Group Title: Department of Computer and Information Science and Engineering Technical Reports
Title: Modeling spatial and temporal dependencies of user mobility in wireless mobile networks
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Title: Modeling spatial and temporal dependencies of user mobility in wireless mobile networks
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Creator: Hsu, Wei-jen
Spryopoulos, Thrasyvoulos
Psounis, Konstantinos
Helmy, Ahmed
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University of Southern California
Unviersity of Florida
Unviersity of Florida
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Modeling Spatial and Temporal Dependencies of

User Mobility in Wireless Mobile Networks

Wei-jen Hsu*, Thrasyvoulos Sp I, 1p, uhil, -t, Konstantinos Psounist and Ahmed Helmy*
*Dept. of Computer and Information Science and Engineering, University of Florida, Gainesville, Florida
32611-6120
tComputer Engineering and Networks Lab, ETH Zurich
1Dept. of Electrical Engineering, University of Southern California, Los Angeles, California 90089-2560
Email: wjhsu@ufl.edu, spyropoulos@tik.ee.ethz.ch, kpsounis@usc.edu, helmy@ufl.edu


Abstract-Realistic mobility models are fundamental to eval-
uate the performance of protocols in mobile ad hoc networks.
Unfortunately, there are no mobility models that capture the
non-homogeneous behaviors in both space and time commonly
found in reality, while at the same time being easy to use and
analyze. Motivated by this, we propose a time-variant community
mobility model, referred to as the TVC model, which realistically
captures spatial and time correlations. We achieve the goal with,
on the one hand, the concept of communities that leads to skewed
location visiting preferences, and, on the other hand, time periods
that allow us to model time dependent behaviors and periodic
re-appearances of nodes at specific locations.
To demonstrate the power and flexibility of the TVC model we
use it to generate synthetic traces that match the characteristics
of a number of qualitatively different mobility traces, including
wireless LAN traces, vehicular mobility traces, and human
encounter traces. More importantly, we show that, despite the
high level of realism achieved, our TVC model is still theoretically
tractable. To establish this, we derive a number of important
quantities related to protocol performance, such as the nodal
spatial distribution, the average node degree, and the hitting
and meeting times for the TVC model, and provide examples of
how to utilize this theory to guide design decisions in routing
protocols.


I. INTRODUCTION
In recent years, there has been an exponential growth in
the popularity of portable computation and communication de-
vices. Advances in wireless communication technologies and
standards have made ubiquitous communication an emerging
reality. With the ever expanding adoption of these wireless-
capable devices, there is an increasing interest in new com-
munication paradigms and applications that do not necessarily
rely on the infrastructure.
Mobile ad hoc networks (MANETs) [9] are self-
organized, infrastructure-less networks that could potentially
support many applications, such as vehicular networking
(VANET) [30], [31], disaster relief [29], wild-life tracking[32],
[33], and Internet provision to rural areas [3], to name a
few. As the devices are easily portable and the scenarios of
deployment are inherently dynamic, mobility becomes one
of the key characteristics in most of these networks. It has
been shown that mobility impacts MANETs in multiple ways,
such as network capacity [8], routing performance [16], and
cluster maintenance [28]. In short, the evaluation of protocols
and services for MANETs seems to be inseparable from the


underlying mobility models. It is, thus, of crucial importance
to have suitable mobility models as the foundation for the
study of ad hoc networks.
Ideally, a good mobility model should achieve a number
of goals: (i) it should first capture realistic mobility patterns
of scenarios in which one wants to eventually operate the
network; (ii) at the same time it is desirable that the model
is mathematically tractable; this is very important to allow
researchers to derive performance bounds and understand the
limitations of various protocols under the given scenario, as
in [5], [6], [8], [26]; (iii) finally, it should be flexible enough
to provide qualitatively and quantitatively different mobility
characteristics by changing some parameters of the model, yet
in a repeatable and scalable manner; designing a new mobility
model for each existing or new scenario is undesirable.
Most existing mobility models excel in one or, less often,
two aspects of the above requirements, but none satisfies all of
them at the same time. The most widely used mobility models
are random mobility models such as random walk, brownian
motion, random direction, and random waypoint [11]. Their
strength is the theoretical tractability but their weakness is the
lack of realism. More complicated synthetic mobility models
(e.g., [21], [48], [49]) improve the realism, but most of the
time at the expense of theoretical tractability. More recently,
a large number of empirical mobility traces from real mobile
users have been collected [12], [13], [14], [15], [26]. Although
one can use such traces directly in an evaluation with excellent
realism, these traces are usually rather inflexible and provide
only a single snapshot of the underlying mobility process. To
address these two issues, trace-based mobility models [18],
[19], [20], [24] have been proposed (i.e. larger, more flexible
\ I lc li traces created from the smaller empirically collected
ones). Yet, most of these models do not possess the necessary
flexibility to match mobility characteristics of traces other than
the ones on which they are based.
The main contribution of this paper is the proposal of a
time-variant community mobility model, referred to as the
TVC model, which is realistic, flexible, and mathematically
tractable. The model captures several important mobility char-
acteristics we, and other researchers, have observed empir-
ically from various WLAN traces [17]. One of the salient
characteristics is location preference. In the TVC model, we
extend the concept of communities from [22] to serve as pop-









ular locations for the nodes. Another important characteristic
is the time-dependent, periodical behavior of many nodes. To
capture this, we implement time periods in which the nodes
move differently [1]. To our best knowledge, this is the
first synthetic mobility model that captures non-homogeneous
behavior in both space and time.
To establish the flexibility of our TVC model we show
that we can match its two prominent properties, location
visiting preferences and periodical re-appearance, with mul-
tiple WLAN traces, collected from environments such as
university campuses [15], [17] and corporate buildings [12].
More interestingly, although we motivate the TVC model
with the observations made on WLAN traces, our model is
generic enough to have wider applicability. We validate this
claim by examples of matching our TVC model with two
additional mobility traces: a vehicle mobility trace[36] and
a human encounter trace[26]. In the former case, we observe
that location visiting preferences and periodical re-appearance
are also prominent characteristics in vehicular movements. In
the later case, we are able to match our TVC model with
some other mobility characteristics, namely the inter meeting
time and encounter duration between different users/devices.
These latter quantities are particularly important for encounter-
based (or "delay-tolerant" [2]) protocols. Despite these char-
acteristics are not explicitly incorporated in our model by its
construction, they can be still realistically reproduced.
Finally, in addition to the improved realism, the TVC model
can be mathematically treated to derive analytical expressions
for important quantities of interest, such as the nodal spatial
distribution, the average node degree, the hitting time (time
required for a mobile node to hit a randomly selected coor-
dinate) and the meeting time (time required for two mobile
nodes to come within communication range of each other).
These quantities are often fundamental to theoretically study
issues such as routing performance, capacity, connectivity, etc.
We show that our theoretical derivations are accurate through
simulation cases with a wide range of parameter sets, and
additionally provide examples of how our theory could be
utilized in actual protocol design. To our best knowledge, this
is the first synthetic mobility model proposed that matches
measurement sets (traces) collected from multiple scenarios,
and has also been theoretically treated to the extent presented
in this paper. Due to its strengths in both flexibility and theo-
retical tractability, the TVC model has two major applications:
to generate realistic mobility patterns under a wide range of
different scenarios, as we show in section IV, and to facilitate
performance analysis and prediction, as we show in section
VII. We also make the code of the TVC model available
at [37].
The remaining of the paper is organized as follows: In
Section II we discuss some related work. Our TVC model
is then introduced in Section III. In Section IV, we show
how to generate realistic mobility scenarios with matching
mobility characteristics in various traces. Then, in Section
V, we embark to present our theoretical framework and
derive generic expressions of various quantities under the TVC
model. The accuracy of these expressions is validated against
simulations in Section VI. Additionally, in Section VII, we


motivate our theoretical framework further, by applying our
analysis to provide guidelines and performance predictions in
protocol design. Finally, we conclude the paper in Section
VIII.


