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Modeling Spatial and Temporal Dependencies ofUser Mobility in Wireless Mobile Networks

Wei-Jen Hsu∗, Thrasyvoulos Spyropoulos†, Konstantinos Psounis‡ and Ahmed Helmy∗∗Dept. of Computer and Information Science and Engineering,University of Florida, Gainesville, Florida

†Computer Engineering and Networks Lab, ETH Zurich‡Dept. of Electrical Engineering, University of Southern California, Los Angeles, California

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 foundin reality, while at the same time being easy to use and analyze.Motivated by this, we propose a time-variant community mobilitymodel, referred to as the TVC model, which realistically capturesspatial and temporal correlations. We devise the communities thatlead to skewed location visiting preferences, and time periodsthat allow us to model time dependent behaviors and periodicre-appearances of nodes at specific locations.

To demonstrate the power and flexibility of the TVC model, weuse it to generate synthetic traces that match the characteristicsof a number of qualitatively different mobility traces, inc ludingwireless LAN traces, vehicular mobility traces, and humanencounter traces. More importantly, we show that, despite thehigh level of realism achieved, our TVC model is still theoreticallytractable. To establish this, we derive a number of importantquantities related to protocol performance, such as the averagenode degree, the hitting time, and the meeting time, and provideexamples of how to utilize this theory to guide design decisionsin routing protocols.

I. I NTRODUCTION

Mobile ad hoc networks(MANETs) are self-organized,infrastructure-less networks that could potentially supportmany applications, such as vehicular networking (VANET) [4],wild-life tracking [19], and Internet provision to rural ar-eas [16], to name a few. Mobility also enables messagedelivery in sparsely connected networks, generally known asdelay tolerant networks (DTNs). As the devices are easilyportable and the scenarios of deployment are inherently dy-namic,mobilitybecomes one of the key characteristics in mostof these networks. It has been shown that mobility impactsMANETs in multiple ways, such as network capacity [9],routing performance [1], and cluster maintenance [24]. Inshort, the evaluation of protocols and services for MANETsseems to be inseparable from the underlying mobility models.It is, thus, of crucial importance to have suitable mobilitymodels as the foundation for the study of ad hoc networks.

Ideally, a good mobility model should achieve a numberof goals: (i) it should first capturerealistic mobility patternsof scenarios in which one wants to eventually operate thenetwork; (ii) at the same time it is desirable that the modelis mathematically tractable; this is very important to allowresearchers to derive performance bounds and understand thelimitations of various protocols under the given scenario,asin [30], [31], [9], [6]; (iii) finally, it should beflexibleenough

to provide qualitatively and quantitatively different mobilitycharacteristics by changing some parameters of the model, yetin a repeatable and scalable manner; designing a new mobilitymodel 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 allof them at the same time. Our goal in this paper is, on onehand, to improve the existingrandom mobility models(e.g.,random walk, random direction, etc.) andsynthetic mobilitymodels (e.g., [12], [11], [17]) on the front ofrealism, byconsidering empirically observed mobility characteristics fromthe traces [14]. On the other hand, the construction of themodel should new model should be simple enough to allowin-depth theoretical analysis, and beflexible enough to havewider applicability than themobility traces(which provideonly a single snapshot of the underlying mobility process)and currenttrace-based mobility models[33], [23], [22] whichfocus mainly on matching mobility characteristics with aspecific class of traces.

The main contribution of this paper is the proposal of atime-variant community mobility model, referred to as theTVC model, which isrealistic, flexible, and mathematicallytractable. One salient characteristic in the TVC model islocation preference. Another important characteristic is thetime-dependent, periodical behaviorof nodes. To our bestknowledge, this is the firstsynthetic mobility modelthatcaptures non-homogeneous behavior in bothspaceand time.

To establish the flexibility of our TVC model we showthat we can match its two prominent properties,locationvisiting preferencesand periodical re-appearance, with mul-tiple WLAN traces, collected from environments such asuniversity campuses [10], [14] and corporate buildings [2].More interestingly, although we motivate the TVC model withthe observations made on WLAN traces, our model is genericenough to have wider applicability. We validate this claim byexamples of matching our TVC model with two additionalmobility traces: a vehicle mobility trace[36] and a humanencounter trace[6]. In the latter case, we are even able tomatch our TVC model with some other mobility characteristicsnot explicitly incorporated in our model by its construction,namely the inter meeting time and encounter duration betweendifferent users/devices.

