequiresextracommunicationforsolvingtwosub-problems.Forthesub-problemS1,eachnodehastoexchangeinformationaboutthecurrentlengthofvirtualqueues(whichisdenotedasq(l)i.)ofitsalloutgoingwirelesslinkinthebeginningoftimeslots.Also,tosolvethesub-problemS2,eachnodedisseminatesinformation(whichisPiintheAlgorithm3.)neededtoruntheAlgorithm3toallothernodes.InthecourseofexecutionAlgorithm3,anodeshouldrunthefractionalknapsackalgorithm,whichisAlgorithm2.NotethatAlgorithm2onlyrequirestheinformationaboutthecurrentqueuelengthoftheneighborsanditisthesamerequiredargumentsfortheAlgorithm1. Sincethesub-problemS2isformulatedasLinearProgrammingproblem,itissolvedbyLinearProgrammingsolversuchasCPLEX.WenowpresenttheexperimentalresultshowinghowefcientourAlgorithm3wouldbewhencomparedtotheLinearProgrammingsolver,whichistheCPLEXinthisexperiment.TheefciencyandsimplicityoftheAlgorithm3isthekeyfactortothedistributedimplementationoftheoverallproblem. 121

PAGE 122

5-1 ,networksizeisrepresentedbythenumberofnodesNandthenumberofsinklocationsL.ThemeasureddataistheratioofthecomputingtimeofCPLEXoverthecomputingtimeofAlgorithm3.Asshowninthetable,Algorithm3showssubstantialenhancementincomputingthesolutionforthesub-problemS2. Ourmethodrequiresaninformationexchangeinthebeginningofthetimeslots.Thisinformationmightbeconveyedintheformofcontrolmessage,whichisnotrelevanttothepurposeofwirelesssensornetwork.Thisoverheadmightnotoccurinthesystemwhichrunscentralizedproblemsolverafterthesystemacquiresthetopologyinformationofthesensornodesattheverybeginningoftheoperationofthesystem.Inouralgorithm,eachnodeshouldbroadcastasinglemessagecontainingthesolutionofAlgorithm2andexchangesthelocalinformationaboutthecurrentlengthofthevirtualqueuestoitsneighbor.Thus,inoverall,NbroadcastmessagesandmaximumN2unicastorpoint-to-pointmessagesareneededasoverheadmessages.Thediscussionaboutefcientwayofbroadcastingmessageisbeyondthescopeofthisresearch.Notethat,ingeneral,thecostofbroadcastismuchhigherthanthecostofunicastandthemessagecomplexityofbroadcastdominatestheoverallmessagecomplexity. 122

PAGE 123

Inthisdissertation,wediscussvariousissuesonefcientenergymanagementschemeinwirelesssensornetwork. First,weshowwhygeneraltechniques,especiallytheroutingprotocolusedinwirednetworkareinappropriateinwirelesssensornetworkthroughextensiveexperiments.Theleastcostroutingalgorithmsthatwirednetworkspreferssometimesaggravatethesituationinwirelesssensornetwork.Thisisbecausetheleastcostroutescanbealsothemostpopularroutessothatthoseroutesarelikelytobeusedfrequentlybyseveralnodes.Thenodesalongtheleastcostroutestendtoexhausttheirenergyinearliertime.Itpartitionsthenetwork,sothatthewholenetworkexpireseventhoughtherestofnodesstillhaveplentyofenergy.Withthehelpofmathematicaloptimization,wecanobtaintheoptimalroutethatwillmaximizethenetworklifetime.Wecomparetheperformancebetweentheleastcostroutingandoptimalroutingintermsofnetworklifetime.Asthesizeofnetworkincreases,thegapbetweentworoutingschemesbecomesbigger. Second,weexaminehowtoapplynon-uniformdeploymentofthesensornodestoresolvetheproblemofunevenenergyconsumptionratesbythenodesortheenergyholeprobleminmulti-hopwirelesssensornetworks.Moregenerally,non-uniformdeploymentwithcarefuldensitycontrolcanbeanimportanttechniqueforachievingadesirablelifetimeandsystem-costtradeoffofthesensornetwork.Ourmaincontributionistopresentamethodforcomputingtherequirednodedensityfunction.Asanexample,weshowthatthemethodenablesustocomputethecorrectdensitiesthatachieveanequalenergyconsumptionrateforallnodes,thereby,extendingthesystemlifetime.Themethodisexpectedtobewidelyapplicabletoothersimilarobjectivesandconstraints. Third,weproposeanewframeworkforimprovingthenetworklifetimebyexploitingsinkmobilityanddelaytolerance.Itisexpectedtobeusefulinapplicationsthatcan 123

PAGE 124

Last,weextendtheourresearchofDT-MSM.Inpreviouswork,weshowthatourmodelissuperiortoothermodelsintermsoflifetime.OneofpossibledirectionoffurtherresearchistodeviseamethodimplementingDT-MSMmodelinadistributedmanner.Amongsubowbasedmodelandqueuebasedmodel,wetargetthequeuebasedmodelbecauseitalwaysproducesabetterresultthantheother.Basicallyweapplythedualmethodbecauseitsometimesrevealsnicestructuralproperties 124

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125

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[1] Ahuja,RavindraK.,Magnanti,ThomasL.,andOrlin,JamesB.NetworkFlows:Theory,Algorithms,andApplications.PrenticeHall,1993. [2] Akyildiz,IF,Su,W,Sankarasubramaniam,Y,andCayirci,E.Wirelesssensornetworks:asurvey.ComputernetworksI(2002). [3] Awerbuch,BaruchandLeighton,Tom.ASimpleLocal-ControlApproximationAlgorithmforMulticommodityFlow.IEEESymposiumonTheoryofComputing.1993,459. [4] .ImprovedApproximationAlgorithmsfortheMulti-CommodityFlowProblemandLocalCompetitiveRoutinginDynamicNetworks.ACMSymposiumonTheoryofComputing.1994,487. [5] Baek,SeungJunanddeVeciana,Gustavo.SpatialEnergyBalancinginLarge-ScaleWirelessMultihopNetworks.INFOCOM2005.2005. [6] Balister,Paul,Bollabas,Bela,Sarkar,Amites,andKumar,Santosh.ReliableDensityEstimatesforCoverageandConnectivityinThinStripsofFiniteLength.MobiHoc'07.2007,75. [7] Basagni,Stefano,Carosi,Alessio,Melachrinoudis,Emanuel,Petrioli,Chiara,andWang,Z.Maria.ANewMILPFormulationandDistributedProtocolsforWirelessSensorNetworksLifetimeMaximization.IEEEInternationalConferenceonCommunications2006.2006,3517. [8] .Controlledsinkmobilityforprolongingwirelesssensornetworkslifetime.WirelessNetworks14(2007).6:831. [9] Bazaraa,MokhtarS.,Jarvis,JohnJ.,andSherali,HanifD.LinearProgrammingandNetworkFlows.Wiley-Interscience,2004. [10] Boyd,StephenandVandenberghe,Lieven.ConvexOptimization.CambridgeUniversityPress,2004. [11] Chang,J.H.andTassiulas,L.Routingformaximumsystemlifetimeinwirelessadhocnetworks.37thAnnualAllertionConf.Communication,Control,andComputing.Monticello,IL,1999. [12] .MaximumLifetimeRoutinginWirelessSensorNetworks.IEEE/ACMTransactionsofNetworking12(2004):609. [13] Fleischer,LisaK.ApproximatingFractionalMulticommodityFlowIndependentoftheNumberofCommodities.SiamJournalofDiscreteMathematics13(2000):505. 126

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Gandham,S.R.,Dawande,M.,Prakash,R.,andVenkatesan,S.Energyefcientschemesforwirelesssensornetworkswithmultiplemobilebasestations.GlobalTelecommunicationsConference,2003.GLOBECOM'03.IEEE.2003. [15] Garg,NaveenandKonemann,Jochen.FasterandSimplerAlgorithmsforMulticommodityFlowandotherFractionalPackingProblems.Proc.39thAnnualSymposiumonFoundationsofComputerScience.1998,300. [16] Gatzianas,MariosandGeorgiadis,Leonidas.ADistributedAlgorithmforMaximumLifetimeRoutinginSensorNetworkswithMobileSink.IEEETransactionsonWirelessCommunications7(2008).3:984. [17] Giridhar,ArvindandKumar,P.R.MaximizingtheFunctionalLifetimeofSensorNetworks.The4thIntl'SymposiumonInformationProcessinginSensorNetworks,2005..2005,512. [18] Heinzelman,WendiBeth.ApplicationSpecicProtocolArchitecturesforWirelessNetworks.Ph.D.thesis,MIT,2000. [19] Heinzelman,WendiRabiner,Chadrakasan,Anantha,andBalakrishnan,Hari.Energy-EfcientCommunicationProtocolforWirelessMicrosensorNetworks.Proc.ofthe33rdHawaiiInternationalConferenceonSystemSciences.2000. [20] Horowitz,Ellis,Sahni,Sartaj,andRajasekaran,Sanguthevar.ComputerAlgori-htms/C++.ComputerSciencePress,1996. [21] JieLian,GordonB.Agnew,KshirasagarNaik.DataCapacityImprovementofWirelessSensorNetworksUsingNon-UniformSensorDistribution.InternationalJournalofDistributedSensorNetworks2(2006). [22] JosephC.Dagher,MarkA.Neifeld,MichaelW.Marcellin.ATheoryformaximizingtheLifetimeosSensorNetworks.IEEETransactionsonCommunications55(2007).2:323332. [23] Li,JianandMohapatra,Prasant.AnAnalyticalModelForTheEnergyHoleProblemInMany-To-OneSensorNetworks.Proc.ofVehicularTechnologyConference.2005,27212725. [24] .Analyticalmodelingandmitigationtechniquesfortheenergyholeprobleminsensornetworks.PervasiveandMobileComputing3(2007).8:233. [25] Luo,JunandHubaux,Jean-Pierre.JointMobilityandRoutingforLifetimeElongationinWirelessSensorNetworks.INFOCOM05.2005. [26] Madan,RiteshandLall,Sanjay.DistributedAlgorithmsforMaximumLifetimeRoutinginWirelessSensorNetworks.Globecom2004.2004,748. 127

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MarkPerillo,WendiHeinzelman,ZhaoCheng.OntheProblemofUnbalancedLoadDistributioninWirelessSensorNetworks.GlobalTelecommunicationsConferenceWorkshops,2004.2004,74. [28] Moiseiwitsch,B.L.IntegralEquations.Longman,1977. [29] Neely,MichaelJ.,Modiano,Eytan,andpingLi,Chih.FairnessandOptimalStochasticControlforHeterogeneousNetworks.IEEETransactionsonNetworking16(2008).2. [30] Olariu,StephanandStojmenovic,Ivan.DesignGuidelinesforMaximizingLifetimeandAvoidingEnergyHolesinSensorNetworkswithUniformDistributionandUniformReporting.INFOCOM2006.2006. [31] Papadimitriou,IoannisandGeorgiadis,Leonidas.MaximumLifetimeRoutingtoMobileSinkinWirelessSensorNetworks.The13thIEEESoftCom,2005.2005. [32] Popa,Lucian,Rostamizadeh,Afshin,Karp,RichardM.,andPapadimitriou,Christos.BalancingTrafcLoadinWirelessNetworkswithCurveballRouting.MobiHoc'07.2007,170. [33] Rappaport,T.S.Wirelesscommunications:principlesandpractice.PrenticeHall,1996. [34] SameerTilak,WendiHeinzelman,NaelB.Abu-Ghazaleh.Infrastructuretradeoffsforsensornetworks.Proceedingsofthe1stACMinternationalworkshoponWirelesssensornetworksandapplications.2002,49. [35] Sankar,ArvindandLiu,Zhen.MaximumLifetimeRoutinginWirelessAd-hocNetworks.INFOCOM2004.2004. [36] Shah,RahulC.,Roy,Sumit,Jain,Sushant,andBrunette,Waylon.DataMULEs:Modelingathree-tierarchitectureforsparsesensornetworks.theFirstIEEEInternationalWorkshoponSensorNetworkProtocolsandApplications,SNPA2003.Anchorage,AK,2003,30. [37] Shahrokhi,FarhadandMatula,D.W.TheMaximumConcurrentFlowProblem.JournalofACM37(1990).2:318. [38] Shi,YiandHou,Y.Thomas.TheoreticalResultsonBaseStationMovementProblemforSensorNetwork.IEEEINFOCOM'08.2008. [39] Wadaa,A.,Olariu,S.,Wilson,L.,Jones,K.,andEltoweissy,M.Trainingasensornetworks.MONET.2005. [40] Wang,Wei,Srinivasan,Vikram,andChua,Kee-Chang.UsingMobileRelaystoProlongtheLifetimeofWirelessSensorNetworks.MobiCom'05.2005,270. 128

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Wang,Z.Maria,Basagni,Stefano,Melachrinoudis,Emanuel,andPetrioli,Chiara.ExploitingSinkMobilityforMaximizingSensorNetworkLifetime.38thHawaiiInternationalConferenceonSystemScience.2005. [42] WeiWang,Kee-ChaingChua,VikramSrinivasan.Trade-offsBetweenMobilityandDensityforCoverageinWirelessSensorNetworks.MobiHoc'07.2007,39. [43] Wu,Xiaobing,Chen,Guihai,andDas,SajalK.AvoidingEnergyHolesinWirelessSensorNetworkswithNonuniformNodeDistribution.IEEETransactionsonParallelandDistributedSystems19(2008).5:710. [44] XiaobingWu,SajalK.Das,GuihaiChen.AvoidingEnergyHolesinWirelessSensorNetworkswithNonuniformNodeDistribution.TobeappearedonIEEETransactionsonParallelandDistributedSystems(2007). [45] Xue,Yuan,Cui,Yi,andNahrstedt,Klara.MaximizingLifetimeforDataAggregationinWirelessSensorNetworks.MobileNetworksandApplications10(2005).6:853864. [46] Yun,YoungSangandXia,Ye.MaximizingtheLifetimeofWirelessSensorNetworkswithMobileSinkinDelay-TolerantApplications.tobeappearedinIEEETransactionsonMobileComputing(????). [47] Zhang,HonghaiandHou,Jennifer.OnDerivingtheUpperBoundof-LifetimeforLargeSensorNetworks.MobiHoc'04.2004,121132. 129

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YoungSangYunwasborninChangHeung,RepublicofKorea,in1970.HereceivedhisBSandMSdegreesincomputerengineeringatPohangUniversityofScienceandTechnologyinKoreain1992and1994,respectively.HeworkedfortheLGElectronics,Inc.,Koreafrom1994to2003.Hewasinvolvedindevelopmentofsystemsoftwarefortelecommunicationequipments,suchasATMswitches.HealsomanagedtheIProutingprotocolsoftwaredevelopmentteam.Since2003,hehasbeenconductingresearchwithDr.YeXiaintheDepartmentofComputerandInformationScienceandEngineeringattheUniversityofFlorida.Hisresearchinterestsarewirelesssensornetworkingandmathematicaloptimization. 130







which is denoted by q(0). When the sink finishes a cycle of visit, the queue at node /

must be cleared. Thus we have qll) = 0. In this model, the flow conservation constraint

is replaced by the queue length dynamics, which is expressed as follows.


z( x )) + q (-1) z,( x/)) q( ) v/ e L; vi e N. (4-50)
k:ieN/(k) jEN/(i)

The energy constraints can be expressed in the same way as in the sub-flow

based MSM. From the above discussion, we have the following optimization problem for

maximizing the lifetime.


Queue-Based DT-MSM


max T


s. t. z)( +) q, x ) = q') V/ e L; Vi c A
k:ieN/(k) jEN/(i)

{Zi S CkxT < E, V T /1 eN/(i) k:ieN/(k)

q(0) = D d, Vi E

ql) 0, Vi E/

x(/) > 0, V/ e L; Vi e RI; Vj e Ni(i)

q1) > 0O, Vi /e ; V/i eL

z/> 0, V/ e

T>0.


The problem shown above can be converted into an LP problem by substituting y/)

for z/1 x~ and introducing the new variable u = 1/T. This linearization method can also

be applied to the sub-flow based MSM.

Discussion on the two delay-tolerant models:


(4-51)

(4-52)

(4-53)


(4-54)


(4-55)

(4-56)

(4-57)

(4-58)

(4-59)

(4-60)









optimal path to maximize the lifetime of the system can be also used in the routing

algorithms.

If the mobility can be applied to the component of WSN, the better performance

might be expected. The mobile sink seems a preferential choice rather than the mobile

sensor nodes. When the sink has a capability of moving, overall energy consumption

of particular sensor nodes can be mitigated as contrasted with the network where the

sink does not move because a heavy relaying burden also tends to follow the sink's

movement.

1.1 Contributions

The contributions of this work are mainly discussed in chapter 3, chapter 4,

and chapter 5. First, we study the density control that makes every sensor node

experience the same rate of energy consumption. The same energy consumption

for all sensor node is a key factor in maximizing the lifetime of the WSN. Our density

control necessitates a non-uniform node deployment where the density at a point is

determined according to the routing protocol used and its distance from the sink node.

We propose a very general routing model that captures various existing routing protocol

strategies. We also propose an iterative method that eventually finds densities of the

ring where the sensor nodes are almost similarly distant from the sink.

Second, we propose new framework that exploits the mobile sinks. The application

we are interested in can tolerate some extent of delay of delivery of information

collection of sensed data. We formulate the Linear Programming problem that reflects

features of our framework. Through the simulation, we also show its performance is

better than other models: Static Sink Model and General Mobile Sink Model.

Third, we devise the adaptive and potentially distributed algorithm for the framework

we propose. Our algorithm can work only with the information about the current queue

size, receiving and transmitting traffic at the sensor node. We show that our algorithm

find a solution which is arbitrarily close to the optimal solution.









any routing strategy used in practice, if one can cast it into a specification in terms of

Fk(j), then one can use (3-8), (3-11) and (3-9) (or, equivalently, (3-13)) to compute
the required node densities. Next, we will consider some simple routing schemes as

examples and later show numerical results about them. Many more routing schemes

can be modeled similarly.