II. RELATED WORK
Mobility models have been long recognized as one of
the fundamental components that impacts the performance of
wireless ad hoc networks. For example, although the per node
capacity of static networks goes to zero as network density
increases [7], it is shown that mobility can improve the scaling
law for the capacity to 0(1) [8]. Different underlying mobility
models also change the performance ordering of various
MANET routing protocols [16]. More recently, mobility has
been utilized as the enabling factor for message delivery in de-
lay tolerant networks (DTNs[2]), where a complete path from
the source node to the destination node does not exist at any
time instant, broadening the scenarios in which communication
networks can be established.
A wide variety of mobility models are available in the
research community (see [11] for a good survey). Among all
mobility models, the popularity of random mobility models
(e.g., random walk, random direction, and random waypoint)
roots in its simplicity. They are not only easy to generate,
tune and scale, but also amenable to mathematical analysis. A
number of important properties for these models have been
studied, such as the stationary nodal distribution [44], the
hitting and meeting times [22], and the meeting duration [45].
These quantities in turn enable routing protocol analysis to
produce performance bounds [4], [5], [6]. However, random
mobility models are based on over-simplified assumptions, and
as has been shown recently and we will also show in the paper,
the resulting mobility characteristics are very different from
real-life scenarios. Hence, it is debatable whether the findings
under these models will directly translate into performances
in real-world implementations of MANETs.
More recently, an array of synthetic mobility models are
proposed to improve the realism of the simple random mobility
models. More complex rules are introduced to make the
nodes follow a popularity distribution when selecting the next
destination [21], stay on designated paths for movements [49],
or move as a group [48]. More variants of mobility rules
can be found in various models [11]. These rules enrich the
scenarios covered by the synthetic mobility models, but at the
same time make theoretical treatment of these models difficult.
In addition, most synthetic mobility models are still limited to
i.i.d. models, and the mobility decisions are also independent
of the current location of nodes and time of simulation.
A different approach to mobility modeling is by empirical
mobility trace collection. Along this line, researchers have ex-
ploited existing wireless network infrastructure, such as wire-
less LANs (e.g., [12], [14], [15]) or cellular phone networks
(e.g., [13]), to track user mobility by monitoring their locations
(through their associations with base stations). Such traces
can be replayed as input mobility patterns for simulations
of network protocols [46], [47]. More recently, DTN-specific
testbeds [26], [27], [30], [32], [33] aim at collecting encounter









events between mobile nodes instead of the actual mobility
patterns. Some initial efforts to mathematically analyze some
properties of these traces can be found in [26], [51]. Yet, due to
the experimental nature of these studies, the size of the traces
and the environments in which the experiments are performed
can not be adjusted at will by the researchers. To improve
the flexibility of traces, the approach of trace-based mobility
models have also been proposed [18], [19], [20], [24]. Based
on the collected traces, these models discover the underlying
mobility rules that lead to the observed properties (such as the
nodal distribution, the duration of stay at locations, the arrival
patterns, etc.) in the traces. Statistical analysis is then used to
determine proper parameters of the model to match it with the
particular trace.
The goal of this work is to combine the strengths of various
approaches to mobility modeling and propose a realistic, flex-
ible, and mathematically tractable synthetic mobility model.
Our work is partly motivated by a previous work of ours [17]
in which we identify several prominent properties that are
common in multiple WLAN traces. Large-scale deployments
of WLANs in university [14], [15] and corporate [12] cam-
puses provide excellent platforms in which huge amount
of user data can be collected and analyzed. Resources for
measurement-based study are available at two network trace
archives [34], [35]. We leverage these traces to understand
empirical user mobility, and propose the TVC model based
on the prominent characteristics observed. This model extends
the concept of communities proposed by us in [22] and also
introduces time-dependent behavior. A preliminary version of
the model has been presented in [1].
We differentiate our work from other trace-based mod-
els [18], [19], [20], [24] in several aspects. First, among all ef-
forts of providing realistic mobility models, to our best knowl-
edge, this is the first work to explicitly capture time-variant
mobility characteristics. Although capturing time-dependent
behavior is suggested in [24], it has not been incorporated in
the particular paper. Second, while previous works emphasize
the capability to truthfully recreate the mobility characteristics
observed from the traces, we also strive to ensure at the same
time the mathematical tractability of the model. Our motiva-
tion is that this additional requirement will largely facilitate
the application of our model for performance prediction of
various communication protocols. Finally, although we start
from mobility characteristics observed from WLAN traces,
we propose a mobility model that has wider applicability in
addition to WLANs, it can be tuned to match with other types
of traces, such as a vehicle mobility trace [36], and even with
other characteristics in other traces of human mobility (e.g.,
the encounter duration and the inter-meeting time [26]).
As a final note, the concept of community is also mentioned
in [25] in a different context. The authors assume the attraction
of a community (i.e., a geographical area) to a mobile node
is derived from the number of friends of this node currently
residing in the community. In our paper we assume that the
nodes follow location-based preference to make movement
decisions, and each node moves independently of the others
(nonetheless, node sharing the same community will exhibit
mobility correlation). Mobility models with inter-node depen-


dency require a solid understanding of the social network
structure, which is an important area under development. In
future work, we intend to include such mobility behaviors
motivated by factors other than location preference (e.g. social
bonds) into our model.

III. TIME-VARIANT MOBILITY MODEL
A. Mobility Characteristics Observed in WLAN Traces
The main objective of the paper is to propose a mobility
model that captures the important mobility characteristics ob-
served in daily life. To better understand this mobility, we have
chosen to study a number of wireless LAN traces collected by
several research groups (e.g., traces available at [34] or [35]).
The reason for this choice is that WLAN traces log information
regarding large numbers of nodes, and thus are more reliable
for statistical analysis. After analyzing a large number of
traces, we have observed two important properties that seem to
be recurrent in all of them: skewed location visiting preferences
and time-dependent mobility behavior [17].
First, by location visiting preference we mean the amount
of time that a node spends associated with a given access point
(AP). In Fig. l(a) we calculate for various traces the fraction
of the total online time an average node spends with its most
favorite AP, its second favorite AP, etc., up to its least favorite
AP. (This is essentially the probability density function of the
association time of a node with an AP, with the APs sorted
in descending order of total association time.) It is clear from
the plot that a node on average spends more than I -.'. of its
online time associated with its favorite AP, and more than I.'.
of its online time at only five APs. We refer to this behavior
by saying that the location visiting preference (or in short
ih ,.iIli, preference") of nodes is skewed.
Second, by time-dependent mobility behavior we refer to
the fact that nodes tend to behave differently during different
times of the day (or even during different days), and most
specifically to exhibit some amount of periodicity in terms of
the locations they choose to visit. In Fig. l(b) we plot the
probability of a node appearing in the same location at some
time in the future, as a function of the difference in time. It
is evident from the plot that nodes appear at the same AP
with a higher probability after a time-gap of integer multiples
of days. This creates the saw-tooth pattern in the curves. A
slightly stronger weekly correlation could also be observed in
some plots (see for example the slightly large peak in the MIT
curve for a time gap of seven days). It is thus clear that nodes
behave differently in different periods in time, and that similar
behaviors tend be repeated on a daily basis.
Unfortunately, most existing mobility models fail to cap-
ture these two properties. For simple random models, like
random direction, random waypoint, random walk, etc., there
is obviously no preference in both space and time. This is
demonstrated in Fig. 1 by a straight line (uniform distri-
bution) for the Random Direction model for the respective
probabilities. Even for more sophisticated models that try to
capture other aspects of mobility, such as group mobility in
the RPGM model [48] or a model considering obstacles and
1.,ilv, .,!i [49], these two properties would also be straight









AP sorted by total am unt of association time
11 21 31 41 51 61 71 81 91 101
1 E+0 -

Random direction model

Co j un ty m od el
1 E -4
.UCSD
E003
SE s


(a) Skewed location visiting preferences.
067 7I------------------


4-i
&


rimegap(days)
(b) Periodical re-appearance at the same location.

Fig. 1. Two important :i....ii. 1. IIiI.. observed from WLAN traces. Labels
of traces used: MIT: trace from [12], Dart: trace from [15], UCSD: trace from
[14], USC: trace from [17].



lines in the plots as spatial and temporal preference is not
a part of these models1. There do exist some more recent
models (e.g., [22], [21], [18], [19], [20]) that aim at capturing
spatial preference explicitly. An example of such a model is
the simpler community model of [22]. As is shown in Fig. l(a),
with appropriately assigned parameters this model is able to
capture the skewed location visiting preference, to some extent.
However, time-dependent behavior is not captured, and thus
the periodical re-appearance property cannot be reproduced,
as shown by the flat curve labeled community model in Fig.
l(b).
It is our goal to design a mobility model that successfully
captures both of these two properties, observed in the majority
of traces. One could argue that a potential shortcoming of
this approach is that WLAN traces do not register contin-
uous movement of the devices, but rather associations of
users/nodes with specific APs. What is more, some devices
are not always on, and typically there are some gaps in the
coverage of access points in these networks. However, we
believe that the two main properties we observed, namely
skewed location preference and time-dependency, are preva-
lent in real-life mobility. This belief is further supported by
observing typical daily activities of humans: most of us tend
to spend most of the time at a handful of frequently visited
locations, and a recurrent daily or weekly schedule is an
inseparable part of our lives. As a result, a model supporting
location-preference and time-dependent mobility should be
able to capture human mobility in many contexts, if carefully
designed. Comparison with non-WLAN traces in Section IV
confirms our argument.