Finally, in addition to the improved realism, the TVC modelcan be mathematically treated to derive analytical expressionsfor important quantities of interest, such as theaverage nodedegree, thehitting timeand themeeting time. These quantities

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are often fundamental to theoretically study issues such asrouting performance, capacity, connectivity, etc. We showthatour theoretical derivations are accurate through simulationcases with a wide range of parameter sets, and additionallyprovide examples of how our theory could be utilized inactual protocol design. To our best knowledge, this is the firstsynthetic mobility model proposed that matches with tracesfrom multiple scenarios, and has also been theoretically treatedto the extent presented in this paper. We make the code of theTVC model available at [40].

The the paper is organized as follows: In Section II wediscuss related work. OurTVC modelis then introduced inSection III. In Section IV, we show how to generate realisticmobility scenarios matched with various traces. Then, inSection V, we present our theoretical framework and derivegeneric expressions of various quantities. Simulation validatesthe accuracy of these expressions in Section VI. Additionally,in Section VII, we motivate our theoretical framework further,by applying our analysis to performance predictions in proto-col design. Finally, we conclude the paper in Section VIII.

II. RELATED WORK

Mobility models have been long recognized as one ofthe fundamental components that impacts the performanceof wireless ad hoc networks. A wide variety of mobilitymodels are available in the research community (see [5] fora good survey). Among all mobility models, the popularityof random mobility models(e.g., random walk, random di-rection, and random waypoint) roots in its simplicity andmathematical tractability. A number of important properties forthese models have been studied, such as the stationary nodaldistribution [3], the hitting and meeting times [29], and themeeting duration [18]. These quantities in turn enable routingprotocol analysis to produce performance bounds [30], [31].However,random mobility modelsare based on over-simplifiedassumptions, and as has been shown recently and we will alsoshow in the paper, the resulting mobility characteristics arevery different from real-life scenarios. Hence, it is debatablewhether the findings under these models will directly translateinto performance in real-world implementations of MANETs.

More recently, an array ofsynthetic mobility modelsareproposed to improve the realism of the simplerandom mobilitymodels. More complex rules are introduced to make thenodes follow a popularity distribution when selecting the nextdestination [12], stay on designated paths for movements [17],or move as a group [11]. These rules enrich the scenarioscovered by thesynthetic mobility models, but at the sametime make theoretical treatment of these models difficult. Inaddition, mostsynthetic mobility modelsare still limited toi.i.d. models, and the mobility decisions are also independentof the current location of nodes and time of simulation.

A different approach to mobility modeling is byempiricalmobility trace collection. Along this line, researchers haveexploited existing wireless network infrastructure, suchaswireless LANs (e.g., [2], [25], [10]) or cellular phone networks(e.g., [7]), to track user mobility by monitoring their locations.Such traces can be replayed as input mobility patterns forsimulations of network protocols [13]. More recently, DTN-specific testbeds [6], [4], [19] aim at collecting encounter

events between mobile nodes instead of the mobility patterns.Some initial efforts to mathematically analyze these tracescan be found in [6], [20]. Yet, the size of the traces andthe environments in which the experiments are performedcan not be adjusted at will by the researchers. To improvethe flexibility of traces, the approach oftrace-based mobilitymodelshave also been proposed [33], [23], [22]. These modelsdiscover the underlying mobility rules that lead to the observedproperties (such as the duration of stay at locations, the arrivalpatterns, etc.) in the traces. Statistical analysis is thenused todetermine proper parameters of the model to match it with theparticular trace.

The goal of this work is to combine the strengths of variousapproaches to mobility modeling and propose arealistic, flex-ible, and mathematically tractable synthetic mobility model.Our work is partly motivated by several prominent, commonproperties in multiple WLAN traces (e.g., traces availablefrompublic archives [38], [37]) we observed in [14], based on whichwe construct the TVC model. This model extends the conceptof communities proposed by us in [29] and also introducestime-dependent behavior. A preliminary version of the modelhas been presented in [15]. In this work we highlight theflexibility of the TVC model by matching the synthetic traceswith two additional, qualitatively different traces to WLANtraces (i.e., vehicular and human encounter traces, in sectionIV). We also extend and present more generic theoreticalresults under the scenario with multiple communities (sectionV), and display its applications on protocol performanceprediction (section VII).

We differentiate our work from other trace-based mod-els [33], [23], [22] in several aspects. First, among all effortsof providing realistic mobility models, to our best knowledge,this is the first work to explicitly capture time-variant mobilitycharacteristics. Although capturing time-dependent behavioris suggested in [22], it has not been incorporated in theparticular paper. Second, while previous works emphasize thecapability to truthfully recreate the mobility characteristicsobserved from the traces, we also strive to ensure at thesame time the mathematical tractability of the model. Ourmotivation is to facilitate the application of our model forperformance prediction of various communication protocols.Finally, most of the other trace-based models have not beenshown as capable to match mobility characteristics of a diverseset of traces, since their focus is mostly on one particular traceor at most a single class of traces (e.g., WLAN trace). We gobeyond that and re-produce matching mobility characteristicsof several qualitatively different traces, including WLAN,vehicle, and human encounter traces.