3.2.3.1 Uniform ring selection

With this scheme, a node finds its next-hop node in the direction to the sink via two

steps. First, the node selects a reachable ring with a uniform probability distribution.

Second, the sending node randomly chooses the next-hop node among the nodes in the

intersection of the selected ring and the sender's communication range. Therefore, the

probability that ring i, (j I) < i < j, is selected as the next-hop ring by a node in ring j

is

F(i) = 1/min(/,j). (3-15)

Note that, in this case, Fj(i) is independent on the node densities. From (3-11), G,

becomes
n-l *
S(n i)
/ min(n, /)
Gj = i- Gn. (3-16)
-1 (j i)a
S( -min(j,/)
i= (Ji-)+

G, is the same as before, equal to K/pn. From (3-13), the node densities can be

computed iteratively from n 1 to 1 by the following expression.

min(n, /) -i/)+ i)
S= min(j, /) n-), (n -
mm En-(nl /)Z
min(n l i (j- i) (2k 1) 1

kj 1 Ki (k- + (k- i)a (2j -1) min(j,) (3-17)
k~ =k-1)









may not be fully packed in the optimal solution when all items with positive rewards are
packed.
Note that this algorithm can be implemented in a distributed and local manner. A
node i only requires the knowledge of ') of each neighbors c NI(i), for all /.

Algorithm 5-2 Fractional Knapsack (2) for Finding f,(2)
sort (i, ,j) in the decreasing order of (ni) )/e,)
(i, I,j) = the first element in the sorted list
U 2E,
while U > 0 do
if (0 ~ i) < 0) then
break
else if (U Me~) < 0 then
M i) U/e^i)
break
else
x i) M
end if
(i, I,j) = the next element of the sorted list
end while


The tricky part in solving subproblem 52 is how to choose the right value for z, so

that the overall objective function, -z -+ N f(z), is maximized. Note that f(z) is a
concave, nondecreasing, and piecewise linear function of z and f(z) is a concave and
piecewise linear function of z. We will search for an optimal solution by increasing z
and we only need to care about those points what mark the beginning or end of a linear
segment. Let z* be the first optimal solution encountered in the search. To the left of z*,
the function f(z) must be increasing (except the trivial case where f(z) is identically 0,
which can be discovered separately); to the right, the function is non-increasing. This is

also a sufficient condition for optimality.
Consider the right derivatives of these piecewise linear functions. Note that the
(right) derivative of the function f(z) can be written as fi(z) = Cv ff(z) 1. The


107









LIST OF FIGURES


Figure

2-1 Graph example 1: node deployment by the uniform distribution

2-2 Graph example 2: node deployment by grid based strategy .

2-3 Lifetime vs. ratio of remaining energy to the initial energy . .


2-4 Performance in various node densities . .


Performance in various number of sinks . . . . . . . . . .

Performance in different number of nodes and different number of sinks .

Performance of different node deployment strategies . . . . . . .

Performance of different communication ranges . . . . . . . .

Performance of different dimension of sensor fields . . . . . . .

Sensor field m odel . . . . . . . . . . . . . . . .

Uniform node selection . . . . . . . . . . . . . . .

Node densities and average per-node energy consumption rates for various


maximum jump sizes, /, under uniform ring selection. a


2. .........


3-4 Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 0. ...........

3-5 Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 1. ...........

3-6 Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 2. ...........

3-7 Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 3. ...........

4-1 Examples of the static sink model (SSM), mobile sink model (MSM), and delay
tolerant mobile sink model (DT-MSM) .. ....................


4-2

4-3

4-4

4-5

5-1


Comparison of lifetimes of MSM and DT-MSM under the various radii of coverage 89

Lifetime against the number of sink locations; maximum coverage; e = 2.0 . 90

Lifetime against the number of sink locations; minimum coverage; e = 3.0 .. 91

Lifetimes versus transmission range, d: INIf = 200, II = 20, e = 2.0 . . 92

Expanded graph of DT-MSM ............................ 97


2-5

2-6

2-7

2-8

2-9

3-1

3-2

3-3


page

. 28

. 30

. 31

. 32

. 33

. 34

. 36

. 37

. 38

.. 46

.. 54









However, it is less generalizable.


Aggregate-Traffic SSM (4-11)

max Z (4-12)

s.t. x,- xki = d, Vi/ AN (4-13)
jEN(i) k:ieN(k)


( C, Cx,+ 7 7.Xki Z < Ei, Vi eV (4-14)
jeN(i) k:iEN(k)
xy > 0, Vi e I; Vj e N(i) (4-15)

Z> 0. (4-16)

Here, xy is the aggregate flow rate of all commodities from node i to node j, i.e., c x,~ =

x.. The equivalence of the problems, (4-5) and (4-11), can be argued as follows.

Clearly, we can always construct a feasible solution to problem (4-11) from any feasible

solution to problem (4-5) by letting xy = c x,f. Conversely, given a feasible solution

{xd} to problem (4-11), one can apply the flow decomposition algorithm [1] to the arc
flows {xy} and obtain path flows for the commodities2 The path flows in turn give the

per-commodity arc flows {x,} feasible to problem (4-5).

Remark: The equivalence is only true for the particular constraints considered here.

The two formulations are not usually equivalent in more general settings, for instance,

if the costs of the commodities (energy per unit of data transmitted or received) are

different, or if some individual commodity rate at some link is upper bounded or lower

bounded by a non-zero value, which in turn might be the result of assigning different

importance levels to different commodities. The per-commodity formulation (4-5) is



2 In general, the flow decomposition algorithm produces both path flows and cycle
flows. However, in any optimal solution of (4-11), the flows on every cycle must be zero.
Hence, we can restrict ourself to the set of feasible flows that can be decomposed into
path flows only.










50
0
-250
0 -100
-a -150
a)
S-200
( -250
E
.2 -300
S-350 3 (2) -
-400 10(3) .............
0.0*100 5.0*105 1.0*106
iteration (k)

Figure 5-5. Long-time average of the difference between outflow and inflow


average solution of sequence {x(k), y(k)} generated by our algorithm eventually fall into

the feasible set.

5.6 Implementation Issues

In this section, we discuss several issues relating to the implementation of our

algorithm as a network control protocol to maximize the lifetime. There are two ways

of implementation for controlling DT-MSM: One is to use the centralized LP solver to

control the network and the other is to adopt the distributed algorithm such as our

proposed one. Both methods must find how much data for a node to send along its

outgoing wireless links in the pursuit of maximizing the network lifetime. The centralized

method will find such a solution and must distribute an obtained solution to each node.

In general, the distributed algorithms have several benefits over the centralized

algorithms. First, there is no single point of failure. In the centralized implementation,

the special node computing the way of controlling the system might be such a point.

Therefore, if such a computing node fails, the whole network stops to work. Second,

the distributed algorithm typically possesses the scaleability. As the network size

becomes larger, the growth rate in computation time of the computing node might be

much higher than one in the distributed algorithm. Third, most distributed algorithms are

also adaptive and dynamic algorithms as our algorithm. As a consequence, it is easy


120









uniformly randomly deployed into a circular area with radius R. Let the center of the disk

be the origin. Each node / is assumed to generate data at a constant rate of d, during

its life span and the initial energy of i is denoted by Ei. Furthermore, the nodes have the

ability of adjusting their transmission power level to match the transmission distance.

Similar to [19], the energy required per unit of time to transmit data at the rate of xU from

node i toj can be determined as follows.

Ed = C x, (4-1)

where C, is the required energy for transmitting one unit of data from node i to j and it

can be modeled as follows [33].

C a + d(i,j)e (4-2)

where d(i,j) is the Euclidean distance between node / and j, a and P are nonnegative

constants, and e is the path loss exponent. Typically, e is in the range of 2 to 6,

depending on the environment. Here, the energy cost per unit of data does not depend

on the link rate, and this is valid for the low rate regime. Hence, we need to assume that

the traffic rate x, is sufficiently small compared to the capacity of the wireless link.

The energy consumed at node i per unit of time for receiving data from node k is

given by [19]

E = 7 ki, (4-3)

where 7 is a given constant. Hence the total energy consumption per unit time at node /

is

E + E, CU- .x + -'k,. (4-4)
jEN kEN jEN kEN
We assume that each sensor node has the same transmission range. Let I denote

the sink. In this paper, we take the convention that the sink is a special node different

from the sensor nodes and I AVf. The required energy for transmitting one unit of









and the number of grids so that too many nodes are not clumped in a single grid. The

location of a node within the grid is determined randomly. With this method, we can get

a graph in which nodes are well-scattered. In later section, we will explain the grid-based

node deployment strategy more detail.

In this chapter, we mainly focus on two types of metrics as performance measures.

One is about the network longevity, and the other is about the energy efficiency of the

algorithm. Of course, the network longevity can be measured by the lifetime, and the

energy efficiency can be represented by the ratio of the total remaining energy to the

total initial energy after simulation is finished. When we say ratio throughout this chapter,

it follows above definition of the ratio.

2.4.2 Lifetime vs. Ratio

Figure 2-3 shows how the ratio changes during the lifetime of sensor network. This

is the result of the SMTE simulation and we assume that the communication range for a

node is sufficiently large so that every node can reach the sink directly. In this figure, We

can observe that the ratio is dropped sharply when the lifetime exceeds a certain point.

We suspect lots of nodes in the sensor network died due to energy exhaustion after the

lifetime exceeds the threshold point, thus the distance to traverse in a single hop might

increase. The more nodes die, the faster energy is used. In fact, we might not get the

same result for all the nodes as the figure 2-3, for example for the node near the sink,

this sudden dropping of the ratio might come at earlier time.On the other hand, for the

node farther from the sink, this dropping happens at the later time, but it is very steep.

2.4.3 Effect Of Node Densities

When the dimension of a sensor field is fixed, we can control the density of the

nodes by changing the number of nodes to be deployed. Figure 2-4A shows the

performance of the simulation under various number of nodes. In this graph, MTE,LP,

and SMTE means the Minimum Transmission Energy simulation, Linear Program

solution, and Sequential MTE, respectively. In our experiments, the number of nodes









CHAPTER 5
A DECOMPOSITION TECHNIQUE FOR DT-MSM

5.1 Overview

we propose an adaptive and potentially decentralized algorithm for the DT-MSM.

The distributed routing algorithms are very important in developing a practical routing

protocol. Distributed algorithms are generally free from the network scalability issues

in several reasons. They do not need to have knowledge about the whole network

configurations and they also do not require the central node to compute the routes for

all nodes in the network. Lagrange multiplier method solves dual of the primal problem.

Dual problem sometimes has a nice structure with which we can decompose the dual

problem into several sub-problems. We use a subgradient projection method to solve the

dual problem and. A sensor node in our method keeps virtual queue which is a scalar

product of the Lagrange multiplier and it is used in solving sub-problems. We propose (a

possibly distributed implementable) decentralized algorithms for solving sub-problems.

Moreover, we analytically show the our algorithm finds a solution arbitrarily close to the

optimal solution of the primal problem. It is verified through the numerical experiments.

5.2 SystemModel and Problem Formulation

The wireless sensor network is modeled as a directed graph, denoted by GO

(A/V, A), where AV = {1,..., N} is the set of vertices representing the sensor nodes and A

is the set of edges representing the wireless links. Each sensor node i generates data at

a constant rate d, and has an initially energy endowment Ei. Let d(i,j) be the Euclidean

distance between nodes i and j.

Let N, denote the set of (downstream) neighbor nodes of node i, i.e., N(i) =

{jl(i,j) e A}. Let c : A R+ be a given cost function on the edge set. The cost c(i,j)
is the required energy to send a unit of data from node i toj and it is usually a function

of the distance between i and j. Let L be the set of the sink locations indexed from 1

through L.










n-1
Y (n i)Fn(i)
i=(n-+)
Gj (n-l) Gn. (3-11)
S j-1
(j_- FJ()
i=-0-I)

Note that Gn is given by (3-8), which depends only on pn. Throughout, pn can be

considered as a given parameter. Hence, if Fk(j) has no dependency on any Pk, one can

compute Gj using (3-11) for all j. The resulting Gj is parameterized by p,. After that, one

can compute the densities pj iteratively using (3-9) for all j from n 1 down to 1.

3.2.2.2 Case of density-dependent routing

The situation becomes more complicated if Fk(j) depends on some Pk. In that case,

G, in (3-11) depends on the unknown pk. One can still start with (3-9) and eliminate all

Gj's from (3-9) by using (3-8) and (3-11). First, note that

j-1

Gk i=(j-1)
Gk- ---) (3-12)
Gj k-1
5 (k i)Fk(i)
i=(k-I)+

Then, (3-9) can be re-written as follows.




Yi = (n-)
=n ~n -i) ( i)aFo(i)
min(nj i) z I) j- i)aF(i) (2k 1)

k= 1 (lk-) (k i)Fk(i) (2j -1)

x Fk(j)pk. (3-13)

The above expression does not imply that, if pn is given, then one can compute

the densities pj for all other. This is because Fk(j) may depend on the densities in

complicated ways. The set of equations in (3-13), for 1 < j < n, is a fairly complex









mitigate the energy hole problem, the sink needs to be moved around the area where

the sensor nodes are deployed. By moving the sink, the traffic concentrated points may

be moved along with the sink because those points are usually occurred near the sink.

Assuming that the application over the WSN can tolerate some extent of delay, We

propose the new framework that extends the lifetime of the WSN. We call it DT-MSM,

Delay Tolerant Mobile Sink Model. We also present the experimental results showing

that our framework outperforms other WSN models.

In chapter 5, We extend our work on the DT-MSM proposed in the chapter 4.

Although the DT-MSM shows substantial improvement of the performance, it requires

solution from the Linear Programming problem. However, there is no way to solve the

Linear Programming problems in the distributed manner and even We need a very

powerful computing node which runs LP solver. To be a practical routing method, it is

necessary to be implementable in a decentralized manner. In this chapter, We propose

a partially distributed algorithm implementing our framework. The proposed algorithm

does not require an LP solver.

In chapter 6, We conclude this dissertation.









eventually solve and is as follows.


min z


L
s.t. ~ e(~')x(') < zEi, Vi c A
I=1 jEN/(i)

x i '- 4 i) +' f-1) ) < 0, V/e L, Vi e A
j:yiEN/(J) jEN/(i)
yfO) = Dd,, (L) = O, Vi E A

x(i) > 0, VI e Vi e Ri, Vj e NI(i)

y() > 0, VieA VI/ {2,3,...,L- 1}

z > 0.


Let be 2 and z* be the optimal value of the formulation (5-28) (5-34) and formulation

(5-17) (5-23) respectively.
Theorem 5.2. Let be 2 and z* be the optimal value of the formulation (5-28) (5-34)

and formulation (5-17) (5-23) respectively. Then, 2 = z*.

Proof. It is obvious that 2 < z*, because the feasible solution set of the formulation

(5-28) (5-34) is bigger than the feasible set of (5-17) (5-23).

Since Theorem 5.1 says that the formulations (5-10) (5-16) and (5-17) (5-23)

are equivalent, we compare the optimal objective value of (5-28) (5-34) to that of

(5-10) (5-16). Let (z*, x*, y*) be one of optimal solution of the formulation (5-10) -

(5-16). The inequality constraint (5-30) can be transformed into equality constraint by

adding slack variables.


5 4
j:iEN(j)


j:iENI(j)
x, ^iU)

++ x i)


x +i)' Yi () Y (11) > 0
jEN/(i)

Y xr) + y') y S'- = s/
jEN/(i)


102


(5-28)

(5-29)


(5-30)


(5-31)

(5-32)

(5-33)

(5-34)


(5-35)


(5-36)









CHAPTER 3
A METHOD FOR DECIDING NODE DENSITY IN NON-UNIFORM DEPLOYMENT OF
WIRELESS SENSOR NETWORKS

3.1 Overview

Wireless sensor networks have diverse applications such as environmental

monitoring (e.g., vehicular traffic, wild life habitat, bridge or earthquake monitoring) and

battlefield surveillance. A sensor network is in general a self-organized infrastructure

that uses multi-hop routing to deliver the collected information to some collection center,

or the sink. The sensor nodes in the network typically face severe energy, computation

and communication constraints. They usually have limited on-board batteries and are

often deployed in harsh environment where human operators cannot access them

easily, making it difficult or impossible to replace the batteries. As a result, much of the

research on sensor networks has focused on the longevity, or lifetime, of the network.

The lifetime of a sensor network has several definitions in the literature. One of the

most popular definitions is the interval from the time when the system starts operation

until the time when the first node exhausts its energy. This is equal to the shortest

lifetime of the nodes. Much work has been published on how to prolong the lifetime of

sensor networks [12] [11] [17] [19] [35] [45]. Some proposed energy efficient routing

strategies, whereas others proposed efficient ways of clustering the nodes to save

energy.

Regardless of the energy-saving strategies used, sensor networks often experience

unbalanced traffic distribution because the multi-hop traffic pattern is typically many-to-one

[24, 27, 30, 32, 34]. Sensor nodes in the network act as data originators and data

relayers. The traffic transmitted by each node typically includes both self-generated

and relayed traffic. Since the entire network traffic flows toward the sink, the nodes

closer to the sink tend to experience more traffic. As a result, their energy consumption

rates tend to be higher than those nodes that are far away from the sink, assuming

the transmission distance is the same. This causes the nodes closer to the sink to die









only need to compute a finite number of node densities, one for each ring. In spite of

its simplicity, the model is still considerably more general than those in [30], [43] (See

the discussion in Section 3.1.), and is already useful for a number of practical cases.

Later in Section 3.4, we will make generalization that allows sensor fields of arbitrary

two-dimensional shapes, routing between two arbitrary locations in the field, and node

density as a function of the precise location.

3.2.1 Sensor Field and Energy Consumption Models

The sensor field in the shape of a disk is shown in Figure 3-1 with the sink at the

center. The communication capability of the nodes is limited so that multi-hop routing is

necessary to deliver the data to the sink. We introduce some definitions and notations.