1In the case of the obstacle-based model, some locations are not allowed to
be visited at all; yet among all the permissible ones, no particular preference
is assigned to any node.


commun...ty.. mode
r r
R ndom rtomoe


TABLE I
PARAMETERS OF THE TIME-VARIANT COMMUNITY MOBILITY MODEL

For all parameters, we follow the convention that the subscript of a quantity represents
its community index, and the superscript represents the time period index.
N Edge length of simulation area
V Number of time periods
Tt Duration of t-th time period
St Number of communities in time period t
C Edge length of community j in time period t
Comm The j-th community during time period t
t Probability that the next epoch is performed in
7FJ community j during time period t
vmi, vmax, v Minimum, maximum, and average speed2
Dmaxj, Dj Maximum and average pause time after each epoch2
Lj Average epoch length for community j
ot |pt Probability that a node is moving | pausing
moe, pause,3 when being in community j during period t
t Fraction of time the node is in
P_ state j (P = Po, +
K Transmission range of nodes
A(aE, bt) The overlapped area between Comm.. of node a
A b and Commr of node b
Wt A specific relationship between a target coordinate
and the communities in time period t
Qt The set of all possible relationships between
a target coordinate and the communities in time period t
P(wt) Probability of a given relationship wt in time period t
A(wt) Area corresponding to the relationship wt
Ph(wt) Unit-time hitting probability
under the specific scenario wt
PH (Wt) Hitting probability for a time period t
under specific scenario wt
Pm Unit-time meeting probability
PM Meeting probability for a time period t
HT(case) Expected hitting time under the given "case"
MT(case) Expected meeting time under the given "case"




B. Construction of the Time-variant Community Model

Skewed location preferences arises naturally due to extended
stay at locations that bear importance to us, such as homes and
offices, cafeterias and libraries. To capture this phenomenon,
we construct popular locations) for the nodes in the simulation
field. These locations, or rather geographical areas, we call
communities, and we make a node visit its own community
more often than other areas. Different nodes can pick different
communities, creating nodes with very diverse behaviors.
Furthermore, multiple communities could be defined if a node
tends to visit multiple locations with high probability, some
of which could also be shared by more than one nodes (e.g.
people working in the same building, libraries, etc.).
Periodical re-appearance at a given location is related to
omnipresent schedules in our lives. Almost everyone follows
recurrent daily or weekly schedules, and different behaviors
based on time-of-day have been observed in many contexts.
To capture time-dependent behaviors, we introduce structure
in the time domain by the use of time periods. For a given
node, we assign several time periods during which it behaves
differently. For example, a node may have different com-
munities during different periods or the same communities
but different mobility parameters to move between them.
To further ensure periodicity, the time period assignment
follows a recurrent structure, with the same "time-period" and
its respective statistical characteristics occurring, say, for all
weekday mornings.
We illustrate the model with an example in Fig. 2. We also









use this example to introduce the notations we use (see Table
I) in the rest of the paper. As shown in the example, there are
V 3 (where V denotes the total number of time periods)
time periods TP1, TP2, and TP3 (of duration T1, T2, and T3,
respectively). During each time period, there are a number
of communities, that is, geographical areas that are heavily
visited. These communities can be chosen differently in each
time period, as shown by the three sub-plots. Within a given
time period t, the j-th community is denoted as Comm .2 This
is a square geographical area of edge length Ct. Note that by
construction the communities can overlap (as in TP1 in Fig.
2), or one community can even contain the other (as in TP2
in Fig. 2). Finally, the number of communities in each time
period may vary. For example, there are 3 communities in total
in the first period, 2 in the second one, and 4 in the third3. The
total number of communities in period t is denoted as St. This
construction allows for maximum flexibility when designing
the simulation setup for nodes with different behaviors. As for
the structure in time domain, we need to arrange time periods
in a re-current sequence (see Fig. 2 or Fig. 3) that corresponds
to a daily or weekly schedule.
We now describe how a node moves inside the above con-
struction. Node movement consists of a sequence of epochs.
Each of these epochs is a Random Direction movement4.
In a typical Random Direction epoch, a node chooses at
the beginning its speed uniformly in [v in, vm a], and a
direction (angle) uniformly in [0, 27]; it also chooses the length
(distance) of movement (usually distributed exponentially with
average in the order of the network dimension), and moves
towards this direction with the chosen speed and for the chosen
distance; at the end of the epoch, the node picks a pause time
randomly and then proceeds to the next epoch.
The difference between our community model and the
Random Direction model is that in addition to all other
parameters, in the community model the node also chooses
randomly the community in which the next epoch will be
performed. That is, with probability 7j the next epoch takes
place inside the node's j-th community (E' Tj 1 1), rather
than moving around the whole simulation area randomly, as in
the standard Random Direction model. (Note that we usually
add superscript t in the notation, i.e. Ti, to denote that these
probabilities might change between time periods.) We say the
node is in state j when it has an epoch in the j-th community.
Further, to ensure that a local move is compatible with the
local community size, we also scale the local epoch length
by drawing it from an exponential distribution with average



2For all parameters used in the paper, we follow the convention that the
subscript of a quantity represents its community index, and the superscript
represents the time period index. Note that all parameters used in the paper can
be set differently for each node. When necessary, we use a pair of parentheses
to include the node ID for a particular parameter, e.g., C denotes the edge
length of the j-th community during time period t for node i.
3To allow a node to move randomly among the whole simulation field
sometimes, we often allocate one community to be the whole simulation field
(e.g. Commr in period TP1 in Fig. 2).
4Note that we could also choose random waypoint or random walk models
for the type of movement during each epoch.


I TP1 TP2 TP3. TP1 TP2. TP3I TP1
Repetitive time period structure, V=3 T 2 I T Time
S 3 S22 S3-4
Time period (TP1) Time period (TP2) Time period (TP3)
Comm Comm

N C c cmm c2 *

Comm2
M3' Co

Comom omm

Fig. 2. Illustration of a generic scenario of time-variant mobility model, with
three time periods and different numbers of communities in each time period.


TP1 TPF TP1 TP? TP1 TPF TP1 TP2 TP1 TPT TP3 TPF TP3 TP2


Weekdays


Weekend


'Time


Fig. 3. An illustration of a simple weekly schedule, where we use time
period 1 (TP1) to capture weekday working hour, TP2 to capture night time,
and TP3 to capture weekend day time.


length Lj, that is, in the order of the given community size5.
It is important to note that a node can still perform some of its
epochs in the whole simulation area, by assigning an additional
community that corresponds to the whole simulation field (e.g.
Common in period TP1 in Fig. 2). We refer to such epochs as
roaming epochs. Finally, after an epoch, a node pauses for
a time uniformly chosen in [0, Dmax,j], where the maximum
pause time is again dependent on the community.
As a final note, one may argue that capturing location
preference and time-dependencies could plausibly be achieved
with a mobility model constructed with different ways than the
one we propose. However, our choices are largely guided by
the fact that most of the building blocks we utilize to create our
mobility model (e.g. random direction epochs, communities,
etc.) are easy to understand, and have been shown to be
amenable to theoretical analysis [22]. This will become evident
in Section V. At the same time, we will show next that these
choices do not compromise our model's ability to accurately
capture real life mobility scenarios.


IV. GENERATION OF MOBILITY SCENARIOS FOR
SIMULATION
The TVC model described in the previous section provides
a general framework to model a wide range of mobility
scenarios, and provides a powerful tool for simulation-based
protocol or service evaluations in MANETs. We have made
our mobility trace generator available at [37]. The tool pro-
vides mobility traces in both ns-2[38] compatible format and
time-location (i.e., (t, x, y)) format.
In this section, our aim is twofold: (i) first, we would
like to demonstrate the model's flexibility and how it can be
configured to generate mobility instances that are represen-
tative of various target wireless networks such as WLANs,
VANETs, etc.; (ii) at the same time, we would like to validate

5To avoid boundary effects, we assume that if the node hits the community
boundary it is re-inserted from the other end of the area (i.e., the boundaries
are "torus" boundaries).









the model's Ic'.liNi or "accuracy" by explicitly comparing
mobility instances produced by our model with real mobility
instances captured in well-known, publicly-available traces.
However, it is important to note that the use of such a model
is not merely to match it with any specific trace instance
available; this is only done for validation and calibration
purposes. Rather, the goal is to be able to reproduce a much
larger range of realistic mobility instances than a single trace
can provide.
We first outline here some general guidelines about how to
use the model in order to construct specific mobility scenarios.
Then, we show how to explicitly configure the TVC model in
order to match the mobility characteristics observed in three
case studies: a wireless LAN trace, a vehicular trace, and a
human-encounter trace. All the parameter values we use in the
examples in this section are also available at [37].
STEP 1: Determine the structure in Space and Time The
first step to construct the TVC model for a given scenario is to
setup the communities and the time period structure. If the map
of the target environment is available, one should observe the
map and identify the points of attractions in the given environ-
ment and how they vary with respect to time (e.g., restaurants
on campus during lunch time, hotels in an amusement park
during nights), and assign the communities/time periods in
the TVC model accordingly. If the map is not available,
alternatively, one could use the general mobility characteristics
observed in typical traces for the particular target network
(as shown, for example, in Fig. 1) as guidelines to assign
the structure in space and time. For example, from location
preference curves like the one in Fig. l(a), one can determine
the number of communities one needs to explicitly create; as
a very simple example, if in most WLAN traces it is observed
that the typical node spends say .' of its time at around 2 to
5 preferred locations (depending on the node), then one could
assign each node to have from 2 to 5 local communities in the
network (with the actual number and locations of communities
randomly chosen for each node), with a larger (roaming)
community representing the rest of the 5% of mobility time6.
Similarly, from curves like the ones in Fig. l(b), one may
observe the re-appearance periodicity and decide on the time
period structure accordingly. If a finer time granularity is
necessary (e.g. time-of-day) one could additionally observe
the mobility characteristics (e.g. location preferences) on an
hour-by-hour basis and identify clear changes in a node's daily
behavior.
STEP 2: Assign Community-related Parameters Ideally,
for a given environment, once the communities are identified,
the related parameters (e.g., ,- D, L', which represent the
probability, average pause time, and average epoch length,
at community j during time period t) could be assigned
according to the mobile nodes' behavior in each community
(e.g., how long does a typical person spend at the cafeteria for
lunch?). Nevertheless, in most cases such information is not
available, or extremely difficult to obtain. Hence, one could

6In reality, one may be able to capture more complex structures with
more communities or structure between them, by combining knowledge from
the actual network area (map), generic mobility characteristics, and other
information about the network.


again resort to measured statistics from typical traces to guide
the assignment of the parameters. It is not difficult to see that,
typically, the attraction of the communities (TF) and the time
spent in each community (related to D', L') determine the
shape of the location visiting preference curve. Thus, one can
use basic probability theory to calculate the expected fraction
of time a node spends in a given community as a function
of these parameters, and derive from it the values needed to
obtain a given location preference profile observed in a curve
in Fig. l(a). We calculate these community stay probabilities
later in Lemma 5.1.
STEP 3: Assign User On-off Behavior Note that the mo-
bility trace generated by the TVC model is an "always-on"
mobility trajectory of the mobile node (i.e., the node is always
present somewhere in the simulation field). Depending on
the target environment, this always-on behavior may not be
realistic. In many empirically collected traces, not all nodes
are present all the time (i.e., some of the nodes are "off" or not
in the observed area sometimes). This is the case in two of the
scenarios we discuss below in the WLAN traces, nodes are
"on" only when they are not moving; in the vehicle mobility
trace, nodes are "on" when they are moving. Thus, before
producing the final synthetic trace, assumptions about when
the user is considered "on" should sometimes be made and
superimposed to the TVC mobility traces generated. We have
applied this step to the traces of the two scenarios mentioned
above.
Next, we look into three specific case studies, namely a
set of WLAN traces, a vehicular trace, and a trace of inter-
node encounters. We show how to apply the fore-mentioned
procedure in each case, and show that \wiilicik mobility
traces produced by the TVC model successfully match the
characteristics observed in the real traces.
1) Matching Mobility (I.,,l,. r a r,. with WLAN Traces:
In the first example, we show that the TVC model can re-
create the location preferences and re-appearance probability
curves observed in WLANs. We construct our synthetic trace
from the TVC model with the following steps: (STEP1) We
divide the simulation area into a 10-by-10 grid and use these
100 grid cells as the locations for the purpose of measuring
mobility statistics for the simulated nodes. For each node, we
assign some of the grids to serve as the node's communities
during each time period, according to the method described
earlier. (STEP2) We use the mobility characteristics obtained
from the WLAN traces (i.e., curves in Fig. l(a)) to calculate
the attraction from the communities (7T) and the pause times
of the node (D') to shape a decaying tail of location visiting
preference. (STEP3) Since devices are usually turned off when
users move them in the real WLANs, we make a similar
assumption that the mobile nodes are considered "on" only
when they are not moving. When the simulated node moves,
we assume that it is not associated with the grid. Note that the
curves in Fig. l(b) represent the probability of an "on" node
associated with the same community after the given time gap,
and the peaks appear when the considered points in time are
in the same type of time period. Therefore, the peak value
is Z J( t)'f(PL )2, where Pt,, denotes the probability









AP sorted by total amount of time associated with it
1 11 21 31 41 51 61 71 81 91


1 E-04
I E-05 I
1 E- del-complex
( 1 E-05 vIT

(a) Skewed location visiting preferences.
S03


o Model-complex
0 2 4 6 8
Tim e gap (days)
(b) Periodical re-appearance at the same location.


Fig. 4. Matching mobility characteristics of the synthetic traces to the MIT
WLAN trace.


a node is considered "on". Hence, the fraction of time nodes
spend on moving (L/v) and pause (Dp) can be adjusted (note
that in this case, Pj D /(D + L /v)) to change the
peak values in the curve of periodical re-appearance property
to match with the curves in Fig. l(b).
We use the MIT WLAN trace (first presented in [12]) as an
example to display the match between the \ illwick trace de-
rived from the TVC model and the real trace. We also achieved
good matching with the USC[17] or the Dartmouth[15] traces,
but do not show it here due to space limitations (see [37]).
We show the skewed location visiting preferences and the
periodical re-appearance properties in Fig. 4 (a) and (b),
respectively. We first try a simple synthetic model (labeled
as to/. .1 i/. \'q.i,/_. using the parameters of Model-1 in Table
II) with one community in two time periods. While this simple
model captures the major trends in the mobility characteristics,
there are several noticeable differences. First, since there is
only one community, the tail in the i.. il. q, / 'lq1't I curve in
Fig. 4(a) is "flat" as opposed to the exponentially diminishing
tail of the MIT curve (notice the Y-axis in Fig. 4(a) is in log
scale). Second, the peaks in the i,. .t I,/ 'li.'im. 1 curve in Fig.
4(b) are of equal heights, due to the simple two-alternating-
time-period structure, as opposed to varying peak values of
the MIT curve. We can improve the matching between the
\N!llidik trace and the real trace by adding complexity in
both space (using more communities) and time (using more
complex schedule, such as the weekly schedule shown in
Fig. 3). In a refined model labeled as Model-complex in Fig.
4, we show that the resulting mobility characteristics match
very closely with the MIT trace. This also demonstrates the
flexibility of our model the user can adjust its complexity
by choosing the number of communities and time periods
needed to achieve a desired level of matching with the mobility
characteristics.
2) Matching Mobility (I,,,,,. r a I. with Vehicle Mobility
Traces: In this example we display that skewed location
visiting preferences and periodical re-appearance are also
prominent mobility properties in vehicle mobility traces. We
obtain a vehicle movement trace from [36], a website that


1i E o6 -




(a) Skewed location visiting (b) Periodical re-appearance at the
preferences, same location.
Fig. 5. Matching mobility characteristics of the synthetic trace to the vehicle
mobility trace.


tracks participating taxis in the greater San Francisco area.
We process a 40-day trace obtained between Sep. 22, 2006
and Nov. 1, 2006 for 549 taxis. We obtain the mobility
characteristics of the taxis by the following steps. For each
taxi, we first identify its movement range within the 40-day
period, then draw a rectangular area that bounds the movement
of the taxi, and divide this area into equal-sized 10-by-10 grids.
We tally the mobility statistics of the taxis using these 100
grids as locations, and show the results in Fig. 5 (a) and (b),
respectively, with the label Vehicle-trace. It is interesting to
observe that the trend of vehicular movements is very similar
to that of WLAN users in terms of these two properties.
We further show that, using the outlined procedures, we
can generate a \llllik trace with similar mobility charac-
teristics as the vehicle mobility trace. After observing the
trace closely, we discover that the taxis are offline (i.e., not
reporting their locations) when not in operation. Hence in
the wldwliL'k trace we make the corresponding assumption
(in STEP3) that the nodes are associated with the current
grid they reside in only when they are moving; we then
consider the pause times as breaks in the taxi operation (hence
P,,j (L /)/(D + L /v) in this case), from which we
can calculate or adjust the respective model parameters. The
curves in Fig. 5 with label Model correspond to the mobility
characteristics of the \' inliLik trace. As a final note, although
vehicular movements are generally constrained by streets and
our TVC model does not capture such microscopic behaviors,
designated paths and other constraints could still be added in
the model's map (for vehicular or human mobility) without
losing its basic properties. We defer this for future work.
3) Matching Contact Characteristics with Encounter-based
Traces: In this example, we show that the TVC model is
generic enough to also reproduce the distributions of the
inter-meeting time and the encounter duration observed from
a human encounter trace [26], by setting up its parameters
properly. Specifically, we tune our mobility model to mimic
the behaviors observed in an experiment performed at INFO-
COM 2005 [26]. In this experiment, wireless devices were
distributed to 41 participants of the conference, with appropri-
ate software installed that could log encounters between nodes
(i.e. coming within Bluetooth communication range), as they
moved around the premises of the conference area.
The inter-meeting time and the encounter duration distri-
butions of all 820 pairs of users obtained from this trace
are shown in Fig. 6 with label Cambridge-INFOCOM-trace.
To mimic such behaviors using our TVC model, we observe
the conference schedule at INFOCOM, and set up a daily












S ]dg IN~F OCO


000001ooo' 0000
(a) Inter-meeting Time. (b) Encounter duration.
Fig. 6. Matching inter-meeting time and encounter duration distributions
with the encounter trace.