* n: the total number of rings.

* Ri: ring i, 0 < i < n. We index the rings in the direction away from the center of the
disk. For convenience, ringe 0, Ro, refers to the center of the disk where the sink is.
Ring 1 is also special. It is a small disk.

w: the width of each ring, R1, .- R,.

Ai: the area of the ring i, 1 < i < n. A, = 7(2i 1)w2.

pi: the node density of ring i, 1 < i < n.

Ni: the number of the nodes in the ring i, 1 < i < n. N, = piA = r(2i 1)w2pi.

The energy consumption/dissipation model of the sensor nodes affects the final

density of each ring. We adopt the energy dissipation model of [19]. The required

transmission energy to send one unit of data to a node at a distance d away from the

sender is given by

Et(d) = 7 + 3d, (3-1)

where 7 is the required energy to operate the transceiver circuitry, 3 is a parameter

characterizing the transceiver amplifiers energy consumption, and a is the so-called

path loss exponent. Normally, a is between 2 to 6 depending on the operating









formulation may be necessary. Let x() be the aggregate flow on link (ij) while the sink

is at stop /. The lifetime maximization problem can be formulated as follows.


Aggregate-Traffic Mobile Sink Model (MSM)


max Z = z 2 + -

s. t. i- -
jEN(i,I) k:iEN(k,


/I 1 jEN(i,I)

x /) > 0, Vi e


I = di,
/)


Vi e N., Vi e


+ 7xk() < E,,
k:iEN(k,l)
'; V/ e L; Vj e N(i, I)


Vi E V


z/ > 0, V/e .


(4-18)

(4-19)

(4-20)


(4-21)


(4-22)

(4-23)


Constraint (4-20) denotes the flow conservation for all nodes when the sink is at /.

Constraint (4-21) says that the total energy consumed at the node i can not exceed the

initial energy E,. By multiplying (4-20) with zi and substituting x,.) z, with a new variable

y we can replace (4-20) with the following new constraint.


y = y z1 *d,
jEN(i,I) k:ieN(k,l)
Similarly, constraint (4-21) can be changed into


Vi e /V; V/ e L.


/S
CE yi) /=1 EN(i,l) k:iEN(k,I)
With the constraints (4-24), (4-25), (4-23), and the non-negativity constraints

for y(), the above optimization problem is converted into an LP problem. Here, y~ is

interpreted as the total traffic volume that node i sends to node j while the sink stays at
I.


(4-24)









Therefore, the average rate of the traffic contributed by ring k to a typical node in ring j is

pk7(2k 1)w2Gk Fk ) pk(2k 1)GkFk(j)
pj7(2j- l)w2 pj(2j- 1)

For each fixed j, 1 < j < n, the rate of the relay traffic at a typical node in ring j, Cj, is the

sum of the above quantity over all rings outside ring j that can reach ring j. That is,
min(n,j+/)
S pk(2k- 1)GkFk()
k=1+l
CJ k= 1 pj(2 1) (3-6)

Recall that, in min(n,j + ), ring n is the outmost ring and I is the maximum jump.

Now we can get the total data transmission rate for a node in ring j, 1 < j < n, by

summing (3-4) and (3-6).
min(nj+/)
K(2j- 1)+ pk(2k 1)GkFk(j)
k=j 1
G =k+ (3-7)
Spj(2j 1)

For the outmost ring n, since Cn = 0 and Sn = K/pn, we have

Gn = K/pn. (3-8)

From (3-7), we can write, for all 1 < j< n,

min(nKj+1)
K m n, ) (2k- 1)GkFk(j)
S= (2j- 1) Pk. (3-9)
k=j+ 1
From (3-9), note that, if Gk is known for all k, pn is known, and Fk(j) is independent

on p for all k andj, then, pj can be computed iteratively forj from n 1 down to 1.

We next show how Gk can be determined. Consider the energy consumption rate

for a typical node in ring j, denoted by Pj. Pj depends on the energy consumption model

discussed in Section 3.2.1. For notational simplicity, we set 7 = 0, which means that we

ignore the energy required for operating the transceiver circuitry and the energy required









In the problem (P(c)) below, the right hand side of (P) is perturbed by C along the
direction Ad.

(P(c)) min cTx

s.t. Ax = d + Ad

x > 0. (5-64)

Let f* and f*(e) be the optimal objective values for problem (P) and (P(e)), respectively.

Lemma 2. f*(e) is continuous, convex, and piecewise linear.

Proof. Let (D) be the dual linear problem of (P).



(D) max dTU

s.t. ATT < c,

where r is the dual variable associated with the constraints of (P). Similarly, the dual

problem for (P(c)) is defined as

(D(e)) max (d + Ad)T7

s.t. AT7 < c.

Since the constraints for (D(e)) are not changed from (D), the feasible solution set

of (D(e)) is the same as that of (D). Let D be this feasible solution set, which does not

depend on c. By the strong duality [9], we know that f*(c) = max{(d + cAd)Te i eD}.
For any c, the optimal objective value can be obtained at one of the extreme points

of D. Let Q be the set of extreme points of D, which is a finite set. Hence, f*(c) =

max{(d + eAd)TT7rl| e Q}. For each 7 e Q, we have

(d + Ad)Tr = dTr + cAdTr, (5-65)


112










9 10
8 ....... 9 ...
S7 a)7 8
E ...w E 7
55 SSMSS
a 4 SSM -X-K-- a 5 SS -
> DT-MSM ...... .... 4 .........MSM ..... .....
3 s3 ( DT-MSM ----***
2 ..S......... ..... .....* ................... 3. DT-MSM .........
1 ---- 0 21-----
1 ::::: : : : 1 ::::: : : :
0 0
5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40
Number of Sink Locations Number of Sink Locations
A 100-node network B 200-node network

Figure 4-4. Lifetime against the number of sink locations; minimum coverage; e = 3.0


are again normalized to the optimal lifetime of the SSM. As shown in Figure 4-3, the

lifetime of the MSM is about 100% ~ 200% greater than that of the SSM. However, the

DT-MSM is 200% ~ 1000% better than the SSM. Moreover, the curves all look linear; the

performance gap can grow even larger with more sink locations.

Interestingly, the lifetime of the MSM increases very slowly with the number of sink

locations. As explained in [38], in the optimal solution, only a few locations from the set

of sink locations are chosen as the true stops for the sink. However, in DT-MSM, the rate

of lifetime increase is substantial as II increases. This is because each node can have

better and better sink location as II increases, and it is not forced to participate in the

communication when the current coverage is not the most favorable for energy saving,

even if the node may belong to that coverage. This is not possible in the MSM because

no matter where the sink stops, every node must participate in the communication.

We wish to make the following remarks. First, our formulations and reported

experiments all use the optimal routing with respect to maximizing the system lifetime.

The routing strategy is important for increasing the system lifetime. For instance, based

on our experiences, when the shortest path routing is used in the static sink model

(results not shown), the lifetime performance is quite inferior to the case of optimal

routing. Second, in our model, the locations of the sink stop candidates are randomly












Lifetime under various communication range
(uniform distribution,single sink)
200000 --
180000 -- i i, **t
-* / *vk
160000 '. x K
140000
120000
100000t
80000
60000
40000


4 5 6 7 8
Communication range
A Lifetime


Ratio under various communication range
(uniform distribution,single sink)


0

07.
0 7

06
05
04
03


0


9 10


4 5 6 7
communication range
B Ratio


Figure 2-8. Performance of different communication ranges



each node prefers the shortest cost hop among its neighbor set. Due to the inclusion

property, this shortest cost hop neighbor is included in the bigger neighbor set with

a larger communication range. Thus, an enlarged neighbor set due to the increased

communication range does not influence the selection of the next hop during routing.

However, too short communication range perhaps destroys the network connectivity. In

conclusion, as long as the connectivity of the network is guaranteed, the communication

range does not have an effect on the performance of the network whose routing scheme

is MTE.

2.5 Experiments on the Minimum Transmission Energy Routing, Sequential
Minimum Transmission Energy Routing, and Optimal Lifetime Routing

For SMTE and LP cases, we observe that their lifetimes increase as the communication

range gets longer, but they seem to converge to a particular value. However, the

quantitative properties of this values are known yet. While the ratio of the LP converges

to the 0, the ratio of the SMTE is still high. We are not sure whether SMTE outperforms

LP or not, if we keep simulating the both two routing scheme beyond 10.

In this section, we study the effects of the different size of sensor fields on the

performance of the sensor networks. In this simulation, we do not fix the number of

nodes to a particular value. Instead, we set the node density to a constant value.


Kc


3 9












L
s. t. Y e()x'0 < zE,, Vi Ar (5-18)
I 1 jeNI(i)

xE ) xE +Fi(i1) yi < 0, v/ eL, Vi
L: ) jN/,(i) (5-19)
x ) M S=1 j:seNIj)
yf) = Ddi, y(L) = O, Vi E A (5-20)

x/) > O, VI e L, Vi e Ri, Vj e Ni(i) (5-21)

yi> 0, VieA, VI {2,3,..., L-l} (5-22)

z > 0. (5-23)

We use an ordered-pair notation (i, /) to denote the index of the flow constraint
(5-19) of the node (except the sink) i when the sink is at /. For a given (x, y), if the flow
constraint (i, /) satisfies equality, we say that constraint (i, /) is binding. (Some authors
may use the term tight or active instead.) That is, the binding constraint (i, /) implies that

EfCEN((i) xji -je) XyF-1) _, y= 0
Lemma 1. (x, y) is feasible to constraint (5-19) if only if all constraint (5-19) for (x, y)
are binding.

Proof. It is clear that any (x, y) satisfies equality in constraint (5-19) is feasible solution.
TO prove the opposite direction of the lemma, let's suppose (x, y) is feasible and some
constraint of (5-19) are not binding. Let K be the set of an ordered pair (i, /), such that

Ej:EN (i) xil) jE(i) + iy'+ ) i) < 0.
We may write a unbinding constraint (i, /) as follows.


Si) x yi( yi()= 0 (5-24)
j:

100


min z


(5-17)









earlier, leaving a hole near the sink and partitioning the whole network while many

remaining nodes still have a plenty of energy. This phenomenon is called the energy

hole problem [23, 24, 43]. In [30], the authors claimed that when the nodes one hop

away from the sink use up all their energy, the remaining nodes have used only 7% of

their energy on average. It has been shown analytically in [23, 24] that the energy hole

problem exists in various sensor networks.

On closer examination, the energy hole is most often observed in networks where

the sensor nodes are homogeneous and uniformly deployed, and they report events

generated at a constant rate to the sink. If we allow non-uniform deployment of the

sensor nodes, by carefully increasing the number of nodes around the sink, we can

prevent the sensor nodes near the sink from depleting their energy faster than others,

and hence, resolve the energy hole problem. Adding more nodes in the sensor field also

has other benefits, such as better connectivity and higher reliability. On the other hand,

adding more nodes means a higher cost. Hence, this solution makes sense in situations

where inexpensive sensors can be mass-produced or having a longer network lifetime

outweighs the cost of the extra sensors. Recent advances in micro-electro-mechanical

and integration technologies make the first situation more and more likely to occur.

This chapter contributes to the research area that seeks to extend network lifetime

by deploying the sensors non-uniformly and by carefully controlling the node densities

in different parts of the sensor field. The main question to be addressed is how many

nodes per unit area (i.e., the node density) should be deployed at each location in

order to achieve a prescribed lifetime-cost objective. The main result of the chapter is a

mathematical method for computing the node densities.

This chapter illustrates the method using a particular example, in fact, a particular

objective. We show how to derive the location-dependent node densities that equalize

the energy dissipation rate of the sensor nodes throughout the network. The result is









Therefore we have the following inequality.


L L
S e'2( f(p)) < E, e'(f(p)(s,/ f(p))) (5-42)
/ 1 p:(ij)Ep,pEP/ /=1

Hence, our new path flow (or equivalent edge flow) can have an objective value, say z,

such that z < 2, which is the contradiction to our premise.

Therefore, 2 = z*. D

Now, we turn our attention to deriving the algorithm by Lagrangian relaxation.

Let ,() be the Lagrange multipliers associated with the constraints in (5-30). The

Lagrangian function of (5-28) is
L N
L(z, x, y,) = z + 7(/)( ) ) i y(-1) yi(/), (5-43)
I=1 i=1 j:iENI() jEN/(i)

where i = (<(), over i e f, I e C.

After grouping the terms based on the primal variables x and y, we get
L
L(z,x,y,w) =z Y (' -')'+

S1 (j(5-44)
N L-1 N
-'(!,+1) (),y,) + D di.
i=l =1 i=l

Since the last term of (5-44) is a constant for a given r, we can ignore it when deriving

the dual function. The Lagrangian dual function 0(7) is now given by

0(7) = min L(z, x, y, 7) (5-45)
L
s. t. x zE, < 0, Vi c i (5-46)
I=1 jENI(i)

x .) > 0, V/ e L, Vi e A, Vj e NI(i) (5-47)
0 < y()_ < M, 1 < / < L 1, Vi/ A (5-48)

z > 0. (5-49)









optimal solution, a node does not pick far-away nodes as the next-hop neighbors even

if the transmission range allows it. The observed fluctuation in the curves is due to

statistical fluctuation in the samples of the random network topologies.












A 1



~k 1

consumption rate
A, densities ---

0 2 4 6 8 101214161820
ring index
A/= 1









consumption rate - AA
densities .. .

0 2 4 6 8 101214161820
ring index
C /=10


consumption rate
densities ---

:I'L 'AI^ L_ .


4 6 8 101214161820
ring index
B =2


35
30
25
20 .
15 S
-0
10
5
n


5
4.5
4
3.5
SAA ,k 3 .a
A 2.5
2 a
1.5
consumption rate 1
densities ------ 0.5
0
0 2 4 6 8 101214161820
ring index
D / = 20


Figure 3-3. Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform ring selection. a = 2.


Hence, its maximum transmission distance is less than / rings away. As a result, it tends

to consume less energy on average than a node in a ring further outside, say Rj for

j > I. The precise situation is complicated, depending on the parameters of the energy

consumption model and the routing probabilities.

3.3.2 Uniform Node Selection

Unlike the case of Uniform Ring Selection, here, the node densities are computed

by successive substitution as in (3-14). The results for the case of a = 0 are given

in Figure 3-4. The average per-node energy consumption rates in all rings are nearly

identical in each of the four plots, which correspond to / = 1,2, 10 and 20, respectively.

When the path loss exponent, a, is 0, it takes a constant amount of energy for a node to


0









ACKNOWLEDGMENTS

I would like to begin by thanking Professor Ye Xia, my dissertation advisor and

mentor during past several years. Dr. Ye Xia has been the greatest advisor, providing

me with with valuable ideas, support, and help during my graduate studies. His wide and

deep knowledge and his enthusiasm always inspires me.

My special thanks also go to Prof. Jonathan Liu, Prof. Alin Dobra, Prof. Shigang

Chen and Prof. Cole J. Smith for their comments and support during my studies.

Last but not least, I want to thank my family for their love while I am studying.

Without their sacrifice, I would not be what I am. I would like to say that I love Chris,

Han, and my wife, KyungHee.









CHAPTER 4
MAXIMIZING THE LIFETIME OF WIRELESS SENSOR NETWORKS WITH MOBILE
SINK IN DELAY-TOLERANT APPLICATIONS

4.1 Overview

A wireless sensor network (WSN) consists of sensor nodes capable of collecting

information from the environment and communicating with each other via wireless

transceivers. The collected data will be delivered to one or more sinks, generally via

multi-hop communication. The sensor nodes are typically expected to operate with

batteries and are often deployed in not-easily-accessible or hostile environments,

sometimes in large quantities. It can be difficult or impossible to replace the batteries

of the sensor nodes. On the other hand, the sink is typically rich in energy. Since the

sensor energy is the most precious resource in a WSN, efficient utilization of the energy

to prolong the network lifetime has been the focus of much of the research on WSNs.

Although the lifetime of a WSN can be defined in many ways, we adopt the widely

used definition, which is the time until the first node exhausts its energy. Much work

has been done during recent years to increase the lifetime of a WSN. Among them,

in spite of the difficulties in realization, taking advantage of mobility in the WSN has

attracted much interest from researchers [7, 14, 16, 25, 31, 36, 38, 40, 41]. We can

take the mobile sink as an example of mobility in a WSN. Communication in a WSN

often has the many-to-one property in that data from a large number of sensor nodes

needs to be concentrated to one or a few sinks. Since multi-hop routing is generally

needed for distant sensor nodes to send data to the sink1 the nodes near the sink can

be burdened with relaying a large amount of traffic from other nodes. This phenomenon

is sometimes called the "crowded center effect" [32] or the "energy hole problem"

[23, 24, 44]. It results in early energy depletion at the nodes near the sink, potentially



1 For ease of discussion, we assume there is only one sink.









extensive simulations performed on the sensor networks. As our intuition, we verified

that lots of factors such as the number of sink nodes or nodes to be deployed, and

the communication ranges have an influence on the lifetime of the sensor networks

through simulations. However, on the contrary to the intuition, we found that there is no

or at most little relationship between the lifetime and deployment strategies in the large

sensor networks. In addition, we measured the lifetime of SMTE, an extended version

of MTE, and discovered the lifetime of SMTE is almost equivalent to the lifetime of the

optimal value that is calculated by Linear Program method.

In our observation, we found that the most effective way to increase the lifetime

of the networks is to put more sink nodes into the sensor networks. The study on the

performance of the sensor networks with multiple sinks might be one of our future

research.











-----------------<
70
60
50
40
30
Consumption rate 20
A, densities ---- 0
10
A4-_AAAAAMAAA 1

0 2 4 6 8 101214161820
ring index
A= 1

-18
16
14
12
consumption rate 1
densities ----- 10
S8
6
,A**A 4
2
0
0 2 4 6 8 101214161820
ring index
C /= 10


45
40
35
Consumption rate 350 n
S densities ---- 30
25
20
15

0 i \


0 2 4 6 8 101214161820
ring index
B =2


0 2 4 6 8 101214161820
ring index
D / = 20


Figure 3-6. Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 2.


sink. Once a transmission reaches inside B(6), it is received by the sink. We let As be

the sensor field with B(6) removed, i.e., As = A\B(6). We wish to find the node density

in A6.