recurrent schedule with five different types of time periods
(STEP 1): technical sessions, coffee breaks, breakfast/lunch
time, evening, and late night (see [37] for the detailed param-
eters). For each time period we set up communities as the
conference rooms, the dining room, etc. We also generate a
community that is far away from the rest of the communities
for each node and make the node sometimes isolated in this
community to mimic the behavior of patrons skipping part of
the conference. It is interesting to note that the inter-meeting
time distribution has a sharp drop (the "knee" in the curve)
at 16 hours, which is approximately the time gap between
the end of the day and the beginning of the subsequent day
at the conference. This suggests the nodes (naturally) meet
with lower probability during the nights, and thus the time-
dependent mobility provided by our TVC model is appropriate.
We can naturally achieve this by assigning nodes to disjoint
communities (i.e., the "hotel i'.Inis ) during the nights. In
STEP2, we use the theory presented in section V to adjust
the parameters and shape the inter-meeting time and encounter
duration curves. For example, a stronger tendency for nodes to
choose roaming epochs (setting larger 7r) would increase the
meeting probability (see, e.g., Eq. (15)), hence reducing inter-
meeting times. Since the devices used to collect the encounter
traces are always-on, we do not apply any changes to the
\ mllclii trace in STEP 3. We randomly generate 820 pairs of
users and obtain their corresponding distributions of the inter-
meeting time and the encounter duration. These distributions
are shown in Fig. 6 with label Model. It is clear that our
TVC model has the capability to reproduce the observed
distributions, even if it is not constructed explicitly to do so.
This displays its success in capturing the decisive factors of
typical human mobility.
It is clear from the cases studied here that, once we observe
the target environment closely and come up with the right
underlying parameters, the TVC model is able to capture the
consequent mobility characteristics well. In addition, with the
respective configuration, it is possible to generate synthetic
traces with much larger scale (i.e., more nodes) than the
empirical ones while maintaining the same mobility charac-
teristics. It is also possible to generate multiple instances of
the i lloclik traces with the same mobility characteristics to
complement the original, empirically collected trace.

V. THEORETICAL ANALYSIS OF THE TVC MODEL
So far, we have established the strength and flexibility of the
TVC model in terms of its ability to reproduce the properties


M111.9% d--aio (1)
1 oo looo loooo looooo0


observed in real mobility traces. Although other proposed
models have also managed to match some sets of collected
measurements [18], [19], [20], [24], none of the existing works
has been shown to capture the variety of qualitatively different
traces (e.g. WLAN, vehicles, inter-contacts) that the TVC
model does. Yet, one of the biggest advantages of our model is
that, in addition to the realism, it is also analytically tractable
with respect to some important quantities which determine
protocol performance. In the rest of this paper, we focus on
demonstrating this last point. We will first derive the various
properties and then validate their accuracy against simulation
results. Then, we will provide some examples of how the
various theoretical quantities thus obtained can be used to
calculate or optimize the performance of popular protocols,
in realistic mobility settings.
We start here by deriving the theoretic expressions of
various properties of the proposed mobility model. We first
calculate the nodal spatial distribution. This can be repre-
sented as a two-dimensional probability density function of
nodes at any given point in the space. The expected number
of nodes in an area is then calculated by integrating the density
function. This property provides a basic demographic profile
of the simulation area, and can also help to evaluate whether
the model reflects the target environment. Then, we derive the
average node degree, which is the average number of nodes
residing within the communication range of a given node. This
is a quantity of interest due to its implication on the success
rate of various tasks in mobile ad hoc networks [41], [42].
Finally, we derive the expected hitting and meeting times for
our model. The hitting time is the time it takes a node, starting
from the stationary distribution, to move within transmission
range of a fixed, randomly chosen target coordinate in the
simulation field. The meeting time is the time until two mobile
nodes, both starting from the stationary distribution, move
into the transmission range of each other. Note that this
definition is different from inter-meeting time used in [23],
[26], which accounts for the time duration between a meeting
event and the subsequent one7. These two quantities are of
interest due to their close relationship to the performance
of DTN routing protocols, or in general the performance of
processes that rely on node encounters. Knowing the meeting
time for a mobility model is, for example, crucial when using
a "mobility-assisted" or "store-carry-and-forward" protocol to
deliver a message [4], [5], [6], while hitting times might be
needed if some nodes in the network are static (e.g. sensors,
base stations, etc.).
We note that a preliminary version of some of the theoretical
derivations presented here appear for a special case of our TVC
model only in [1] (that model included one community and
two time periods only). Here, we generalize all derivations
for any community and time-period structure. Moreover, we
present some additional results regarding the spatial distribu-
tion and the average node degree that are relevant to various
wireless communication protocols, as we show in Section VII.
We start with a useful lemma that calculates the probability

7Although not shown in this paper, we could use similar techniques as in
[45] to derive inter-meeting times








of a node to reside in a particular state.
Lemma 5.1: The probability that a node moves or pauses
(after the completion of an epoch) in state j, at any given time
instant during time period t, is:

t( 7Lt vt t
t (1)
l "k^^kulc IycJ


Pt
pause,j


7t t
Ei k k + k )


Proof: The result follows from the ratio of the average
durations of the moving part (L /vt) and the pause part (Dt)
of each state, weighted by the probabilities of choosing the
state. U
Note that the above stationary probabilities can be calculated
for each time period and node separately. We use Pt(i) to
denote the probability that node i is in state j during time
period t (i.e., P (i) = P (i) + P i))
3 M V6 3 move(+ pause,j

A. Derivation of the Nodal Spatial Distribution
We start with the derivation of the nodal spatial distribution.
This becomes relatively straight-forward after we observe that
a node follows a basic random mobility model (i.e., random
direction) in each community. Hence, when a given node is in
state j, it appears equally likely in any point within Commj.
Theorem 5.2: For a given area A, the probability for a node
to appear in A at any given time instant during time period t


Sp(x, y) dxdy,

where the function p(x, y) is the spatial density function,


p(x, y)


E t/ t2
P /C .o
:l(, c ... i *


(4)


Proof: A node could appear at a given point in space
when it is in state j if and only if the j-th community includes
the point. Within the community, the appearance probability of
the node is uniformly distributed. Considering a given point
(x, y), the probability for a node to appear at the point is
the sum of the contributions from all of its communities that
contain the point. 0
Note that the nodal spatial distribution for each time period
is independent, hence can be calculated separately with the
above Theorem.

B. Derivation of the Average Node Degree
The average node degree of a node is defined as the expected
number of nodes falling within its communication range. Each
node contributes to the average node degree independently, as
nodes make independent movement decisions.
Lemma 5.3: Consider a pair of nodes, a and b. Assume
further that, in time period t, community j of node a and
community k of node b overlap with each other for an area
A(a, It ), as illustrated in Fig. 7. Then, the contribution of
node b to the average node degree of node a, when a resides


Fig. 7. Illustration of overlapping communities in general case.


in its j-th community and b resides in its k-th community, is
given by
7K'2 A(a,'t)
t 3, (5)
C (a) C (b) (5)
where K is the communication range of the nodes.
Proof: Since nodes are uniformly distributed within each
community, the probability for node b to fall in the j-th
community of node a is simply the ratio of the overlapped area
over the size of the k-th community of node b. Node a covers
any given point in its community equal-likely, hence given
node b is in the overlapped area, it is within the communication
range of node a with probability wK2/Ct2 (a).
Following the same principle in Lemma 5.3, we include all
community pairs and arrive at the following Theorem.
Theorem 5.4: The average node degree of a given node a


SPcomm E E
vCot .(a) Vb vCo.mm.(b


rK2 A(a,' t)
PJ(c 2 (a) c()) (6)


Proof: Eq. (6) is simply a weighted average of the node
degree of node a conditioning on its states. For each state with
probability PJ(a), the expected node degree is a sum over all
other nodes' probability of being within the communication
range of node a, again conditioning on all possible states. U
Corollary 5.5: In the special case when all nodes choose
their communities uniformly at random among the simulation
field, Eq. (6) degenerates to


t (b) 2
Vb VComm (b) C (b)
7K2
NV72 5 P (b)
Vb VComE (b)
Vb VCoTm (b)


2b)
N2


i7K2
N2.
Vb


Proof: This result follows from that a randomly chosen
community is anywhere in the simulation field equal likely.
If nodes pick their communities randomly and independently,
the actual location of node a would not make any difference in
its average node degree. Regardless of the location of node a,
it falls within the k-th community of node b with probability
C2 (b)/N2. Within the community, node b appears uniformly,
contributing probability 7K2/C2 (b) of appearance within
node a's communication range. Note that the equation reduces
to each node b contributing i7K2/N2 to the average node de-
gree, the same as if node b roams around the whole simulation
area without any preference in space (i.e., communities). U


C15









Similar to the nodal spatial distribution, the average node
degree can be calculated for each time period separately.


C. Derivation of the Hitting Time
The sketch of the derivation of the hitting time is as
follows: (1) We first condition on the relative location of
the target coordinate with respect to a node's communities
(e.g. target inside community i, target outside community,
etc.). We thus have to derive the hitting time for each sub-
case separately. (2) We then derive the unit-step probabilities
of hitting a target, Ph, for a given subcase. The unit-step
probability is the probability of encountering the target exactly
within the next time-unit (rather than within the duration of a
whole epoch). In other words, we approximate the continuous
mobility with a discrete version of it where nodes move
in discrete steps. It has been shown in [22] that the latter
provides a good approximation for the continuous version, and
is easier to analyze for our purposes. (3) The expected hitting
probability for a whole time period, PH, is then calculated
for each sub-case from the unit-step probability, by assuming
IIIlling" occurs independently in each time step8. (4) Finally,
taking the weighted average of each sub-case (i.e. weighted
by probability of a given target being located inside a given
community) we get the overall hitting time.
The most influential factor for the hitting time is whether the
target coordinate is chosen inside the node's communities. We
denote the possible relationships between the target location
and the set up of communities during time period t as the set
Qt. Note that the cardinality of set Qt is at most 2st (i.e. for
each of the St communities, the target coordinate is either in
or out of it). Also, not all of the 2s combinations are always
valid. For example, in the set up of time period 2 in Fig. 2,
the communities are overlapped, hence if the target is within
Comm1 it must be within Conmm .
Lemma 5.6: By the law of total probability, the average
hitting time can be written as


E
wl Q1 l ....WV-QV


P(wl,...,wV)HT(wl,...,wV), (8)


where w w2, ..., wv denote one particular relationship (i.e. a
combination of {out, in}s ) between the target coordinate and
the community set up during time period 1, 2, ..., V, respec-
tively. Functions P(.) and HT(-) denote the corresponding
probability for this scenario and the conditional hitting time
under this scenario, respectively. Note that each sub-case
{w1, w2, ..., wv} is disjoint from all other sub-cases.
To evaluate Eq. (8), we need to calculate P(w1, ..., wv) and
HT(w1, ..., wv) for each possible sub-case (w1, ..., wV).
Lemma 5.7: If the target coordinate is chosen independent
of the communities and the communities in each time period
are chosen independently from other periods, then


P(W"..., WV)


n 1P(wt),


8This assumption of independence is shown in [22] to be a good approx-
imation, when the expected length of an epoch is in the order of the square
root of the area of the community the epoch takes place in.


where P(wt) A(wt)/N2, i.e., the probability of a sub-
case w' is proportional to the area A(wt) that corresponds
to the specific scenario wt, which is a series of conditions
of the following type: ({target E commr }, {target
comm }, ..., {target E commrs).
Proof: The result follows from simple geometric argu-
ments. U
The first step for calculating HT(w1, ..., wv) is to derive
the unit-time hitting probability in time period t under target
coordinate-community relationship wt, denoted as P/(wt).
Lemma 5.8: For a given time period t and a specific
scenario wt,
St
P (wt) I(target e Comm |wt)P .t.o,2Kv /C ,
j31
(10)
where I(.) is the indicator function.
Proof: The overall unit-time hitting probability is the sum
of the hitting probabilities contributed by epochs in each state.
Note that the hitting event can only occur when the node is
physically moving, and the node can hit the target when it
is moving in its j-th community only if the target coordinate
is within the community9. When a node moves with average
speed vj in community j, on average it covers a new area of
2K-j in unit time. Since a node following random direction
movements visits the area it moves about with equal proba-
bility, and the target coordinate is chosen at random, it falls
in this newly covered area with probability 2Kv~/Ct [22].
Hence the contribution to the unit-time hitting probability by
movements made in state j is P t ov, j2Kvj/C0 i.e., when
the node moves in community j and the target is in the newly
covered area in the time unit. 0
Note that the movement made in each time unit does not
increase or decrease the probability of hitting the target in
the subsequent time units, therefore each time unit can be
considered as an independent Bernoulli trial with success
probability given in Eq. (10). The corollary below immediately
follows.
Corollary 5.9: The probability for at least one hitting event
to occur during time period t under scenario wt is

(t) = 1 (1 P(wt))t. (11)

Finally, using the law of total probability, we derive the
conditional hitting time under a specific target-community
relationship, HT(w1, ..., wV).
Theorem 5.10:


HT(wl, ..., wV)


v
5 HT(w', ..., w Ifirst hit in period t).
t=1


P(wl,..., wV, first hit in period t),


where the probability for the first hitting event to happen in

9We neglect the small ii..i.i".1I1 that the target is chosen out of the
community but close to it, and make the contributions from epochs in state
j zero if the chosen target coordinate is not in community j.









time period t is
P(wl,..., wV, first hit in period t)
Tn-1(1 P(w1) p()
P
and the hitting time under this specific condition is
HT(wl, ..., wV first hit in period t)


1 (~
Ti=
P=1


t1 1
1) + T'+ T p(W t)
i=1 h[(U' )


(13)





(14)


where P 1 - II1( Ph,(wt)) is the hitting probability
for one full cycle of time periods.
Proof: Eq. (13) holds as each cycle of time periods
follows the same repetitive structure, and for the first hitting
event to occur in time period t it must not occur in time period
1, ..., (t 1). The first term in Eq. (14) corresponds to the
expected duration of full time period cycles until the hitting
event occurs. Since for each cycle the success probability of
hitting the target is P, in expectation it takes 1/P cycles to
hit the target, and there are 1/P 1 full cycles. The second
term in Eq. (14) is the sum of duration of time periods before
the time period t in which the hitting event occurs in the last
cycle. Finally, the third term is the fraction of the last time
period before the hitting event occurs. Note that the last part is
an approximation which holds if the time periods we consider
are much longer than unit-time. 0


D. Derivation of the Meeting Time
The procedures of the derivation of the meeting time is
similar to that of the hitting time detailed in the last section.
In short, we derive the unit-step (or unit-time) meeting prob-
ability, P,, and the meeting probability for each type of time
period, PM, and put them together to get the overall meeting
time in a similar fashion as in Theorem 5.10.
Similar to Lemma 5.8, we add up the contributions to the
meeting probability from all community pairs from node a
and b in the following Lemma.
Lemma 5.11: Let community j of node a and community k
of node b overlap with each other for an area A(at, .' ) in
time period t, as illustrated in Fig. 7. Then, the conditional
unit-time meeting probability in time period t when node a
and b are in its community j and k, respectively, is



t 2Kv A(at .t) A(at ,.t)
t t% (' K')
.* '. s,' . . i

SPt at 2Kv A(a, )A(a, I (15)
M+ ) o, A(at,.) C2(a) C 2(b)
S2K A(a, 't) A(a, t) 15)
Pt aip
stop,() o ve (b)A(a, I.') C2(a) C0((b)

Proof: Equation (15) consists of two parts:
(I) Both of the nodes are moving within the overlapped area.
This adds the first term in Eq. (15) to the meeting probability.


_A(a'k) A(at' k)
The two ratios, (a;- ,and 'bt capture the probabilities
> "a) Ck (b)
that the nodes are in the overlapped area of the communities.
The contribution to the unit-time meeting probability is the
product of probabilities of both nodes moving within the
overlapped area and the term 2n 2 which reflects the
covered area in unit time. We use the fact that when both
nodes move according to the random direction model, one can
calculate the effective (extra) area covered by assuming that
one node is static, and the other is moving with the (higher)
relative speed between the two. This difference is capture with
the multiplicative factor v [22].
(II) One node is moving in the overlapped area, and the
other one pauses within the area. This adds the remaining two
terms in Eq. (15) to the unit-time meeting probability. These
terms follow similar rationale as the previous one, with the
difference that now only one node is moving. The second term
corresponds to the case when node a moves (and b is static),
and the third term corresponds to the case when node b moves
(and a is static).
The derivation of the unit-time meeting probability between
nodes a and b for time period t includes all possible scenarios
of community overlap. If node a has St(a) communities
and node b has St(b) communities, there can be at most
St(a)St(b) community-overlapping scenarios in time period
t. U

Note that (15) is the general form of Equation (13) and (14)
in [1]. If we assume perfect overlap and a single community
from both nodes, we arrive at (14). If we assume no over-
lap, we result in (13). Also note in the general expressions
presented in this paper, the whole simulation area is also
considered as a community. Therefore we do not have to
include a separate term to capture the roaming epochs.
Corollary 5.12: The probability for at least one meeting
event to occur during time period t is

PM 1 E {Pov(a' b). (1 -Pt(a't t } (16)
-" -~ ~(js kci (16)
v(j,c)
where P,, (at, '' ) is the probability that the community j of
node a overlaps with community k of node b. This quantity
is simply 1 if the communities have fixed assignments and
A(a', I.) 7 0. If the communities are chosen randomly, this
probability can be derived by Lemma 4.5 in [1]. Due to space
constraint, the Lemma is not reproduced here.
Finally, similarly to Theorem 5.10, the expected meeting
time can be calculated using the results in the Lemmas in this
section.
Theorem 5.13: The expected meeting time is
v
MT MT(meet in period t)P(meet in period t).
t=l
(17)
Where the quantities in the above equation are calculated by


, P P(18)


P(meet in period t)








1'-tier
Community Community-1

2nd-tier cente
Commu ity er Community-2

(a) Concentric multiple-tier (b) Multiple randomly placed
communities setting, communities setting.
Fig. 8. Illustration of the community setup for the generic cases of time-
variant community model.