Consider a point y c As. Let g(y) be the total traffic rate of a node at y. Let c(y) be

the rate of the traffic to be relayed by a node at y. We again assume that the rate of the

locally generated traffic at each point is inversely proportional to the node density at that

point. Then, this rate at a node is K/p(y) for some constant K > 0. We have,


g(y) = c(y) + K/p(y).


(3-22)









[41] Wang, Z. Maria, Basagni, Stefano, Melachrinoudis, Emanuel, and Petrioli, Chiara.
"Exploiting Sink Mobility for Maximizing Sensor Network Lifetime." 38th Hawaii
International Conference on System Science. 2005.

[42] Wei Wang, Kee-Chaing Chua, Vikram Srinivasan. "Trade-offs Between Mobility and
Density for Coverage in Wireless Sensor Networks." MobiHoc '07. 2007, 39-50.

[43] Wu, Xiaobing, Chen, Guihai, and Das, Sajal K. "Avoiding Energy Holes in Wireless
Sensor Networks with Nonuniform Node Distribution." IEEE Transactions on
Parallel and Distributed Systems 19 (2008).5: 710-720.

[44] Xiaobing Wu, Sajal K. Das, Guihai Chen. "Avoiding Energy Holes in Wireless
Sensor Networks with Nonuniform Node Distribution." To be appeared on IEEE
Transactions on Parallel and Distributed Systems (2007).

[45] Xue, Yuan, Cui, Yi, and Nahrstedt, Klara. "Maximizing Lifetime for Data Aggregation
in Wireless Sensor Networks." Mobile Networks and Applications 10 (2005).6: 853
864.

[46] Yun, YoungSang and Xia, Ye. "Maximizing the Lifetime of Wireless Sensor
Networks with Mobile Sink in Delay-Tolerant Applications." to be appeared in
IEEE Transactions on Mobile Computing (????).

[47] Zhang, Honghai and Hou, Jennifer. "On Deriving the Upper Bound of a-Lifetime for
Large Sensor Networks." MobiHoc '04. 2004, 121 132.


129









[14] Gandham, S.R., Dawande, M., Prakash, R., and Venkatesan, S. "Energy efficient
schemes for wireless sensor networks with multiple mobile base stations." Global
Telecommunications Conference, 2003. GLOBECOM '03. IEEE. 2003.

[15] Garg, Naveen and Konemann, Jochen. "Faster and Simpler Algorithms for
Multicommodity Flow and other Fractional Packing Problems." Proc. 39th Annual
Symposium on Foundations of Computer Science. 1998, 300-309.

[16] Gatzianas, Marios and Georgiadis, Leonidas. "A Distributed Algorithm for Maximum
Lifetime Routing in Sensor Networks with Mobile Sink." IEEE Transactions on
Wireless Communications 7 (2008).3: 984-994.

[17] Giridhar, Arvind and Kumar, P.R. "Maximizing the Functional Lifetime of Sensor
Networks." The 4th Intl' Symposium on Information Processing in Sensor Networks,
2005.. 2005, 5 -12.

[18] Heinzelman, Wendi Beth. Application Specific Protocol Architectures for Wireless
Networks. Ph.D. thesis, MIT, 2000.

[19] Heinzelman, Wendi Rabiner, Chadrakasan, Anantha, and Balakrishnan, Hari.
"Energy-Efficient Communication Protocol for Wireless Microsensor Networks."
Proc. of the 33rd Hawaii International Conference on System Sciences. 2000.

[20] Horowitz, Ellis, Sahni, Sartaj, and Rajasekaran, Sanguthevar. ComputerAlgori-
htms/C++. Computer Science Press, 1996.

[21] Jie Lian, Gordon B. Agnew, Kshirasagar Naik. "Data Capacity Improvement of
Wireless Sensor Networks Using Non-Uniform Sensor Distribution." International
Journal of Distributed Sensor Networks 2 (2006).

[22] Joseph C. Dagher, Mark A. Neifeld, Michael W. Marcellin. "A Theory for maximizing
the Lifetime os Sensor Networks." IEEE Transactions on Communications 55
(2007).2: 323 332.

[23] Li, Jian and Mohapatra, Prasant. "An Analytical Model For The Energy Hole
Problem In Many-To-One Sensor Networks." Proc. of Vehicular Technology
Conference. 2005, 2721- 2725.

[24] "Analytical modeling and mitigation techniques for the energy hole problem
in sensor networks." Pervasive and Mobile Computing 3 (2007).8: 233-254.

[25] Luo, Jun and Hubaux, Jean-Pierre. "Joint Mobility and Routing for Lifetime
Elongation in Wireless Sensor Networks." INFOCOM 05. 2005.

[26] Madan, Ritesh and Lall, Sanjay. "Distributed Algorithms for Maximum Lifetime
Routing in Wireless Sensor Networks." Globecom 2004. 2004, 748-753.


127













of nodes (400 nodes) which are deployed by the uniform distribution and the number


of sinks are increased from 1 to 60. In general, as the number of sinks is increased,


average length of the paths and average number of hops becomes smaller. In addition,


traffic relay load of the nodes is perhaps reduced. The result is shown in Figure 2-5A


and figure 2-5B. As we see in figure 2-5A, the number of sinks becomes larger, SMTE


Lifetime according to the number of sinks
in 400 nodes (uniform distribution)


10 20 30 40 50
Number of sinks

A Lifetime (uniform distribution)


Remaining enerOg rat o according to the number of sinks
In 4U nodes (uniform dlstrbution)


08 o : -


*** *


Ilt4%g. X -*-
MTE -1--
SMTE---
LP X


* i


10 20 30 40
Number of sinks

B Ratio (uniform distribution)


Lifetime according to the number of sinks in 400 nodes (grid)
3 5e+06 MTE
MTE ------- :
SSMTE x
3e+06 LP
2 5e+06 --~ ~--'*:m--- -- ;.-
2 5e+06

2e+06 -'""

1 5e+06 -..-':

le+06

500000

0 10 20 30 40 50 60
Number of sinks

C Lifetime (grid based deployment)


Remaining energy ratio according to the number of sinks in 400 nodes (grid)
1.


08\

06

04 .

02


I E
SMTE --X--
ILP ---


10 20 30 40 50
Number of sinks

D Ratio (grid based deployment)


Figure 2-5. Performance in various number of sinks




outperforms the LP. Ratios of all three routing cases tends to be constant or slightly


increasing after a certain number of sinks. Interestingly, we find that it is more effective


to increase the number of the sinks rather than to increase the node density for the


network lifetime. Note that, when the node density increases node has shorter links with


3 5e+06

3e+06

2 5e+06

2e+06

1 5e+06

le+06

500000


I I I I









disconnecting the sink from the remaining sensors that still have plenty of energy.

However, by moving the sink in the sensor field, one can avoid or mitigate the energy

hole problem and expect an increased network lifetime.

This chapter proposes a framework to maximize the lifetime of a WSN by taking

advantage of sink mobility. Compared with other mobile-sink proposals, the main

novelty is that we consider the case where the underlying applications tolerate

delayed information delivery to the sink. One of the application examples is battle

field surveillance, where sensor nodes are deployed to monitor the movement of enemy

vehicles or troops. A mobile sink attached to an unmanned aerial vehicle flies over

the monitored region regularly to harvest the collected intelligence. To avoid being

intercepted or detected by enemy forces, the mobile sink needs to operate in only a few

safe locations within a limited operation time. Another example is habitat monitoring

where a mobile robot is used to collect information from the sensor nodes in the field. If

much of the habitat area is not be accessible by the robot or if it is desirable to minimize

disturbance to the targeted animal species, the mobile robot will trace predetermined

paths and stop by a set of pre-arranged locations regularly for data collection.

In our proposal, within a prescribed delay tolerance level, each node does not need

to send the data immediately as it becomes available. Instead, the node can store the

data temporarily and transmit it when the mobile sink is at the location most favorable

for achieving the longest network lifetime. To find the best solution within the proposed

framework, we formulate optimization problems that maximize the lifetime of the WSN

subject to the delay bound constraints, node energy constraints and flow conservation

constraints. Another one of our contributions is that we compare our proposal with

several other lifetime-maximization proposals and quantify the performance differences

among them. Our computational experiments have shown that our proposal increases

the lifetime significantly when compared to not only the stationary sink model but also

more traditional mobile sink models without delay tolerance.









Our proposal is more sophisticated than most previous lifetime-improvement

proposals that we know of. It integrates the following energy-saving techniques,

multipath routing, a mobile sink, delayed data delivery and active region control, into

a single optimization problem. Such sophistication comes at a cost. Whether the

proposal should be adopted in practice will depend on the tradeoff between the lifetime

gain and the actual system cost. The latter includes all costs/complexity in implementing

the proposal and in actual operations. These may include extra communication protocols

for coordination and control, e.g., new routing and rate control protocols, extra memory

for keeping delayed data and memory-management costs, and application-level costs

incurred by delayed information delivery. Even if the decision is not to adopt it due to

a high cost or high complexity, the framework in the chapter is still useful because it

can supply the practitioners with a performance benchmark, e.g., how much lifetime

improvement opportunity there is. By also formulating the optimization problems

related to other proposals and providing cross comparison, the chapter provides extra

convenience for comparing and understanding different proposals.

Being one of the early papers on extending the network lifetime with mobility

and delay tolerance, the chapter focuses on formulating several simple and typical

lifetime-maximization problems and evaluating the lifetime improvement. There can

be many variants of the problem formulation, some of which can be very difficult, often

involving NP-hard combinatorial sub-problems. The degree of lifetime improvement

demonstrated by this chapter can justify further work on more difficult problems.

We now briefly review the most relevant work on how to exploit mobility to increase

the network lifetime. In [36], the authors introduced mobile agents, which move around

and collect data from nearby sensor nodes on behalf of the immobile sink. When the

mobile agents move to the vicinity of the sink, they forward the collected data to the

sink. In that framework, communication occurs only from the sensor nodes to the mobile

agents or from the mobile agents to the sink via a single hop; the sensor nodes do not









richer in energy reservoir. The sinks sometimes connected to the wired network like the

Internet, thus played as a gateway to the outside world for the wireless sensor networks.

Due to the limitation of available energy or constrained capability of wireless

transceiver, it is not feasible for sensor node to communicate with the sink directly.

Instead, a sensor node uses other sensor nodes as next hops in order to get to the

sink. For this reason, the communication behavior in the wireless sensor network shows

multi-hop pattern. Therefore the individual wireless sensor node plays the roles of a

source of the data and a relayer of the data for other nodes. This feature of WSN makes

it very similar to the Mobile Ad-hoc Networks (MANET) in that communication is done

in multi-hop fashion. However, there is a huge difference between these two types

of network. While the connectivity is the ultimate goal of the MANET, the longevity of

network is the primary objective of the WSN.

The functions of sensors are very diverse: seismic, magnetic, thermal, acoustic,

visual, and radioactive. As the variety of the functions of sensor, the potential applications

of the WSN is also boundless. According to [2], the applications of WSN can be

categorized into the military applications, environmental applications, health care

applications and home applications. But this classification can be broaden into more

categories, such as applications in space exploration or disaster relief.

Since sensor nodes reply on the embedded battery power, sometimes replenishment

of battery is very costly. When considering the number of sensor nodes deployed it does

not seem possible. Therefore clever management of energy reservoir is required to

extend the lifetime or improve the throughput of the WSN. There have been several

challenging research issues on the efficient energy management.

Typically sensor node is able to alter its transmission power and as a consequence,

it can change reachable transmission distance. By changing the transmission power,

one can make a sensor node have the different set of neighbor nodes at times. In

the perspective of network, different topologies can be obtained by the adjustment












Main Algorithm
N L-1
y(k)= argmin { -(q' 1)(k) qj'(k))yj()} (5-55)
y,)E[O,M(e)], i=1 /=1
iEAN,1 L
(z(k), x(k)) arg min {f ( (q (k)- q i(k))xi)} (5-56)
'E[o,M(c)], i eA,1E j N,(JE )( / (i,j), A'
EL1 EjC N,(i) e/) U <-zE,,ieN"
z>O
qi)(k 1) = [ )(k)- ( x xi)(k) +yi')(k)- x')(k) y('-1)(k))],
jeN~ (i) j:ieN (j)

Vi c Af, V/ c L (5-57)

qs(k+ 1) 0. (5-58)

Note that (5-55) and (5-56) are solved by Algorithm 5-1 and 5-3, respectively, with

suitable modification of the notations. We can consider qi') as a virtual queue at node i(')

in Figure 5-1, and (5-57) can be understood as the queue dynamic. That is, the queue

length of the node at time slot k + 1 is equal to the queue length at time slot k plus the

new arrivals (E ij:ij) x y (-1)) and minus the total service (EEN/(i) xI) y(I)). Since

qk 1) = 0, all flow reaching the sink should be drained out.

5.4 Performance Analysis

In this section, we show that our algorithm converges to the optimal solution and

the virtual queues are bounded, both in the long-run average sense. The analytical

technique is in part borrowed from [29].


110



















Figure 4-1. Examples of the static sink model (SSM), mobile sink model (MSM), and
delay tolerant mobile sink model (DT-MSM)

4.3 Lifetime Maximization in Delay Tolerant Mobile Sink Model

In this section ,we consider how to maximize the lifetime of WSN with the mobile

sink in applications that can tolerate a certain amount of delay. We call the resulting

WSN model delay tolerant mobile sink model (DT-MSM). In this setting, each nodes

can postpone the transmission of data until the sink is at the stop most favorable for

extending the network lifetime. This way, the nodes can collectively achieve a longer

network lifetime. In contrast, the SSM and MSM do not exploit this possibility.

Let D be the maximum tolerable delay, or the delay tolerance level. We assume that

the sink finishes one round of visit to all the stops (where the sink stays for a positive

duration to collect data) in D time units, and then, repeats with another round again and

again. Note that two consecutive visits to the same stop takes a time D.

Let's take an example to show how our framework can outperform other ones.

Consider the two-node example shown in Figure 4-1. N1 and N2 are two sensor nodes

and L1 and L2 are the candidate stops of the mobile sink. Suppose we ignore the

receiving energy requirement and suppose the transmission energy per unit of data is

equal to the square of the distance between the sender and the receiver. Both nodes

NI and N2 generate data at 1 bps and have 100 units of energy initially. If the sink is

located at 0 in the SSM, both nodes spend 4 units of energy for sending a bit of data.

It is obvious that the optimal lifetime is 25 seconds. In the MSM with sink locations


Ni Li 0 L2 N2
0 0 0 0 0
I, 1 >|< 1 >|< 1 >|< 1 >|













Comparison of lifetimes for the single sink
600000 ME
SMTE .
500000 LP *
---~------+-- *f /

400000

300000

200000

100000

0
200 300 400 500 600 700 800
Number of nodes

A Lifetime in the single sink


1 4e+06 MTE --
SMTE---- X----
1 2e+06 LP --- .

le+06

800000

600000 '

400000 -*

200000

n


200 300 400 500 600 700 800
Number of nodes

C Lifetime in the multiple sinks


Figure 2-4. Performance in various node dens


,Comparison of ratio of remaining energy to initial energy for the single sink
(1
S09

-x07 *. --

S06
S 05 "
2 04 _-
03 SMTE X
I LP ..... X : ^ .
02
o 02 L i i i i I
(Y 200 300 400 500 600 700 800
Number of nodes

B Ratio in the single sink





07 "'" -- "-
0 6 X
- 0 x -

a)
06 --
05
E 05 :
a) '* :
04
0o 03 -
02 : '
01MTE -MT
0LP ). ----.
200 300 400 500 600 700 800
Number of nodes

D Ratio in the multiple sinks


scheme that may make full advantage of the increase of node density. The lifetime for


SMTE is much longer that MTE and its energy efficiency is better than LP


2.4.4 Effect Of The Number Of Sinks


In this section, we focus on how the number of sinks would affect the performance


of the sensor networks. When the number of the sink is 1, its position is fixed at the


center of the sensor field. However, if there are several sinks in the network, their


locations are randomly selected. Furthermore, in the multiple sink system, every sensor


nodes should select only one of the sinks as a destination of the data depending on the


routing algorithm. Thus, if the MTE used, node may select the sink which it can reach


with the smallest energy. We observe the performance of the network with fixed number


a)
E

a)
E
F-


























Figure 3-1. Sensor field model

environment. Some amount of energy is also required to receive a unit of data. The

energy consumed at a receiver is as follows.


E(-) = 7, (3-2)

where 7 is the same as in (3-1). Normally, the receiving energy requirement does not

depend on the distance from the transmitter. For ease of presentation, in the analysis to

be shown later, we sometimes ignore the receiving energy.
The maximum transmission range of a sensor node is also an important parameter.

In this section, we assume this range is / rings, 1 < / < n. That is, a sensor node

can transmit data up to I rings away without relaying. We call this maximum range the

maximum jump. If the maximum jump is equal to n, then every sensor node is able to

send data to the sink directly, which is the assumption of [30].
The routing strategy has major impact on the final densities. We will consider

several routing strategies later.






























Figure 5-3. Lyapunov drift









.'


5.0*1011

4.0*1011

3.0*1011

2.0*1011

1.0*1011

0.0*100

-1.0*1011

93 fl*flh


A(q(k)) -


1.4*107 1.5*107 1.5*107 1.6*107 1.6*107
iteration (k)

of the algorithm over time


1.0*109
9.0*108
8.0*108
7.0*108
6.0*108
5.0*108
4.0*108
3.0*108
2.0*108
1.0*1 08
0.0*10 0
0.0*100 1.5*107


iteration (k)

Figure 5-4. Time average of total virtual queue size over time


We also measure the Lyapunov drift, A(k) = V(q(k + 1)) V(q(k)), at every

iteration. As expected by Lemma 3, we can observe that the drift is bounded from

above.

Figure 5-4 shows the time averaged value of the total queue size, q> ). By
/ i
the second part of Theorem 5.5, this value is bounded from above, which is verified

here.

Figure 5-5 show the long-time average value of g(i, I, k) = x')(k) xjx )(k)

y'(-1)(k) y()(k) for a few selected i and /. As shown in the figure, it can be observed

that long-time average value of g(i, I, k) converges to 0. It means that the long-time


119









traffic that needs to be forwarded to nodej. from node i when the sink is at /. That is,

S= x V V/ V, Vj e R. (4-26)
CE R

Since at node i e /iV, the commodity or sub-flow of other nodes c e RI, c 4 i must

be forwarded as soon as it has been received, we must have


Xk' x1c) (4-27)
k:iEN/(k) jEN/(i)

Here, we define NI(i) = Ri n N(i, I), where N(i, I) is as given in (4-17). The flow

conservation at node i can be expressed as follows, which is the same as in the MSM

except that the amounts of traffic originated from node / itself, (w('); I C C, i C R), are

now decision variables.


zi xi X Wi ). (4-28)
JEN(i) k:iN/(k) )

The data buffered during the previous sink-movement cycle must be cleared in the

current cycle. This requirement can be written as


Swi) = D d,. (4-29)
I:iERi









rate of each node. Finally, we compute the average per-node energy consumption rate

for each ring. The goal is to verify whether the calculated densities result in an even

energy consumption rate in all rings.

3.3.1 Uniform Ring Selection

The results for various maximum jump sizes are shown in Figure 3-3, where the

path loss exponent, a, is 2. We have conducted extensive experiments for other values

of a; but the results are omitted for brevity. In Figure 3-3, we show both the average

per-node energy consumption rate and the calculated node density in each of the rings.

Several observations can be made. First, the average per-node energy consumption

rates of the rings are nearly identical. This demonstrates that our modeling approach

and analytical method are highly accurate, and that correct node densities can be

derived from the resulting mathematical expressions. Second, the shape of the density

function, as a function of the ring index, is somewhat surprising in some cases. The

functions are not even monotonic in the case of I = 10 or / = 20.

In the cases of I = 1 or / = 2, the density function is monotonic and increases very

fast as the ring gets closer to the sink. It is easy to explain the case of / = 1. Since the

maximum jump size is 1, all the traffic of a node must flow through the adjacent ring on

the inside. Therefore, the traffic load becomes heavier as the ring gets closer to the sink.

It is necessary to deploy more nodes in the rings closer to the sink so as to balance

the energy dissipation rates across the rings. As it approaches the sink, the area of

the ring decreases while the number of nodes in the ring increases. Hence, the density

increases fast.

For larger values of the maximum jump, e.g., / = 10, it is not necessarily true that

higher node densities are required for rings closer to the sink. This is more due to the

"boundary effect". In this case, each node can directly transmit its traffic to multiple

inside rings. However, longer transmission distance requires more energy. A node in

one of the / inner-most rings (R,, 1 < i < /) has fewer than / rings left on the inside.









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

THE RESEARCH ON EXTENDING THE LIFETIME OF WIRELESS SENSOR
NETWORKS

By

YoungSang Yun

August 2010

Chair: Ye Xia
Major: Computer Engineering

We study various problems efficient energy management for wireless sensor

networks. First, we research the energy efficient deployment of wireless sensor nodes

so that energy consumption rates of all nodes are equal during the lifetime of the

wireless sensor network. If the sensors are deployed uniformly across the network,

they experience different traffic intensities and energy depletion rates depending on

their locations. Usually, the sensors near the sink tend to deplete their energy sooner;

when enough of them exhaust their energy, they leave holes in the network, causing

the remaining nodes to be disconnected from the sink. One of the solutions to this

energy-hole problem is to deploy the sensors non-uniformly. Moreover, we describe

a method for deciding the sensor deployment densities so as to equalize the energy

consumption rates of all nodes. The method is general and can be applied to other

objectives and constraints.

Second, we propose a framework to maximize the lifetime of the wireless sensor

networks by using a mobile sink when the underlying applications tolerate delayed

information delivery to the sink. Within a prescribed delay tolerance level, each node

does not need to send the data immediately as it becomes available. Instead, the

node can store the data temporarily and transmit it when the mobile sink is at the

most favorable location for achieving the longest WSN lifetime. We call the proposed

framework as Delay-Tolerant Mobile Sink Wireless Sensor Network. To find the best









{L1, L2}, due to the symmetry of the structure, the sink stays at both L1 and L2 for the
same amount of time to achieve the maximum lifetime. Each node spends 1 or 9 units of

energy for sending 1 bit of data depending whether the sink is at L, or L2. The average

energy consumption per bit is 5 units. Thus, the lifetime is 20 seconds. In the DT-MSM,

we assume that the sink alternates between the two stops and stays for 1 second at

each stop in each cycle. Hence, this is the case that D = 2 seconds. When the sink

stays at L1, only N1 sends 2 bits of data to the sink; when the sink moves to L2, only N2

transmits 2 bits of data (N2 keeps its data while the sink is at Lj). Both nodes spend 2

units of energy every 2 seconds or 1 unit of energy per second on average. Thus, the

lifetime is 100 seconds, a significant increase compared to the SSM and MSM. This

is because, in the DT-MSM, the nodes do not always participate in communication for

all the sink stops; they each wait until the sink's location is most favorable for energy

saving, and then send data at the higher rate. Recall that we have assumed that the

traffic rate is sufficiently small compared to the capacity of the wireless link, and hence,

sending data at a higher rate does not alter the per-bit energy consumption.

Unlike the MSM or SSM, the sink in the DT-MSM can collect data from only a subset

of the set of all sensor nodes, Vf, at each stop. Let Ri be the subset of Vf such that

only nodes in Ri can participate in the communication toward the sink when the sink is

at / e L. We call Ri the coverage of the sink location /. Note that the union of Ri over

/ e L must be the set of all sensor nodes, iV. In other words, any sensor node should

be covered by at least one sink location. When the node / is in RI, node / is said to be

active at / e L. Although we can construct R in many ways depending on the application

of interest, in this paper, a very simple method of constructing R is considered. Fix

a positive number r. We call r the radius of coverage of the sink. For each / C , if

d(i, I) < r, where i e AV, then i e Ri. Here, the radius of coverage of the sink (r) should

be large enough so that every sensor node belongs to at least one R1. Note that the

minimum r depends on the locations of the sink stops.









The flow that maximizes the network lifetime T can be obtained by solving an

following linear program.


S kNksR(i )
- EkEN,k s R(i)


Vi e N, i 4 s

otherwise


(2-18)


(2-19)


Vi E N (2-20)


V(i,j) E L


(2-21)


Note that this is the problem that maximizing the minimum lifetime of source nodes. The

flow conservation in the network is represented as (2-19), and inequalities (2-20) mean

the energy constraints on each node. By changing the variable T to 1/z, we can obtain

a minimization problem but equivalent to the above maximization problem.


(P) min


Vi e N, i 4 s


jENi


Sxji
j:iEN


SkE s R (i)
- EkeN,k s R(i)


e. ex. < zE(i)
jEN x
x > 0


(2-22)


(2-23)


otherwise

Vi E N (2-24)


V(i,j) c L (2-25)


Note that in this problem, we must determine x = {xi} and z.

2.4 Experiments

2.4.1 Graph Generation

We assume that the dimension of the sensor field is fixed as 50m x 50m and n

nodes are randomly created or deployed with a certain strategy in a sensor field. If there

exists only one sink in the network, its position is also fixed to the exact center of the


(Po)


max


s.t. x
jENi


j:iENj


T eyxy < E(i)
jENi


X > 0









the experiments, we also observe that when the first node dies due to the depletion of

energy, almost 95 percent of energy of the remaining nodes is still unused.

Now let's change the definition of the network lifetime from the time until the first

node died to the time until network is disconnected. This means that we can consider

that the network is still operational at the time the first node died, unless remaining

network is partitioned. Hence, MTE needs to be modified and extended as follows: For a

given network in which the edge cost is the transmission cost, every source nodes need

to find the shortest path to the sink and this path is used as a the minimum transmission

cost path for a node. When a certain node has drained its energy, by deleting this node

and all the edges incident to this node, we may get a reduced network. We can check

whether the reduced graph is strongly connected component or not. If the graph is still

a strongly connected component and that means the every source nodes must have

a connectivity to the sink. Thus, MTE can be applied on the reduced graph again. We

can apply this process over and over until the resulting network is no longer strongly

connected component. We call such a repeated MTE as a Sequential MTE (SMTE).

Note that above definition of the lifetime of the network in the SMTE is equivalent to the

time until network is partitioned. In fact, we do not consider the situation such that parts

of the sensing area become uncovered due to the outage of the sensor nodes.

2.3.3 Linear Programming Model

Let's denote xy, called a network flow, the rate of data from node i to nodej. To

transmit an unit bit of data from node i to node j, e, amount of energy is used. The

lifespan of node / under flow x = {xi} is given by



T(x) = E() (2-16)
ZJENV etxY
The network lifetime T under flow x is the time until the first node used up its

energy. That is,

T(x) = min T,(x) (2-17)
iEN








which is a linear function of c. Therefore, f*(c) = max{(d + cAd)T'r r E D} is a
continuous, convex, and piecewise linear function [10]. D

Theorem 5.3. Let z* () be the optimal value for the problem in (5-59) (5-62) and z* be
the optimal value for the unperturbed problem. Then, z*(c) z* as c 0.

Proof. This is a direct consequence of Lemma 2. O

Next, we want to prove our algorithm converges to the optimal objective value in
the time average sense. Let us define a Lyapunov function of the queues by V(q) =

EiEn ,EY(qi'))2. Let A(k) V(q(k 1))- V(q(k)).
Lemma 3. There exists a positive constant B such that for any small positive C and 6,
the following condition holds for any time slot k and for any q(k),
2 2 (1)
A(k) z(k) < B+ 2z(c) 2c q, (5-66)
/EC iEN"
where 2(c) is part of an optimal solution of the c-perturbed problem.

Proof. By squaring (5-57) and arranging it, we get

(q(')(k + 1))2 (qi)(k))2 < g2(i, /, k) 2q(')(k)g(i, /, k),

where g(i, I, k) = EjN(i) x, ')(k) EL:iCN) xj')(k) yi')(k) yi-1)(k). Note that
g(i, I, k) < NM because EjeN(), x'(k) < (N 1)M and yi')(k) < M for all k.


113









We first define an c-perturbed problem, which will be used later. Here, e is the same

small positive constant in the definition of M(c).

min z (5-59)
L
s. t. e)x, U < zEi, Vi E (5-60)
I=1 jEN/(i)


yiENI) (I)_g NI
j:iLEN(j) jENN(i) (5-61)
( ) = M +NLe
=1 j:seNI(j)
yO) = Dd,, y(L) = 0, Vi e A. (5-62)

The usual non-negativity constraints of the variables are still required. In the above

problem, we inject extra supply in the amount c at each node i(/), i e iV, I e C. The

demand at the destination node s is now M + NLc, so that there exists a feasible flow.

Remark: The crucial fact is that any feasible flow to the perturbed problem still satisfies

yi() [0,M(c)],i e NA, i < / < L- 1;x,~) e [0,M(c)],i e NA, Ie c,j e Ni(i);

L1 jCEN,(i) e~( ) < zEi, i e A/; and z > 0. Hence, the vector y is feasible to the
optimization problem in (5-55), and (z, x) is feasible to (5-56).

In the following lemma, we discuss the properties of the optimal objective value

function of the c-perturbed problem. For simplicity of discussion, we consider the

standard linear programming problem:

(P) min cTx

s.t. Ax = d

x > 0. (5-63)









Summing the above inequality over all i and /, we have


1) 2


(,


L Ni
1= 1 i= 1


,i jEN
L
=B- 2 (') (k) xj() (fk)
/=1 (j,s)EA/


N L-1

=1 /) 1


g 2(i, /, k)-
I,i

- xi')(k)
j:iENI(j)


2 q') (k)g(i, k)
I, i

-y0) (k) + yi(-1)(k))


L
2 (qI) (k) q(k) x (k)
/-1 (ij)EA;j#s


(5-67)


2i ql)(k)yi0 )(k),
i= i


where B A LN3M2. (5-67) can be obtained by regrouping the terms based on variables

x and y. Note that the third term in the last equality excludes links to the sink. Adding

2qs(k)(, (j,s)EA xj(s)(k)) = 0 to (5-67), we have


N
A(k) i=1
N L-1
+2 / qi(+1) (k)
i=1 1=1


2 (qj( k)
/ (ij)EA'


Note that the third term now includes the links to the sink. We also used the fact

yi) (k) = Dd,.
Adding (2/6)z(k) to both sides of inequality (5-68), we get


N
2 u q(1)(k)yi)(k)
i= 1


z(k)
0


S (qji) (k)
/ (i,j)EA'


N L-1
2/i


S (qiE (k)
/ (ij)EA'


(5-68)


A(k) + z(k) < B
6


N
< B +2 qf )(k)(0)(e)
i=i
N L-1
+2 (q+ 1)(- q
i=1 I= 1


2


(5-69)


q,) (k)) yi(') (k)


q1')(k)) xU')(k)


q,) (k)) yi(') (k).


q) (k))) ')(k)


') (k)) i) (C),


))2] <


xx )(k) _
U)i-
(i)


q(')(k))x,(')(k)


q(') (k)) ) (c)










Ratio of Rema' inn enerav
to initial energy during lifetime ot sensor network

0.9

0.8
.E-
0.7 -
0
0.6 -
I-
0.5

0.4 -
C
0.3

0.2 -
I-

0
0 0.1 i -


S 0 20000 40000 60000 80000 100000 120000 140000 160000
Time (lifetime)

Figure 2-3. Lifetime vs. ratio of remaining energy to the initial energy


are increased by 25 from 200 to 800 and we run 100 times for each configuration. We

assume that there is only one sink at the very center of the sensor field and we set the

7 of the energy model to 4. All three simulations show that lifetime increase with steady

paces as the number of nodes increase. Note that the lifetimes of the SMTE and LP

are almost the same. We observe the decrease in the ratio for the LP case, as the node

density increases. However, in cases of MTE and SMTE, the ratio seems to be constant

without regard to the increase of node density. Hence, we conclude that for the MTE, the

increase of the node density may not contribute to the lifetime that much. As the node

density increases, average distance between nodes get shorter, thus each node may

get energy saving due to the shorter distance hop. However, increased number of nodes

might also put the relaying stress on the nodes and this stress might be stronger to the

nodes closer to the sink than the nodes farther from the sink. This relaying burden cuts

down the benefit of the shortened average distance between nodes. For the LP cases,

the decreased ratio may account for the increased lifetime of the network. SMTE is very









1.2 Organization of the Study

Rest of this dissertation consists as follows: In chapter 2, as a preliminary work,

We compare the lifetimes of several routing protocols which might be used in the WSN.

The simplest one might be the Minimum Transmission Energy (MTE) routing in which

every sensor nodes greedily choose the path along which total energy consumption in

transmission is minimal. On the other hand, We can use the maximal lifetime routing

in which routing paths are calculated by solving the optimization problem with the

global view of the network topology. In the MTE, after the first node dies due to energy

exhaustion, a large number of nodes still possess plenty of energy. Thus, rather than

to pause the operation of the whole system, the network can continue working as long

as the connectivity to the sink is preserved from all remaining sensor nodes. In the

Sequential Minimum Transmission Energy protocol (SMTE), alive sensor nodes keep

routing based on MTE after the failed sensor node is removed from the network topology

until the sink is isolated from the network. These three routing protocols are compared

through extensive experiments with various simulation settings.

In chapter 3, We discuss non-uniform node deployment strategies. When nodes

are uniformly distributed in the region, the energy hole problem near the sink may not

be an avoidable phenomenon. The rationale of non-uniform deployment is to put more

nodes to the place where relaying burden is excessive. The area nearby the sink is such

a place. In this chapter, We solve the problem of how many nodes should be deployed

in a specific area of the sensor network. The sending (or outgoing ) traffic from a node

is composed of relaying traffic and self-generated traffic. We establish the recurrence

relations showing how the outgoing traffic of nodes is related according to the distance

from the sink, when a geographical routing is used. In addition, We mention the way to

obtain the solution that satisfying the system of recurrence relation.

In chapter 4, We study how the mobility would improve the lifetime of the WSN.

Especially the mobility of the sink, or the mobile sink, is considered in this chapter. To









Unlike the case with node density independent routing, (3-35) is not a typical linear

integral equation and the mathematical theory on the existence and uniqueness of the

solution is unknown at this point. However, the earlier model with the ring structure and

energy-dependent routing in Section 3.2.2.2 is a special discrete analog of this and,

there, our computation experience has shown that successive substitution always finds

a solution. There are good reasons to believe that a solution to (3-35) often exists and

can be found by successive substitution.









Table 4-1. Experimental parameters and their values
# of sensor nodes {100, 200}
# of possible sink locations {5, 6, 7, 8, 9, 10, 15, 20, 30, 40}
path loss exponent (e) {2.0, 3.0}
transmission range {5, 6, 7, 8, 9, 10, 15, 20, 30, 40, 50}
a 10 pJ/bit/m2
0 0.0013 pJ/bit/m4
Initial Energy (E,) 500 J
Data generation rate (d,) 500 bps

3.5
3
E 2.5
*4-t
2
> 1.5
'a 1
0.5 MSM -
DT-MSM ----.--......
0 --------------
10 12 14 16 18 20 22 24 26
Radius of the Coverage of the Sink

Figure 4-2. Comparison of lifetimes of MSM and DT-MSM under the various radii of
coverage


SDelay-Tolerant Mobile Sink Model (DT-MSM): When the mobile sink is at a stop,
a subset of the sensor nodes can participate in the communication. We use the
queue-based variant of this model to evaluate the performance.