V1 1t
MT(meet in period t) = T 1) + T' + ,
i=1 i=1 Pm
(19)
where Q 1 Hni1 (1 Pj) is the meeting probability for
one full cycle of time periods.
Proof: The proof is parallel to that of Theorem 5.10.
Eq. (18) holds as each cycle of time periods follows the
same repetitive structure, and for the first meeting event to
occur in time period t it must not occur in time period
1, ..., (t 1). The first term in Eq. (19) corresponds to the
expected duration of full time period cycles until the meeting
event occurs. Since for each cycle the success probability of
the two nodes meeting each other is Q, in expectation it takes
1/Q cycles for them to meet for the first time, and there are
1/Q 1 full cycles. The second term in Eq. (19) is the sum
of duration of time periods before the time period t in which
the meeting event occurs in the last cycle. Finally, the third
term is the fraction of the last time period before the meeting
event occurs. Note that the last part is an approximation which
holds if the time periods we consider are much longer than
unit-time. U

VI. VALIDATION OF THE THEORY WITH SIMULATIONS
In this section, we compare the theoretical derivations of the
previous section against the corresponding simulation results,
for various parameter settings. Through extensive simulations
with multiple scenarios and parameter settings, we establish
the accuracy of the theoretical framework.
We summarize the parameters for the tested scenarios in
Table II. Table II (a) lists the parameters we use for a simplified
model (two time periods with two communities in each time
period, where one of the communities is the whole simulation
field). For more complex models, we try out the setup of tiered
communities (as shown in Fig. 8 (a)) and multiple randomly
placed communities (as shown in Fig. 8 (b)). In the tiered
communities layout, a randomly chosen point in the simulation
field serves as the center of the communities, and multiple tiers
of communities with different sizes share the same center. This
construction is suggested by a common observation from our
daily lives: People visit the vicinity area of locations that bear
importance to them more often than roam far away. When we
assign the tiered community structure, it naturally makes sense
to have the node visit the outer tiers less frequently than the
inner tiers, although this is not required for the theoretical
derivation. In the simulations, we use two alternative time
periods with a two-tier local community in each time period,
and the parameters are listed in Table II (b). In the multiple


randomly placed communities layout, multiple communities
are instantiated randomly to show that our theory is not limited
to a single community. We use two time periods with two
randomly placed communities each for this scenario. Other
than the difference in community setup and sizes, we again
use the parameters in Table II (b) for this case. Our discrete-
time simulator is written in C++, and nodes move as described
in Section III. More details about the simulator, as well as the
simulator code, can be found at [37].

A. The Nodal Spatial Distribution
To observe the nodal spatial distribution, we divide the
simulation area into a 20-by-20 grid and count the average
number of nodes in each grid block during the simulation.
The results presented in this subsection is the average of 5000
runs of independent simulations.
If the communities are randomly chosen, the node should
appear at each of the 400 evenly divided grid equal-likely,
with probability 1/400. We observe that the spatial distribution
of node varies a bit about this value in the simulation, as
shown in Fig. 9 (a). The minor discrepancy is due to the finite
number of samples. To make the scenario more interesting,
we also generate the spatial distribution for nodes when the
communities are fixed. We use the parameter sets of Model-1
(one community in each time period) and Model-5 (two-tier
community in each time period) from table II, and assign the
center of the community at either (300, 300) or (700,700) with
one half probability. The resulting nodal spatial distributions
are shown in Fig. 9 (b) and (c), respectively. The node appears
with higher probability where the communities are assigned.
From Eq. (3), for the scenario in Fig. 9 (b), the node appears
in the community with probability 0.0864 and in other area
with probability 0.0008, respectively. For the scenario in Fig. 9
(c), the node appears in the first-tier community, the second-
tier community, and the other area with probability 0.0759,
0.0039, and 0.0004, respectively. In both cases the simulation
results follow the theoretical results reasonably well, within
about 10% error for the area in the communities.

B. The Average Node Degree
For the average node degree, we create simulation scenarios
with 50 nodes in the simulation area, and calculate the average
node degree of each node by taking the time average across
snapshots taken every second during the simulation, and then
average across all nodes. All the runs last for 60000 seconds
in this subsection.
As we show in Corollary 5.5, when the communities are
randomly chosen, the average node degree turns out to be the
average number of nodes falling in the communication range
of a given node, as if all nodes are uniformly distributed.
Hence the average node degree does not depend on the exact
choices of community setup (i.e. single, multiple, or multi-tier
communities) or other mobility parameters. We illustrate this
scenario in Fig. 10 (a), where we compare the evolution of
the theoretical average node degree versus the communication
range (K) to the simulation results for some of the models
listed in Table II. The simulation curves follow the prediction











TABLE II
PARAMETERS FOR THE SCENARIOS IN THE SIMULATION


Common parameters: For simplicity, we use the same movement speed for all node: vmax 15 and vmin 5 in all scenarios. In all cases we use two time periods and they
are named as time period 1 and 2 for consistency. In the simple model we use a single local community (with subscript 1) in each time period. For the generic model, we test with
two different configurations: (1) A two-tier community in each time period, as illustrated in Fig. 8 (a). In this scenario the inner tier community and the outer tier community has
edge length C11 and C02, respectively. (2) Two randomly placed communities in each time period, as illustrated in Fig. 8 (b). In this scenario the communities both have edge
length C11, but the parameters correspond to the two communities are different (i.e., correspond to subscript 11 and 12 in the table). In all cases, there is also a roaming state (with
subscript r) in which the node moves about the whole simulation area (i.e. the whole simulation area is a community).

(a) The simple model.


Description
Match with the MIT trace
Highly attractive communities
Not attractive communities
Large-size communities


1000 100 100
1000 200 50
1000 100 100


Dm_,l Dma,/
100 50
100 200
50 200
50 100
(b) he eneric model.
Dmaxzli Dmax,12
25 15
50 20


80 520 0.714 0.286 0.8
52 520 0.667 0.333 0.889
80 800 0.5 0.5 0.667


D15

15


D .2 1,
30
30


Dm.ax,12
25
20


7F,
0.2
0.111
0.333
0.111


5760
3000
2000


T'
2880
2000
1000
1000


3
30


Model 7 1200 160 480 160 480 50 20 15 30 20 30
Model name LL L L1 L L 71 7t1 t22 l 2t T1 T2
Model 5 300 500 1000 200 300 1000 0.6 0.3 0.1 0.85 0.1 0.05 5760 2880
Model 6 100 300 1000 200 500 1000 0.5 0.35 0.15 0.7 0.2 0.1 5760 2880
Model 7 140 600 1500 200 500 1600 0.8 0.15 0.05 0.7 0.2 0.1 5760 2880


0 200 4 7 0
600 800


(a) Randomly placed community. (b) Single-tier community centered at (300, 300) (c) Two-tier community centered at (300, 300) or
or (700, 700) with one half ,.... ...i. (700, 700) with one half ,.... ..hI".

Fig. 9. Spatial distribution of the node (shown as the -',.. ..iiii' for a node to appear in each 50x50 grid block).


C(omuny lacedton conn(K)
(a) Randomly placed community.


(b) Relative error for scenarios with fixed
single-tier community.


(c) Relative error for scenarios with fixed two-tier
communities.


Fig. 10. Comparison of theoretical and simulation results (the average node degree).


(a) Hitting time, simple model.


d x 20 40 o60

1
Communication range (K)

(d) Meeting time, simple model.


(b) Hitting time, multi-tier communities.
0 04













0 02

-0 02 t 2 40 60
0 04
e 12
6 1--
-0 12 Model 5


Communication range(K)

(b) Hitting time, multi-tier communities.

0 04 -
002

-o 02 40 60


-o 0 6
12-
-0 14
Communlcatlon range (K)

(e) Meeting time, multi-tier communities.


(c) Hitting time, multiple random communities.


(f) Meeting time, multiple random communities.


Fig. 11. Relative error between theoretical and simulation results (the hitting time and the meeting time).


Model name
Model 1
Model 2
Model 3
Model 4


Model name
Model 5
Model 6


i;,



J/ a


os
40
02

o o 150 30-0 410









of the theory well. Other configurations we tried (not listed
here) also show similar trends.
Again, to make the scenario a bit more realistic, we simulate
some more scenarios when the communities are fixed. Among
the 50 nodes, we make 25 of them pick the community
centered at (300, 300) and the other 25 pick the community
centered at (700, 700). We simulate scenarios for all seven
sets of parameters listed in Table II. Models 1 through 4
correspond to scenarios with single-tier communities in each
time period, and models 5 through 7 correspond to scenarios
with multi-tier communities. We show the relative errors,
calculated as Error = (Theory-Simulation)/Simulation,
in Fig. 10 (b) and (c). A positive error indicates the theoretical
value is larger than the simulation result, while a negative
error indicates the converse. In the simulations, when the
communication ranges are small as compared to the edge of
the communities, the relative errors are low, typically below
11' except for Model-3, indicating a good match between
the theory and the simulation. However, as the communication
range increases, the area covered by the communication disk
becomes comparable to the size of the community and Eq. (5)
is no longer accurate since the communication disk extends
out of the overlapped area in most cases. That is the reason
we start to observe the discrepancies between the theory and
simulation. Besides Model-3, we observe at most _'i '. of
relative error when the communication disk is less than 2_i '.
the size of the inner-most community, indicating that our
theory is valid when the communication range is relatively
small.