We have experimented with different parameters extensively, such as the number

of nodes, the number of possible sink locations and the parameters for the energy

consumption model. Only a small subset of the results are reported here for brevity.

In Table 4-1, we provide the system parameters and their values for the reported

experiments in this paper. We adopt the data for the last four parameters from [18]. In all

experiments, we use GLPK for solving the linear programming problem.

First, we would like to mention the impact of the radius of coverage of the sink

on the performance of the DT-MSM. For this experiment, the positions for 100 nodes

and 20 mobile-sink locations are randomly generated (I|/VI = 100, I1 = 20) in a









2.2 Related Works

In [19], the authors propose the LEACH (Low-Energy Adaptive Clustering

Hierarchy) to minimize global energy consumption by distributing the relaying burden

to all the nodes. Sensor nodes are divided into several clusters and in each cluster

there is a cluster head which is responsible for relaying the all the traffic generated in

the corresponding cluster to the sink. In addition, the role of cluster head is not fixed to

the specific node, that is, the cluster head is assigned to the highest energy node in a

adaptive manner. which improves the lifetime of sensor network significantly.

In [12], the authors define the sensor network lifetime as the time until the node

drains out its energy at the first time. They also shows that minimum energy routes

which minimizes energy consumption as possible while relaying the traffic at each node

are not good in the perspective of the lifetime of the sensor networks due to uneven

energy depletion behavior. That is, the minimum energy routes causes the fast energy

consumption rate of some nodes which are on the frequently used routes. The authors

formulate the routing problem as an optimization problem where the objective is to find

the flows that maximize the sensor network lifetime.

Formulation of maximizing the lifetime as a Linear Program can be found in many

literatures. In [26], the authors propose two distributed algorithms for the same problem

as [12]. The key idea of their algorithms is to solve dual problem instead of solving the

primal problem directly. By changing the objective functions of dual problem slightly (In

this case, the object function is no more linear function), We can make the separable

nonlinear program problem, and this nice structure of the problem make easy to devise

a distributed algorithm for solving maximum lifetime problem. One of their algorithms

is a partially distributed one, the other can be implemented in fully distributed manner.

However, both algorithms use a well-known sub-gradient algorithm. [35] also defines

the maximum lifetime routing problem as a variant of maximum concurrent flow problem

(MCFP), a sort of linear program problem. The original MCFP defined in [37] is a









of buffered data (i.e., queue size) at node / just after the sink leaves location /. Thus,
y('-1) y) is the change in the buffered data at node i while the sink is at location /.

In addition, at the sink (node s), all arrival traffic must be drained. Thus, we have
L N
E S x Dd, = 0. (5-3)
/=1 j:sEN/(j) i=1

The problem we address in this paper is to maximize the number of rounds (or

cycles), T, made by the mobile sink while maintaining the flow conservation (5-2)
and (5-3), subject to the energy constraints at the sensor nodes. More precisely, the

problem can be written as follows.

max T (5-4)

s. t. (5 -2), (5 -3) (5-5)


(ei0x) T E,, V/ei ci (5-6)
( jN(i) L

x ) > 0, VI e , Vi e R,, Vj e Ni(i) (5-7)

yi') > 0, Vi/ A',V/eI (5-8)

T > 0. (5-9)

Constraint (5-6) means that the total energy expenditure at a node during T rounds

should be less than or equal to the node's initial energy endowment. The above problem
can be easily transformed into a linear programming problem, which will be shown next.


5.3 Decomposition by the Lagrange Method

In this section, we illustrate a distributed algorithm to solve the problem defined in

Section 5.2. The following is the equivalent linear problem, which is obtained from the
maximization problem of (5-4) (5-9) by replacing 1/T with z. For convenience, we

also define M = iN1 Dd, y,(O) = Dd, and yL) = 0 for all i e f.









LIST OF TABLES
Table page

2-1 System parameters used in the simulation . . . . . . . ..... . 27

4-1 Experimental parameters and their values . . . . . . . ..... . 89

5-1 Performance comparison between CPLEX and Algorithm 3 . . . . ... 121








Now, we turn to the subproblem 52. For the ease of exposition, we intentionally
change the original 52 to the maximization problem. Suppose, for a moment, z is fixed at
a certain point 2 and define f(2) as follows:



f(2) =max z (
i 1 =1 j:jENI(i)

s.t. O < x') < M, Vi e V/ e Vj e NI(i)
L
Se')_< E2Eii, V/c Ae .
I11 jENI(i)

Let f(2) = max{E 1 E j N,( (7i) -1))l)} subject to x() e [0, M] for all
I e ,j c NI(i) and i1 jiEN/(i) e (') < 2Ei. Then, f(2) = -2 + 1 f(2).
The last equality states that the maximization problem for finding f(2) can be further
decomposed into smaller maximization problems in which each node i tries to find f,(2).
The problem to find f,(2) at each node i corresponds to the fractional knapsack problem,
which has a polynomial time greedy algorithm [20].
Suppose there are N knapsacks and knapsack i has a weight capacity of 2Ei. For
knapsack i, we can pack items, denoted by (i, I,j) such that e {jl(i,j) e A', V1
}. We assume that each item can be infinitely divisible. Suppose there is a reward
(ji') _- i)) when we pack a unit of item (i, I,j). Also, consider eg') as the weight of one
unit of item (i, I,j). Again, recall that the maximum available amount of an item is limited
by M. The profit of an item (i, I,j) is defined as the reward per unit weight of that item, or
(~i) _- ij)/e0). The fractional knapsack problem is to select the items to pack subject to
the capacity constraint of the knapsack such that the total reward is maximized.
The solution is straightforward. We greedily pack the most profitable item among the
remaining ones until that item is exhausted or the knapsack capacity is reached. This
operation is repeated until all profitable (that is, with positive profit) items are packed or
the knapsack is full. Details are listed in Algorithm 5-2. In some cases, the knapsack


106









difference between head queue length and tail queue length for a link) at each iteration.

However, while the latter has energy constraints on the nodes, the former have capacity

constraints on links. The algorithm proposed in [13] is almost identical to that of [15], but

by changing the termination condition, running time of the [13] is now independent of the

number of commodities. [45] also solves the MCFP based on the algorithm of [15],but

the authors uses aggregation tree, which is an aggregated structure of unicast routes

from all sources to the common sink. The running time to find aggregation tree for a

common sink is not that much larger than that of finding the shortest path between an

individual source to the common sink. Thus, instead of calculating the shortest path for

every sources, [45] tries to find the aggregation tree for the common sink.

In other hands, [17] solves given Linear Program for a simple and regular topologies,

for example linear array topology, with an analytical manner. However, since their

analysis is restricted to the regular topology, it makes no sense to apply their analysis in

real world.

Some researchers study how the multi-path routing can contribute to reduce

the imbalance of energy burden of the sensor nodes. For example, [5] explains the

optimizing trade-offs between the energy cost of spreading traffic and the improved

spatial balance of energy burden. The authors propose the multi-path routing based on

the node proximity. Their algorithm, in fact, is an heuristic approach to prevent energy

hole around the sink node from occurring at a rather early time.

2.3 System Model

Consider a wireless sensor network with n sensors and base stations (or sinks).

Assume that the location of the sensors and sinks are fixed and known in advance. We

can model such a sensor network as a undirected graph G(N, L) where N is the set of

sensor nodes and L is the set of edges between some two nodes. And also an individual

node is represented as ni and link between node ni and nj is denoted by e,.

































To my loving family









As stated earlier, we assume that the traveling time between the sink locations is

negligible. With this assumption, the orders of visits to the sink locations does not matter

for the optimization problem. Thus, for simplicity of explanation, we assume that the

order of visits is given as 1 2 -+ ... L.

DT-MSM was introduced in [46]. It is suitable for an application that can tolerate a

certain amount of delay. In DT-MSM, each node can postpone the transmission of data

until the sink is at the location most favorable for extending the system lifetime. However,

there is usually a maximum delay that the application can tolerate. This maximum delay

tolerance is denoted by D. The sink must complete one of its tours from node 1 to L and

back to 1 within D time units and then repeats the same tour in the next round.

Since each tour takes D time units, the problem of maximizing the system lifetime

is to maximize the number of tours, which is denoted as T. The actual lifetime is T D.

The decision variables are how much time the sink stays at each location / e within

each tour, denoted by ti, and what the rate of data transmission from node i to j will be,

denoted by a(l). Note that in an optimal solution, the mobile sink does not necessarily

visit all the sink locations. In that case, we still let the sink visit such a node; but the time

of stay is 0.

It turns out, in the problem formulation, t, and agl) always show up together in the

form ta). We can define x) = ta(') to replace ta('). Clearly, x/) can be interpreted as

the traffic volume on the link (i,j) when the sink is at /. We will take the view of traffic

volume in the following discussion.

Unlike the non-delay-tolerant mobile sink model, where, regardless where the sink

is, every sensor node i must transmit all newly generated data at the data-generation

rate d,, a sensor node in DT-MSM can temporarily delay data transmission by storing

the data in the local buffer. With delay tolerance and data buffering, there is another

flexibility of similar nature that one can take advantage of. When a sink is at location /, it

may be required to collect data only from nearby sensors. This set of sensors is denoted









more generalizable. The aggregate-traffic formulation (4-11) is not necessarily useful

if one wishes to incorporate more constraints. But it is useful in this paper because it is

easier to compute.

4.2.2 Mobile Sink Model

In the mobile sink model (MSM), we assume that the sink can move around within

the sensor field and stop at certain locations to gather the data from the sensor nodes.

Let L be the set of possible locations where the sink can stop (also known as sink

stops). The sink does not necessarily stop at (i.e., stays for a positive duration) all

locations in L in the interest of maximizing the network lifetime [38, 41].

As previous authors [38], throughout the paper, we make the assumption that

the traveling time of the sink between locations is negligible. This way, the resulting

problem formulations are simple enough for us to obtain precise numerical solutions

for evaluation purpose. The assumption is appropriate when the traveling time is much

smaller than the time spent by the sink to collect data in each location.

In this model, the order of visit to the stops has no effect on the network lifetime

and can be arbitrary. The sink sojourn time at a location / e L is denoted by z1; it is

the time that the sink spends at / to collect data from the sensor nodes. The overall

network lifetime is Z = ECL z1. When the sink is at stop /, we denote the (downstream)

neighbors of node i as

N(i, )= {j U {Af}{d(i,j) < d}. (4-17)

To find the optimal network lifetime, we need to consider the routing of the traffic as well

as the duration of the sink's sojourn time at each stop (also see [16, 31, 38, 41]).

Similar to the case of the static sink model in Section 4.2.1, there is a per-commodity-based

formulation of the lifetime-maximization problem, and there is an equivalent, simpler,

aggregate-traffic-based formulation. For brevity, we will only present the latter. However,
we re-iterate that, if additional constraints are present, the per-commodity-based












0.2

0.15

.0 0.1
0
0.05

Primal Optimal ---
0
0.0*100 1.0*107 2.0*107
iteration (k)

Figure 5-2. Convergence to the optimal value, z*


value for the problem in (5-17) (5-23). Next, we show how the Lyapunov drift and the

queue size evolve.

For the simulation experiments, we randomly place 50 sensor nodes in a circular

region with a radius 25m. We also generate 6 sink locations in the same region for

the mobile sink to visit. The cost of transmission between two nodes depends on the

distance between them.

c(i,j) = 3d(i,j)2, (5-84)

where = 0.0013pJ/bit/m2 [19]. The data generation rate of node / is randomly

selected from [0, 500] bps and each node has 500 J of initial energy.

Transmission can only occur within a limited range, which is defined to be 7.5m in

our setting. In the subgradient projection method, we use a constant step size, which is

6 = 10-8. In all simulations, the perturbation parameter is c = 10-8.

Figure 5-2 shows the convergence result of our algorithm to the primal optimal

value. As a reference, the optimal solution of the primal problem (5-17) (5-23) is

obtained by the CPLEX linear programming solver. The curve labeled as 2(k) is the time

average value of z(k) at iteration k. Figure 5-2 verifies the first part of our main theorem,

Theorem 5.5.


118


















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.,.


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I--


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id.


Ir;

I I I ,
S. .
C .- I-
* -.- ** ,,--




r .' '.
,', / : -


_ ,_.-. .


^ _- ^,- ., ,
1, --. -

*- I,
Lr .i2'.

*- :-



4.-- 7 j~ r 1


A Initial Network


B Resulting Network : after running MTE


-- '


- t ,-


* -?

--K-
11I\


S;^





S1'
*-r


L- tT.. 2-r -


L ~-~- h'I
fl*.


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L* .j


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i-.--

A


"7


Wi


Sa


-- ". J'
^TIh- -.



v
*1


, *- --







- ". /
/^-..
I-.I

.4


,^

?- 1-
j-.w
'I


C Resulting Network : after running SMTE


D Resulting Network : after solving LP


Figure 2-1. Graph example 1: node deployment by the uniform distribution


r


f


&


'I


m


^


u ^


.*


L









CHAPTER 2
EXPERIMENTS ON THE MINIMUM TRANSMISSION ENERGY ROUTING,
SEQUENTIAL MINIMUM TRANSMISSION ENERGY ROUTING, AND OPTIMAL
LIFETIME ROUTING

2.1 Overview

Recent advances in micro-electro-mechanical technology, wireless networking,

and integration technology provide new applications in various areas. Wireless sensor

network is the one of these areas and may be perhaps the most active research issue.

Sensor nodes can be implemented by integrating various sensing capabilities and

wireless communication into low cost and small form-factor embedded system without

paying much cost in these days. Wireless sensor networks which are the communication
infrastructure formed from massively deployed such sensor nodes in ad-hoc manner

have a variety range of application. For examples, environment monitoring (vehicular

traffic monitoring, wild life habitat monitoring, etc.) and military surveillance.

Sensor nodes typically operate with a limited on-board battery and it's very hard

or even impossible to replace the battery or replenish the energy of the nodes. Thus,

the longevity of wireless sensor networks has been considered as a primary goal in

wireless sensor network researches and many works to prolong the lifetime of wireless

sensor networks have been published [12] [11] [19] [35] [45]. On the other hand,

some researchers focus on how to apply the proposed algorithms to the real world, for

instance, distributed implementation of the linear programming approaches ( refer to [26]

[35]) or aprroximated algorithms (refer to [35] [15] [3] [45]).
In general, there are many factors that defines the lifetime of the sensor network,

for example, the size of the sensor field, the number of sensor nodes, the number of the

sink, etc. In practice, knowing how these parameters affect the lifetime of the networks.

However, investigating this problem taking all these into account is so difficult task. As

a preliminary step of this research, we do extensive simulations to get some intuition on

the relationships that maybe exist between the lifetime and these factors.









Let's denote Ni as the set of neighbor nodes which node i can reach with a certain

transmit power level. Each node / is assumed to have a limited energy level to begin

with or an initial energy level, denoted by E, which never be replenished during its

lifetime and let R(i) be the data generation rate which means the amount of data that

needs to be forwarded to the sink during unit time interval. Note that for the single sink

system, incoming data rate for the sink, denoted as s, should be equal to the aggregate

of the data generation rate of all nodes except the sink. We can express a foregoing

statement as R(s) = --eiN,,isR(i).

2.3.1 Energy Model

Energy required for a node i to transmit a unit of data to a node j, denoted as e is

given by,


e, = a + Od (2-10)

while the energy consumed by a node / in receiving a unit of data from a node j,

denoted as ej is given by,


e, =a (2-11)

where do is the Euclidean distance between node i and node j, and a is a

consumed energy to run the transmitter or receiver circuitry in the sensor, and 3

is the required energy to run the transmitter amplifier[19]. Although 6 should be

determined from various environmental factors, typically it is a constant between 2

and 4. For example, in free space propagation model, 6 is considered as 2, so that

energy consumption for the transmission of a single unit of data is proportional to the

square of the distance [33].

As for the initial setting of networks, we assume that every source has a connectivity

to the sink, that is, there are at least one path to the sink from every sources.









For the energy constraint (4-44), we can apply a similar procedure. Hence, we can

conclude that any feasible solution to the problem P(NV, L, ri), after suitable extension, is

also a feasible solution to the problem P(AV, L, r2).

Next, the queue-based model is less constraining than the sub-flow-based model;
this results in lifetime gains in the former model. The following theorem formalizes the

fact that the queue-based model always outperforms the sub-flow-based model.
Theorem 2. Let T be the optimal objective value to problem (4-41), and T* be the

optimal objective value to problem (4-51) with the same configuration (i", ) and the

same radius of coverage r. Then T < T*.

Proof. Let ((x')), (i)), (2k), T) be the optimal solution to problem (4-41). We

will prove this theorem by constructing a feasible solution to problem (4-51) with

((xi'), (wi()), ), ( ) and showing that under this feasible solution, the objective value
of problem (4-51) is T.
We now define a vector w as follows. For each I e , we let w(i) = () if i e Ri, and
w() = 0 otherwise. Then, we let q(0) = D d,, and q(') = ('-1) wi() for all i/ A. We

have the following sequence of assignments for the qi).


q(1)




q ,ll
;(2)



(H)


q(0) -_ w ) = D d,- w

) W(2)
q -w,

q(L1 _1W)(IL


By summing up above assignments for all/ e L, we have q) = D d, Y1 wl =

D d, D d, = 0 by (4-45) and the construction of w. Hence, (4-56) is satisfied. Since
the configuration and radius of coverage r for problem (4-41) are the same as those for

problem (4-51), NI(i), i e Ni, I e L are the same for both problems. Because of this and
by (4-43) and w() = q(l-) q(), (4-53) is satisfied. The energy constraints (4-44) and









From basic knowledge of geometry, we can find the area of the intersection of two

disks. If the distance between the centers of two disks of radii r and R, respectively, is d,

the area of the intersection is given by

A(r, R, d)
d2 +r2R d R22 -r2
=r2 cos- ( + R2 COS-1 +
2dr 2dR
(d r+ R)(d+ r- R)(d- r R)(d r R).

Now, we can find the area of Ri n Q. Let d be the distance of node X from the

sink and note that Iw is the communication range of node X. The area of Ri n Q, for

(j /) < i < j, is obtained by,

A(Q n Ri) = A(w, iw, d) A(/w, (i 1)w, d). (3-19)

The area of RO-I)+ n Q is given by

A(R0-/)+ n Q) = A(/w, w(j -/)+, d). (3-20)

The probability Fj(i) can be obtained by plugging equations (3-19) and (3-20) into

(3-18). By applying the expressions for Fj(i) to (3-13), we derive a set of equations

in pj only, for different j. From (3-18), note that Fj(i) depends on various pi. Hence,
the resulting equations cannot be solved by deriving each pi for i = n 1 down to 1

iteratively using (3-13). But, they may be solved by successive substitution, which is to

iterate (pt)) over t as in (3-14).

Note also that, previously, we have assumed that the distance between node X and

its next-hop node is determined by the rings in which the two nodes lie. This assumption

is not accurate in the current model. The distance to the next-hop node depends on

where the next-hop node lies in its ring. We will later describe an extended formulation

that incorporates the accurate distances between nodes, but will do so in a much more

general setting with respect to other aspects as well (Section 3.4). The price to pay is









The description of Algorithm 5-3 is ambiguous on how to make the algorithm
distributed and (partially) local. There are different possibilities. In one version,

each node i executes the outer for-loop in parallel, and for that, it requires only local
information. After the one-dimensional array Pi[] is computed, node i can broadcast this
array to all other nodes. After a node collects the complete two-dimensional array, it can

compute z* by itself. Another possibility is that each node i sends the array P,[] to the
sink; the sink computes z* and sends it back to every node, which goes on to execute
Algorithm 5-2.

5.3.2 Main Algorithm

We now assume that the system operates in a time-slotted way. The Lagrange dual

problem of (5-28) is

Dual: max 0(7)

s.t 7 > 0 (5-53)

Consider the subgradient projection method to solve problem (5-53). The update of

r at each iteration is given by the following equations.


()(k +1) = [7(/)(k) 6( x,')(k) + yj()(k)-
jiEN(i)
Sx')(k) y,'-l) (k))], Vi c A/,V c. (5-54)


where [b] = max{0, b} and 6(> 0) is a sufficiently small number.

Our algorithm is motivated by the subgradient algorithm, but not exactly identical.

The standard convergence results of the subgradient algorithm do not apply. In Section
5.4, we will use a different analytical framework to prove the optimality of the algorithm.

Let 6qi')(k) = (/)(k). We have the following algorithm. For technical reasons, the
upper bound of the flow variables is modified from M to M(c) M + NLe, where c is a
small positive value.


109









data from a sensor node i to the sink / is denoted by Ci, and it is given by (4-2) with

j replaced by /. We define the (downstream) neighbors of node i as N(i) = {j

AP U {I} d(i,j) < d}, when the transmission range is d. Note that the neighbors may

include the sink.

The paper does not consider MAC-layer contention. It is assumed that contention

is resolved by some MAC-layer protocol. The operation of the MAC-layer protocol

determines the link rates, which are assumed to be large enough so that they do not

impose a constraint on the data rates. Future work may try to relax these assumptions.

Conversely, if the data rates are small, then even simple MAC-layer protocols will be

able to deliver the required link rates. In other words, it can be easy to design one of

such protocols.

4.2.1 Static Sink Model

In the static sink model (SSM), the sink is located at the origin and remains

stationary during the operation of the WSN. Data originated from the sensor nodes

flows into the sink in a multi-hop fashion. As soon as the data becomes available at

a node, it gets transmitted toward the sink. Typically, the rate at which each sensor

node i harvests data from the outside world is a constant. We denote it by d,. The data

generated by a source is sometimes called a commodity or a sub-flow [1, 15]. Let x,

be the rate assignment from node i to the nodej for the traffic generated by node c

(commodity c). The problem of maximizing the lifetime in this model is formulated as

follows [11, 12].











Aggregate-Traffic Sub-Flow-Based DT-MSM


max T (4-42)

s. t. zi ( x ) k = wE() V/e /L; Vi e R (4-43)
\jeN/I(i) k:ie (k)
SII
zi/ C:E ) 7 +- x T Ei, Vi/E (4-44)
/=1 eNI(i) k:iN/(k)
w,(i) = D d,, Vi E AN (4-45)
I/:ERf
x ) > 0, VI e L; Vi e RI; Vj e Ni(i) (4-46)

w(') > 0, V/l LV/i R/ (4-47)

z > 0, V/e (4-48)

T > 0. (4-49)

The equivalence of the two formulations can be argued as follows. First, it is clear
that the feasibility set of the per-commodity-based formulation is inside the feasibility
set of the aggregate-traffic-based formulation. Conversely, given a feasible solution
{xu), w(i), zi} to the latter formulation, we can treat w,')/zi as the supply at each node
i when the sink is at stop /. For each /, we can decompose the arc flows {xf}i) into
path flows for all the commodities, {x, c)} (again, without loss of generality, we assume
the decomposition does not lead to positive cycle flows), satisfying (4-32), (4-33) and
(4-34).
Hence, {x('' w(}), z/} is a feasible solution to the per-commodity-based formulation.
4.3.2 Queue-Based Model
In the queue-based model, each sensor node can buffer data originated from any
node. Let q(') be the queue length at node / just before the sink moves from location /
to / + 1. Assume that each node i has D d, amount of data at the beginning of a cycle,


(4-41)








We then have,
c(y) = p(Zg(z)f(z,y)dz, y e A6. (3-23)
JA, p(Y)
Note that c(y) depends on the node density function p. But, we suppress it in the
notation for now. Then, we have

(y) g(z)f (z, y)dz+ y E A6. (3-24)
p(y) p(y)
After rearrangement,

p) = g f(z, y)p(z)dz + ye A6. (3-25)
/ g(y) g(y)

Let P(y) be the expected energy consumption for a node at y E As.

P(y) = (7 + | y xll)g(y)f(y,x)dx, ye .4 (3-26)

Let y* be an arbitrary point on the boundary of A, whose node density is assumed to be
a known parameter. Our objective is to have P(y) = P(y*) for all y e As, where

P(y*) = p (7 + y*- x l)f(y*,x)dx. (3-27)
p(y*) A
This gives the following.

P(y*)
g(y) = j y, ye .4. (3-28)
fA(7 + P I|y x||a)f(y, x) dx
In the above, we assume f is independent on the node density function. p(y*) is a
constant (parameter). Then, g(y) can be determined for all y e A6. Then, (3-25) is a
linear integral equation with the unknown function p. It is known as a Fredholm equation
of the second kind [28], which has the following general form.

(x) () (x) Ai k(x, s) (s)ds. (3-29)

In (3-29), A is a known constant, O(x) is an unknown function and f(x) is a known
function, f(x) / 0, where 0, f : U R for some U c Rm. The function k(x, s) is called a









of coverage. In the next theorem, we prove that the bigger the radius of coverage,the
longer the optimal lifetime is.
Theorem 1. If ro < r, < r2, then the optimal objective value for the problem P(AV, L, r2) is

greater than or equal to that for the problem P(.V, L, ri).

Proof. Consider the two optimization problems P(.V, L, ri) and P(jV, L, r2) with r, < r2.

It is obvious that NI(i, ri) c NI(i, r2) for all i E AV, I E . Therefore, we can split the larger
set Ni(i, r2) into two sets A and A, where A = Ni(i, ri), and A = Ni(i, r2)\A. Similarly,
we can also split the upstream neighbor set for node / into B = {k e fiVli e Ni(k, ri)}

and B = {k e f/l i e Ni(k, r2)}\B. In other words, A and B are the sets of additional
downstream and upstream neighbors for node i, respectively, as the radius of coverage
increases from r, to r2.
Suppose that (x, w, 2, T) is a feasible solution to the problem P(/V, L, ri). Now,
consider equation (4-43) for the optimization problem P(A/, L, r2). For V/ e L, Vi e

Ri(r2),

z )- M w(l). (4-61)
E\JE N(i, r2) k:iENi(k,r2) (
We have the following by separating the neighbor sets into A, A, B, and B.


z xM +) X ) x - X) = w(). (4-62)
JEA JEA kEB kEB

Fix / e L. Suppose i e R/(ri). We extend the vector 2 so that ') = 0 when e A and
Xk, = 0 when k e B. Then, for such I and i, the extended vector (x, w, 2, T) satisfies

(4-62) since the original vector satisfies (4-43) for i e Ri(r1).

Next suppose i e R/(r2)\R/(ri). Then, we can extend the vector x further by setting

x() = 0 for all j c NI(i, r2) and x') = 0 for all k such that i e Ni(k, r2). Furthermore,
we extend w by setting w,') = 0. After such extension, (x, w, 2, T) satisfies (4-62) for
i E Ri(r2)\Rl(r1) (since all terms are zero).









where s' is a nonnegative slack variable for (i, /). Let (2, y, ) be the optimal solution
of the formulation (5-28) (5-34) with constraint (5-30) replaced by the above equality

constraint with slack variables.

Suppose that 2 < z*. From a given optimal solution (2,, 2, ) that satisfies (5-36)
we can construct a path flow. Let P' be the set of path flows who have the (i, /) as the
origin. In the path flow formulation, (5-36) can be rephrase as follows.


f(p) = d + sI, (5-37)
pP/P

where f(p) is the amount of flow along the path p and di = Dd, if I = 1, otherwise dl = 0.

Now we construct another path flow f from f by the following flow assignment.

f(p)= f(p)(1 s'/ (p)) (5-38)

Here, if s' = 0, then inequality constraint becomes equality constraint. Thus, we suppose
that s' > 0. The above assignment satisfies the following property.


Sf(p) = d (5-39)
pEPi

If we construct edge flow from the new path flow assignment, f, the edge flow

assignment satisfies the flow conservation constraint as (5-12).
The energy constraint (5-29) can be also represented as a path flow as follows.
L
e)( f(p)) < 2E, (5-40)
I=1 p:(ij)ep,pEP/

Putting f(p) = f(p)+ f(p)(s,'/ ^ f(p)) into the energy constraint, we get
L L
i fp)) /=1 p :(ij)Ep,pEP /=1


103









CHAPTER 6
CONCLUSION

In this dissertation, we discuss various issues on efficient energy management

scheme in wireless sensor network.

First, we show why general techniques,especially the routing protocol used in wired

network are inappropriate in wireless sensor network through extensive experiments.

The least cost routing algorithms that wired networks prefers sometimes aggravate

the situation in wireless sensor network. This is because the least cost routes can be

also the most popular routes so that those routes are likely to be used frequently by

several nodes. The nodes along the least cost routes tend to exhaust their energy in

earlier time. It partitions the network, so that the whole network expires even though

the rest of nodes still have plenty of energy. With the help of mathematical optimization,

we can obtain the optimal route that will maximize the network lifetime. We compare

the performance between the least cost routing and optimal routing in terms of network

lifetime. As the size of network increases, the gap between two routing schemes

becomes bigger.

Second, we examine how to apply non-uniform deployment of the sensor nodes to

resolve the problem of uneven energy consumption rates by the nodes or the energy

hole problem in multi-hop wireless sensor networks. More generally, non-uniform

deployment with careful density control can be an important technique for achieving a

desirable lifetime and system-cost tradeoff of the sensor network. Our main contribution

is to present a method for computing the required node density function. As an example,

we show that the method enables us to compute the correct densities that achieve an

equal energy consumption rate for all nodes, thereby, extending the system lifetime. The

method is expected to be widely applicable to other similar objectives and constraints.

Third, we propose a new framework for improving the network lifetime by exploiting

sink mobility and delay tolerance. It is expected to be useful in applications that can


123









2.3.2 Simulation Model Of Minimum Transmission Energy Routing

Several routing algorithms for wireless sensor networks have been proposed.

Among them, minimum transmission energy(MTE) routing has an objective to minimize

the total transmission energy during lifetime, which is the time until the first node died.

The typical approach to the minimum transmission energy routing is to apply the

shortest path algorithm to the graph in which the edge cost is the energy that is required

to transmit a unit of data between two nodes [45]. Refer to 2.3.1 for more details. In fact,

MTE is another optimization problem and we can formulate it as a Linear Program:

min etxiJ (2-12)
iEN jeN

R(i) Vi E N, i 4 s
s.t. x, x (2-13)
jeNi J:eNj EkeNv,ks R(i) otherwise

T exu < E(i) Vi/eN (2-14)
jENi
xU > 0 V(i,j) L (2-15)

If the 5 defined in the 2.3.1 equals to 1, then MTE is equivalent to Minimum Hop

Routing. Due to the triangular inequality, MTE prefers the longer edges to get to the

neighbor close to the sink. If the 6 is greater than 1, MTE prefers the paths which consist

of many shorter edges to the paths of few longer edges.

However, the lifetime of the MTE strategy can perform arbitrarily badly, because this

does not consider the residual energy of the intermediate nodes along the paths, thus

can cause an unbalanced energy consumption distribution: the nodes on the minimum

energy path to the sink are drained their energy very fast, so that entire network can be

partitioned [11],[12]. [30] and [39] argue that by the time the sensors close to the sink is

drained its energy, sensors farther away from the sink still have plentiful energy. From














_. 4.. -
2- -. ,- "- -j '-
r > - ,- .' r4 ..
*_ r-











T - a ; -, ' ^ i i \ r
I .' r
"I* '. i -1 t'* i "* V j / _
S.. Nt. -. I, ,
* 4' A. 1.



... 4 -., '









, .'V -. -- ^ 7 .1. './ Z, ^
*"~ ~ i 1 \
-4-- '"-t

4 ..... ,-r-*'.r
- ' ' *_ T '. I - -\.













.' : _- .' '* 1 - -- -I \
'. : ,

,-s. r- A I
II 4



r. 4 4-, .. ,

--- /-... 4' r-- 7
... 4,_ - J: _
ct~ .r
I ---V '






A Initial Network B Resulting Network: after running MTE




F I-2. Graph *example 2:I ne e

'' "I "4 '. ,
N> "
J41- t *--
.I4'4 I 4
r7
I. I *1 I






1,.4 p *-- _.
t.- 4 I. -,
/ 4? j
-\-- : -"I



C~ ~ Resultin Network fe unigST Resulting Network after solving LPE
Fgr 2 Ga eap 2m
A 4-7 p I
t 4 r-..4~~

4.. SI..f'" -. 'I
*r 4 I




I -. ,,
-- I
4.. .
4 .4- 4-

C~ Reutn Newok afte runin SMEDRsligNtor:atrsligL
Fiur 2-.Gaheape2 oedplyetbrdbsdsrtg


















*consumption rate
densities
'I

0 2 4 6 8 101214161820
ring index
A/= 1


0 2 4 6 8 101214161820
ring index
B =2


4000
3500
3000
2500
2000
1500
1000
500
0


U consumption rate
1 , densities -.--- --
0 2 4 6 8 101214161820
ring index
C /= 10


0 2 4 6 8 101214161820
ring index
D / = 20


Figure 3-7. Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 3.


Let f(y, x) be the probability density function for a node at y to choose a node at

location x as the next-hop node, x e A. It satisfies Jf f(y, x)dx = 1 for every y.3 Note

that a node at y in general may not be able to reach directly everywhere in the whole

sensor field. If the region that it can reach directly is denoted by A(y), where A(y) c A,

we can assume that f(y, x) is non-zero only on A(y), and fA(y) f(y, x)dx = 1.




3 We allow the next-hop node to be the sink. Hence, we assume f(y, x) = 0 for x e
B(6) and x / 0. That is, if the next-hop node is inside the small neighborhood of the
sink, B(6), it must be the sink.


3000
2500
2000
1500
1000
500
0









5-2 Convergence to the optimal value, z* . . . . . . . ..... ....... 118

5-3 Lyapunov drift of the algorithm over time . . . . . . . ... . . 119

5-4 Time average of total virtual queue size over time . . . . . . . . 119

5-5 Long-time average of the difference between outflow and inflow . . . ... 120









3.2.3.2 Uniform node selection

In this routing model, a node X can select any node with the same probability as

long as the target node resides in X's communication range and is closer to the sink

than X is. This scheme is motivated by geographical routing.




S



o X



T



Rj

Figure 3-2. Uniform node selection


Figure 3-2 illustrates the underlying geometry for Uniform Node Selection. Node

X can choose any node in the shaded part in the figure as the next-hop node. Let

the region within the communication range of node X be denoted by Q. As shown in

Figure 3-2, node X can choose a next-hop node in the intersection of Q and the inner

rings that X can reach, i.e, L, -) (Rk n Q). When the nodes in ring k are uniformly

distributed, the number of nodes in Rk n Q is pkA(Rk n Q), where the notation A(S)

represents the area of a region S. The number of possible next-hop nodes for node X is

ZYk /j-i) PkA(Rk n Q). Hence, the probability F)(i) is, for ( /)+ < /
F (i)= p(R ) (3-18)
j-1
Spk-A(RknQ)
k=(j-l)+











Per-Commodity Static Sink Model (SSM)


max Z (4-6)

di, if i = c
s. t. x- X = Vi, Vc GA (4-7)
jEN(i) k:iN(k) 0, otherwise


E C x -x Z EN(i) c k:ieN(k) c

xV > 0, Vi, Vc A; Vj e N(i) (4-9)

Z> 0. (4-10)

The constraint (4-7) is the "flow conservation constraint", which states that, at

a node i, the sum of all outgoing flows for a commodity c is equal to the sum of all

incoming flows for the commodity c. If i = c, the incoming flows should include the

flows generated at node / itself, or d,. The inequality (4-8) is the energy constraint and it

means that the total energy consumed by a node during the lifetime (Z) cannot exceed

the initial energy of the node. With this formulation, the routing is dynamic and allows

multipath communications. There is no assumption on fixed-path routing, such as the

shortest path routing. The above optimization problem can be easily converted into a

linear programming (LP) problem.