C. The Hitting Time and the Meeting Time
We perform simulations for the hitting and the meeting
times for 50,000 independent iterations for each scenario
listed in Table II, and compare the average results with the
theoretical values derived from the corresponding equations
(i.e. (8) and (17)). To find out the hitting or the meeting time,
we move the nodes in the simulator indefinitely until they hit
the target or meet with each other, respectively.
Again we show the relative errors between the theoretical
values and the simulation results for various scenarios in Fig.
11. We see that for all the scenarios, the relative errors are
within acceptable range. These results display the accuracy
of our theory under a wide range of parameter settings.
The absolute values for the error are within 1.'. for the
hitting time and within 2_I' for the meeting time. For more
than 71' of the tested scenarios, the error is below 10%.
The errors between the theoretical and simulation results are
mainly due to some of the approximations we made in the
various derivations. For example, there exist some border
effects with respect to the hitting and meeting probabilities
within a community. When a node is close to the border of
a community, it could also "see" some other nodes outside
of the community if its transmission range is large enough.
However, we have chosen to ignore such occurrences to keep
our analysis simpler. Furthermore, the approximation of the
hitting and meeting processes with discrete Bernoulli trials is
valid only for the epochs that are large enough (in the order


0.86

0.4
0.2 -Modell 200nodes
Model3 760nodes
0 1
10 20 30 40
Nodal communication range

Fig. 12. Geographic routing success rate under different mobility parameter
sets and node numbers.


of community size). Nevertheless, as shown in the figures,
the errors are always within acceptable ranges, justifying our
simplifying assumptions.


VII. USING THEORY FOR PERFORMANCE PREDICTION

Although the various theoretical quantities derived for the
TVC model in Section V are interesting in their own merit,
they are particularly useful in predicting protocol performance,
which in turn can guide the decisions of system operation. We
illustrate this point with two examples in this section.



A. Estimation of the Number of Nodes Needed for Geographic
Routing

It has been shown in geographic routing that the average
node degree determines the success rate of messages deliv-
ered [42]. Thus, using the analytical results of Section V-B
we can estimate the number of nodes (as a function of the
average node degree) needed to achieve a target performance
for geographic routing, for a given scenario.
We consider the same setup as in Section VI-B, where
half of the nodes are assigned to a community centered at
(300, 300) and the other half are assigned to another com-
munity centered at (700,700). We are interested in routing
messages across one of the communities, from coordinate
(250, 250) to coordinate (350,350) with simple geographic
routing (i.e., greedy forwarding only, without face routing
[43]). Using simulations we obtain the success rate of geo-
graphic routing under various communication ranges when 200
nodes move according to the mobility parameters of Model-
1 (Table II). Results are shown in Fig. 12 (each point is the
percentage of success out of 2000 trials). If we assume now
that the mobility model was different, say Model-3, we would
like to know how many nodes we would need to achieve
similar performance. Using Eq. (6) we find that 760 nodes
are needed to create a similar average node degree for Model-
3. To validate this, we also simulate geographic routing for
a scenario where 760 nodes follow Model-3. Comparing the
resulting message delivery ratio for this scenario to the original
scenario i- 'i" nodes with Model-i) in Fig. 12, we see that
similar success rates are achieved in both scenarios under the
same transmission range, which confirms the accuracy of our
analysis.









B. Predicting Message Delivery Delay with Epidemic Routing

Epidemic routing is a simple and popular protocol that has
been proposed for networks where connectivity is only inter-
mittent (often referred to as Delay Tolerant Networks) [39].
It has been shown that message propagation under epidemic
routing can be modeled with sufficient accuracy (assuming the
number of nodes is large enough) using a simple fluid-based
model [40]. (Note that its performance has also been analyzed
using Markov Chain [33], [23] and Random Walk [5] models.)
This fluid model has been borrowed from the Mathematical
Biology community, and is usually referred to as the SI
(Susceptible-Infected) epidemic model. The gist of the SI
model is that the rate by which the number of "infected"
nodes increases ("infected" nodes here are nodes who have
received a copy of the message) can be approximated by the
product of three quantities: the number of already infected
nodes, the number of susceptible (not yet infected) nodes,
and the pair-wise contact rate, 3 (the implicit assumption
there of course being that nodes meet independently). This
contact rate in the SI model is equivalent to the unit-step
meeting probabilities calculated in Section V-D. Thus, one
could in essence plug-in these meeting probabilities into the SI
model equations and calculate the delay for epidemic routing.
Yet, in the TVC model (and often in real life) there are
multiple groups of nodes with different communities, and thus
different pair-wise contact rates that depend on the community
setup. For example, nodes with the same or overlapping
communities tend to meet much more often than nodes in
far away communities. For this reason, we extend the basic
SI model to a more general scenario that is applicable to the
TVC model.
We consider the following setup in the case study: We use
Model-3 (Table II) for the mobility parameters. A total of M =
50 nodes are divided into two groups of 25 nodes each. One
group has its community centered at (300, 300) and the other
at (700, 700). One packet starts from a randomly picked source
node and the time needed until it reaches all other nodes in the
network using epidemic routing is calculated. The propagation
of the message can be described by the following equations:


dt= o I((t) S (t) + ooI2(t2 (t)Sl(t)
dl2(t) j
dt 3o 2(t)S2(t) + 3 n, 1t(t)S2(t) (20)
S(t) + I(t) = M/2
S2(t) +1(t)= M/2.
where S, (t) and Ix(t) denote the number of susceptible and
infected nodes at time t in group x, respectively. Parameters
3o, and /3,no_o represent the pair-wise unit-time meeting
probability when the communities are overlapped (i.e., for
nodes in the same group) and not overlapped (i.e., nodes
in different groups), respectively. We use Eq. (15) to obtain
these quantities. This model is an extension from the standard
SI model [40] and similar extensions can be made for more
than two groups [50]. The first equation governs the change
of infected nodes in the first group. Notice that the infection
to susceptible nodes in the group (Sl(t)) can come from the
infected nodes in the same group (I (t)) or the other group


S20
10 Th o (SI mod )
0 Sinultion


500 1000 1500
Simulaton time


2000


Fig. 13. Packet propagation with epidemic routing. The total population is
divided into 2 groups with different community.


(I2 (t)). We can solve the system of equations in (20) to get the
evolution of the total infected nodes in the network. As can
be seen in Fig. 13, the theory curve closely follows the trend
in simulation curve (the non-perfect matching between the
two curves, is due to the fact that fluid models become more
accurate approximations of the actual stochastic spreading for
large numbers of nodes). This indicates first that scenarios
generated by our mobility model are still amenable to fluid
model based mathematical analysis (SI), despite the increased
complexity introduced by the concept of communities. It also
shows that results produced thus can be used by a system
designer to predict how fast messages propagate in a given
network environment. This might, for example, determine
if extra nodes are needed in a wireless content distribution
network to speed up message dissemination.
As a final note, in addition to epidemic routing, the theoret-
ical results for the hitting and meeting times could be applied
to predict the delay of various other DTN routing protocols
(see e.g. [22], [5], [40]), for a large range of mobility scenarios
that can be captured by the TVC model.


VIII. CONCLUSION AND FUTURE WORK

We have proposed a time-variant community mobility model
for wireless mobile networks. Our model preserves common
mobility characteristics, namely skewed location visiting pref-
erences and periodical re-appearance at the same location.
We have tuned the TVC model to match with the mobility
characteristics of various traces (WLAN traces, a vehicle
mobility trace, and an encounter trace of moving human
beings), displaying its flexibility and generality. A mobility
trace generator of our model is available at [37]. In addition
to providing realistic mobility patterns, the TVC model can
be mathematically analyzed to derive several quantities of
interest: the nodal spatial distribution, the average node de-
gree, the hitting time and the meeting time. Through extensive
simulation studies, we have verified the accuracy of our theory.
The TVC model can be easily generalized to provide sce-
narios in which nodes display more heterogeneous behavior.
Nodes may have different set of parameters and even the time
period structure can be different for different nodes. With these
extensions, we have a mobility model to describe an environ-
ment including users with diverse mobility characteristics. We
believe such a model is a very important building block for
understanding protocol performances in real-life settings. To
demonstrate this last point, we also provide some examples










of how our theory can be used in practice to predict protocol
performance and guide design decisions.
In the future we would like to further analyze the perfor-
mance of various routing protocols (e.g., [5], [6]) under the
time-variant community mobility model. We also would like
to construct a systematic way to automatically generate the
configuration files, such that the communities and time periods
of nodes are set to capture the inter-node encounter properties
we observe in various traces (for example, the Small World
encounter patterns observed in WLAN traces [46]).


ACKNOWLEDGEMENT

The authors thank Michal Piorkowski for pointing us to
the cab spotting website and sharing archived data. This work
is supported by NSF awards CNS-0520017 and career award
0134650.


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17


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