The particular formulation above is equivalent to the following formulation, where

the flows of the commodities are aggregated into a single arc flow. The new formulation

has much reduced complexity and is useful for finding numerical solutions quickly.


(4-5)













Comparisond f ifetrie random node dstributlon and
gri based distribution in single sinK
600000 MTE grid-based
MTE gnd-based --
1.. .. ... -,- i


a)
E


E
H-


02


300 400 500 600
Number of nodes

A Lifetime (single sink)


700 800


Comparison of ra random node distribution and
grid -ased distribution in single sinK




03 3 E ED
LWWD r
"i ii


SMTE- grd-based --X---
LP grd-based -
TE- random -
SMTE random ----
LP random -0-


300 400 500 600
Number of nodes

B Ratio (single sink)


ir time of uifo m node d loment and
grid-based nod e deployment in multiple sinks
1 6e+06 TE-grid ----
MTE grid
MTE -gnd -)
14e+06 : : :

1 2e+06 i ..

le+06

800000


400000

200000

200 300 400 500 600 700 800
Number of nodes

C Lifetime (multiple sink)


.2


Companion of ratio ran om node distribut on and
grid-based dstributlon in multiple sinks


0 8 '-X-- 4 -. ,
--


04 .

MTE grid
02 SMTE grid
LP grid -
MTE Uniform I 0
SMTE Uniform '-MWl ,*
LP Uniform I
200 300 400 500 600 700 800
Number of nodes

D Ratio (multiple sink)


Figure 2-7. Performance of different node deployment strategies



equal to the communication range, the neighbor set Ni of node is is the set of all nodes


reside within this circle. Thus, larger communication range means the bigger neighbor


set. In addition, neighbor sets with different communication ranges have an inclusion


property, that is, the neighbor set with a larger communication range must include the


neighbor set with a smaller communication range.


Figure 2-8 shows the lifetimes and ratios of the simulations of various communication


ranges. In these simulations, the number of nodes is fixed to 400 and the communication


range varies from 3.5 meters to 10 meters. In each simulation, we use the same network


topology for the fairness. Note that the lifetimes and ratios remain constant for the case


of MTE without regard to the changes of the communication range. In the MTE routing,


T


700 800










transmit one unit of traffic to any receiver in its range, regardless of the distance. Many

wireless devices do not have the capability to adjust the transmission power level, and

hence, fit into the case of a = 0.


6 70 4 120
0 6 3.5
C. 5 60 100
E 5E 3
S 4 consumption rate 2.5 80 v
densities -- --- 40 C a5
3 i 2 consumption rate 60
30 15 densities -- --
So -1.5 no
2 40
S20 c 1
S 1 10 0.5 20
01 0 0 'AA 0
g 0 2 4 6 8 101214161820 g 0 2 4 6 8 101214161820
ring index ring index
A /= B =2

1.4 300 2.5 .200
0 0
LP 1.2 250 2
C2 2 -4 **-- --.'- -2--- -,------'.--150
E 1 E
t 200 1
S1.5
0 0.8
o 0.8 150 100 "
S0.6 consumption rate -*- 1 -
0.4 densities 100 0 consumption rate -
< 0.5 densities -- 50
S 0.2 50 a
S0 0 -AA A 0 0
0 2 4 6 8 101214161820 0 2 4 6 8 101214161820
ring index ring index
C / = 10 D / = 20

Figure 3-4. Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 0.


The results for a = 1, 2 and 3 are shown in Figure 3-5, 3-6 and 3-7, respectively.

In all plots, the curve for the average per-node energy consumption rate is flat. This

means that, if we deploy the nodes according to the computed densities, we can achieve

an even energy dissipation rate in all rings. These results indicate that our modeling

approach, analytical method and numerical solutions are all accurate or sound. Observe

the curves for the node densities, which can be quite oscillatory or irregular. We see that

it is hard to predict the deployment densities without precise computation.










14 A, 80 14 80
0 12 ******************* 70 0 70
LP 12 P70 12
E 10 60 E10 60
0 10
t 50 t 50
o 8 i o 8 consumption rate -
i \ 40 a densities 40 E
> 6 6
consumption rate 30 30 -
4 densities 20 20
S2 10 2 10
a 0 '-A 0 0
> 0 2 4 6 8 101214161820 0 2 4 6 8 101214161820
ring index ring index
A/= 1 B =2

14 80 14 80
S12 70 12 70

S50 S 50
10 8 E 80
o 40 o 40 "
> 6 6
Consumption rate 30 consumption rate 30
densities -- -- 4 densities 20
a, 2 10 a 2 10
___o_______________________ o '0AOAkAA AAAAA o
0 0 A0 0
> 0 2 4 6 8 101214161820 > 0 2 4 6 8 101214161820
ring index ring index
C / = 10 D / = 20

Figure 3-5. Node densities and average per-node energy consumption rates for various
maximum jump sizes, /, under uniform node selection. a = 1.


3.4 General Sensor Field and Routing Models

In this section, we consider general models where the shape of the sensor field

is arbitrary and the routing depends on the precise node location. The resulting node

density may vary continuously over the sensor field. For brevity, we omit some details in

the derivation, which is similar to the case with rings.

3.4.1 General Two-Dimensional Model

3.4.1.1 Node-density independent routing

Let A c R2 denote the whole region of the sensor field and let the sink be at the

origin. The region near the sink is an anomaly for the model. Let B(6) = {y e R2 ||lyll <

6} for some small 6 > 0. We assume that there are no other nodes in B(6) except the









If we put (-)x, z, and ( ) T in the places for x*(D'), z*(D') and T*(D') on the left hand
side of constraint (4-54), we have


I zI C u -. x u ) D 7 + D T ( (4 6 4 )
=1 eNi(i) k:ieN(k)

After canceling D and D', it is easy to see that the new solution (x, q, z, T) satisfies the

energy constraint of the problem P(D).
From the constraint (4-55) for the problem P(D') we have q,(0)(D') = D'd,. Since
q (D)q*(D'), q((D')= q(0)( ) = D'dJ(Vi e n). Therefore we have


q(0) = Dd,. (4-65)

From above argument, we have shown that new solution (x, q, z, T) is feasible to the

problem P(D). Hence, we have

T*(D) > T = T*(D').

Thus, it must be that T*(D)D > T*(D')D'.
Using a similar argument, we can also conclude that T*(D)D < T*(D')D'. Hence,
T*(D)D must equal to T*(D')D'

D

4.4 Experimental Results

In this section, we will present the results from numerical experiments. In particular

we have compared the network lifetimes of the following models.

* Static Sink Model (SSM): The stationary sink is located at the origin. We take the
performance of this model as the reference for comparison.

Mobile Sink Model (MSM): The sink can move to several locations to collect data.
When the sink is at each location, all sensors participate in the communication,
sending and relaying traffic to the sink.








kernel, where k : U x U -- R. In the integral of (3-29), V is a subset of U. For our case
of (3-25), A = 1, the unknown function is p, the known function is K/g, and the kernel is

k(z, y) = f(z, y)
g(y)
(7 xll)f(y,x)dxf() (3-30)
= z f (z, y). (3-30)
f, (7 z x |a)f(z, x)dx
It has been shown by the theorems known as Fredholm Alternatives that the
solution to (3-29) nearly always exists and is nearly always unique.
3.4.1.2 Node-density dependent routing
In general, we can assume f depends on the density function, and write f(y, x, p),
where p is a function on A6. As an example, f(y, x, p) may be proportional to the node
density at x and some other properties at y and x, denoted by h(y, x), if x is in the
region that a node at y can reach. That is,

h(y,x)p(x) X A(y)
f(y, x, p)= A(y) h(y,x)p(x)dx (3-31)
0 otherwise.

Then, we can write,

g(y, p) = c(y, p) K/p(y), y E As. (3-32)

c(y, p) = P) g(z, p)f(z, y, p)dz, y e As. (3-33)
JA p(Y)
If it is required that the per-node energy consumption rate is equal everywhere, the
outgoing traffic from a node at y satisfies the following, which also depends on p.

g() (7 + y*- xll)f(y*,x,p)dx K
f( (7+ P ly- xH1')f(y,x,p)dx p(y*)' y
The function p satisfies the following functional equation.

(y) = f (z, y, p)p(z)dz y y As. (3-35)
A g(y,p) g(y, p)









problem can be expressed as follows:


max A (2-1)

subject to > x(P) < c(e) for all e c E (2-2)
P:eEP

> Ad(j) for all j c V (2-3)
PEP,

x(P) > 0 for all P (2-4)


Now, the dual problem is like:


min c(e)l(e) (2-5)
e
subject to I(e) > zj for all j c V, P e Pj (2-6)
eEP
k
d()zy > 1 (2-7)
j 1

I(e) > 0 for all e c E (2-8)

zj> 0 for all j e V (2-9)


Note that I(e), zj are dual variable associated to the constraints for the edges and

nodes in the primal problem, respectively. In the development of the approximation

algorithm, the dual problem can be considered as an assignment of lengths ( : E R)

to the edges such that D(/)/a(I) is minimized, where a(/) = Ej mincost(/) and

D(I) = Zel(e)c(e). mincostj(/) is the minimum cost of d(j) units of commodity to flow

from source of j to sink of j. Note that I(e) means the cost of one unit of commodity

to flow along edge e and actually is the dual variable of the dual problem. The authors

also show that if the shortest path algorithm can be used as a subroutine in each

iteration instead of finding the minimum cost flow for a single commodity, convergence

time of the algorithm can be improved. [3, 4] is very similar to [35] in that they are

iterative algorithms trying to balancing the queue length for a single link (minimizing the








system of nonlinear equations with pj as the variables. However, (3-13) does suggest
a different kind of iterative method to compute each pj, which will be called successive
substitution. Suppose, we initialize the iteration at some constant p0) for all 1 < j < n.
This gives F(O)(j), for different k andj. Then, one can substitute p0) and F()(j) into the
right hand side of (3-13) and derive lp ) for all 1 < j < n. In a general iteration step t,
the following iteration occurs, for 1 < j < n.

't+1) Y~ J-l (j i)F(Ot)(i)
i (n-I ++
n (, -i (n i)a Ft)(i)
min(nj/) (j1 / -i)j Ft)(i) (2k 1)

k j 1 (k- ~ (k -i) aF t)(i) (2 1)
xF(t) )p t), (3-14)

where, for a fixed t, (Fkt)(j)) is computed using (p')). The process can be continued
until pjt) converges, for 1 < j < n. We can take values in the limit as the solutions of
the equations in (3-13). We will show by experiments in Section 3.3 that this procedure
indeed works.
Note that the constant K does not show up in (3-13). Hence, for the purpose of
computing the node densities, we can set K = 1 without affecting the solution, provided
p, is given. Suppose (Fk(j)) is independent of the scaling of the node density functions.
That is, Fk(j) remains unchanged for all k andj when pi is scaled by a constant factor
S> for all i, 1 < i < n. It can be observed that if (pj)n-1 is a solution to (3-13)
given pn, then (tpj)n 1 is a solution to (3-13) given Kpn. In this case, one can decide
the node densities as follows: Choose an arbitrary positive value for p,; compute all pj
for 1 < j < n; and find a suitable constant K and use (Kpjn 1 as the node densities for
deployment so that every ring has sufficient nodes to satisfy the monitoring need.
3.2.3 Models of Routing/Node Selection
We now have general equations that the node densities must satisfy to achieve an
equal energy dissipation rate for all nodes under a generic routing model (Fk(j)). For









Table 5-1. Performance comparison between CPLEX and Algorithm 3
network size (# of sensor nodes, # of sink locations) ratio of computing time
(50,5) 30.333
(100,5) 25.644
(200,5) 23.110
(50,10) 34.474
(100,10) 24.714
(200,10) 21.701


for the distributed algorithm to cope with the change of the network. However, when

the network changes, the computing node must redo the computation. In other words,

the distributed system ensures a certain degree of fault tolerance in nature. For those

reasons, distributed algorithm sometimes is preferable as a network control algorithm for

the large scale networks.

However, these benefits come at the cost. In our distributed algorithm, every node

runs three algorithms presented in the previous sections. Each node requires extra

communication for solving two sub-problems. For the sub-problem S1, each node has

to exchange information about the current length of virtual queues (which is denoted

as q(().) of its all outgoing wireless link in the beginning of time slots. Also, to solve the

sub-problem 52, each node disseminates information (which is Pi in the Algorithm 3.)

needed to run the Algorithm 3 to all other nodes. In the course of execution Algorithm

3, a node should run the fractional knapsack algorithm, which is Algorithm 2. Note

that Algorithm 2 only requires the information about the current queue length of the

neighbors and it is the same required arguments for the Algorithm 1.

Since the sub-problem 52 is formulated as Linear Programming problem, it is solved

by Linear Programming solver such as CPLEX. We now present the experimental

result showing how efficient our Algorithm 3 would be when compared to the Linear

Programming solver, which is the CPLEX in this experiment. The efficiency and

simplicity of the Algorithm 3 is the key factor to the distributed implementation of the

overall problem.










TABLE OF CONTENTS


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

LIST O FTABLES ..................... ................. 7

LIST OF FIGURES .................... ................. 8

ABSTRACT .................... ................... .. 10

CHAPTER

1 INTRODUCTION .................... ............... 12

1.1 Contributions ................... ............... 15
1.2 Organization of the Study ........................... 16

2 EXPERIMENTS ON THE MINIMUM TRANSMISSION ENERGY ROUTING,
SEQUENTIAL MINIMUM TRANSMISSION ENERGY ROUTING, AND OPTIMAL
LIFETIME ROUTING .................... ............ 18


2.1 O verview . . . . . . . . . . . .
2.2 Related Works ..................
2.3 System Model ...................
2.3.1 Energy Model .. ...............
2.3.2 Simulation Model Of Minimum Transmission
2.3.3 Linear Programming Model .........
2.4 Experim ents . . .. . .. . .. . .. .
2.4.1 Graph Generation .. ............
2.4.2 Lifetime vs. Ratio .. .............
2.4.3 Effect Of Node Densities ......
2.4.4 Effect Of The Number Of Sinks .......
2.4.5 Effect Of Node Densities And The Number C
2.4.6 Effect Of Node Deployment Strategies . .
2.4.7 Effect Of Communication Ranges ......


Energy


Routing


Sinks


2.5 Experiments on the Minimum Transmission Energy Routing, Sequential
Minimum Transmission Energy Routing, and Optimal Lifetime Routing .37
2.6 Summary .................... ................ 38

3 A METHOD FOR DECIDING NODE DENSITY IN NON-UNIFORM DEPLOYMENT
OF WIRELESS SENSOR NETWORKS ....................... 40


3.1
3.2


O verview . . . . . . . . . . . . .
Models with Discrete Ring Structure ..........
3.2.1 Sensor Field and Energy Consumption Models
3.2.2 Deriving the Node Densities of the Rings .
3.2.2.1 Case of density-independent routing .


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









tolerate a certain amount of delay in data delivery. We present the mathematical

formulations for optimizing the network lifetime under the proposed framework. We

identify several properties that our models possess. To validate the framework's ability

for improving network lifetime, we conduct extensive experiments and found that the

framework is superior to the models published previously, including the static sink

model and the mobile sink model without delay tolerance. The lifetime gain of the

proposed model is significant when compared to the previous models. Furthermore,

as the number of sink locations increases, the optimal network lifetime increases

substantially. The results of the paper can both be applied to practical situations and

be used as benchmarks for studying energy-efficient network designs. Furthermore,

we can point out three interesting future work directions. The paper has not touched

upon the issue of finding efficient algorithms to solve the optimization problems, but

has relied on standard, centralized algorithms. The first direction is to find simpler,

preferably distributed, algorithms, which are clearly more generally applicable. The goal

is likely to be attainable since the problems formulated in this paper are extension of

the network-flow problems and many efficient algorithms are known for such problems.

The second direction is to relax some of the simplifying assumptions of the formulations.

For instance, we can bring the non-zero traveling time by the sink and/or the finite link

transmission rate into the formulations. Either one seems to make the problems very

difficult, but more relevant at the same time. The third direction is to consider where to

choose for sink stops so that the network lifetime can be optimized.

Last, we extend the our research of DT-MSM. In previous work, we show that our

model is superior to other models in terms of life time. One of possible direction of

further research is to devise a method implementing DT-MSM model in a distributed

manner. Among sub flow based model and queue based model, we target the queue

based model because it always produces a better result than the other. Basically

we apply the dual method because it sometimes reveals nice structural properties









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128









CHAPTER 1
INTRODUCTION

The technological breakthroughs in MEMS (Micro Electro Mechanical Systems),

DSP (Digital Signal Processing), integrated circuit technologies, microprocessor

hardware, and wireless communication techniques have been phenomenal since last

few decades. In addition, the technological advance that researchers and engineers

have achieved in ad-hoc networking routing and protocol, pervasive computing,

embedded system technologies make it possible to mass-manufacture the low cost,

small-sized form factored, and versatile sensor nodes. They integrate general purpose

processors, wide varieties of sensing devices, and wireless communication devices.

A Wireless Sensor Networks typically consists of these cheap sensor nodes deployed

into the targeted area to be monitored. However, to make the cost of deploying wireless

sensor network to be low, a sensor node has an on-board battery, as well as a low

power processor and a limited memory space. Therefore it is necessary to make sensor

nodes collaborate with other sensor nodes to overcome this limitation. Since a Wireless

Sensor Network (WSN) is constructed with a huge number of sensor nodes, which are

densely and sometimes randomly deployed into the region of our interest, the location

of sensor nodes is not known at the time of deployment. Occasionally the sensor

nodes need to be deployed randomly into the hostile or hazardous terrain so that it is

not easy or even impossible to access that region Thus, the protocols or algorithms

used in the WSN should have the ability of organizing the network autonomously.

Furthermore, since the number of deployed sensor nodes is often tremendous, an

untethered operation for the individual sensor node is required.

The mission imposed on the sensor nodes is gathering information about the

surrounding environment, processing sensed raw data such as compression or

quantization, and transferring processed data to special locations called the sinks

for further processing. The sinks are typically more powerful in processing power and