As a final note, in [26], the authors assume the attractionof a community (i.e., a geographical area) to a mobile nodeis derived from the number of friends of this node currentlyresiding in the community. In our paper we assume that thenodes make movement decisions independently of the others(nonetheless, node sharing the same community will exhibitmobility correlation, capturing the social feature indirectly).Mobility models with inter-node dependency require a solidunderstanding of the social network structure, which is animportant area under development. We plan to work furtherin this direction in the future.

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III. T IME-VARIANT COMMUNITY MOBILITY MODEL

A. Mobility Characteristics Observed in WLAN Traces

The main objective of this paper is to propose a mobilitymodel that captures the important mobility characteristicsobserved in daily life. To better understand this mobility,wehave conducted extensive analysis of a number of wirelessLAN traces collected by several research groups (e.g., tracesavailable at [38] or [37]). The reason for this choice is thatWLAN traces log information regarding large numbers ofnodes, and thus are reliable for statistical analysis. Afteranalyzing a large number of traces, we have observed twoimportant properties that are common in all of them: (a)skewedlocation visiting preferencesand (b)time-dependent mobilitybehavior[14].

More specifically, thelocation visiting preferencerefersto the percentage of time a node spends at a given accesspoint (AP). We refer to the coverage area of an access pointas a location. In Fig. 1(a), we draw the probability densityfunction of the percentages of online time an average userspends at each location, ranking the locations from the mostfavorite place to the least for various traces. The distributionappears highlyskewed; more than95% of user’s online timeis spent at only top five APs. Thetime-dependent mobilitybehavior refers to the observation that nodes visit differentlocations, depending the time of the day. In Fig. 1(b) we plotthe probability of a node re-appearing at the same location atsome time in the future, as a function of the elapsed time. It isclear that this probability displays some amount of periodicity,as the mobile nodes have stronger tendency to re-appear at apreviously visited location after a time gap of integer multiplesof days. A slightly higher peak on the 7th day, suggesting astronger weekly correlation in location visiting preferences,could also be observed in some curves (e.g., MIT).

Unfortunately, these two prominent realistic mobility char-acteristics are not captured by commonly used simple randommodels, as they do not possess any space or time dependentfeatures in user mobility. This is demonstrated in Fig. 1 by astraight line (uniform distribution) for the Random Directionmodel. The same could be obtained from Random Waypoint,Random walk, etc., or even more sophisticated models withoutspatial-temporal preferences (e.g., [11], [17]). There are somemore recent models (e.g., [29], [12], [33], [23]) that aim atcapturing spatial preference explicitly. As shown in Fig. 1(a)using the simple community model [29], with appropriatelyassigned parameters this model is able to capture theskewedlocation visiting preference, to some extent. However, time-dependent behavior is not captured, and thus theperiodicalre-appearanceproperty cannot be reproduced, as shown bythe flat curve labeledcommunity modelin Fig. 1(b).

It is our goal to design a mobility model that successfullycaptures theskewed location preferenceand time-dependencymobility properties observed in the tracesin an analyticallytractable fashion. We believe that although the above obser-vations are made based on WLAN traces, the two propertiesin question are indeed prevalent in real-life mobility. Thisbelief is supported by typical daily activities of humans: mostof us tend to spend most time at a handful of frequentlyvisited locations, and a recurrent daily or weekly scheduleis an inseparable part of our lives. It is essential to design

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(b) Periodical re-appearance at thesame location.

Fig. 1. Two important mobility features observed from WLAN traces. Labelsof traces used: MIT: trace from [2], Dart: trace from [10], UCSD: trace from[25], USC: trace from [14].

TABLE IPARAMETERS OF THE TIME-VARIANT COMMUNITY MOBILITY MODEL 1

N Edge length of simulation areaV Number of time periodsT t Duration of t-th time periodSt Number of communities in time periodtCt

j Edge length of communityj in time periodtCommt

j The j-th community during time periodt

pti,j

The probability to choose communityj whenthe previous community isi, during time periodt

πtj

Stationary probability of an epoch incommunityj during time periodt

vmin, vmax, v Minimum, maximum, and average speed1

Dmax,j , Dj Maximum and average pause time after each epoch1

Lj Average epoch length for communityj

P tmove,j |P

tpause,j

Probability that a node is moving| pausingwhen being in communityj during periodt

P tj

Fraction of time the node is instatej (P t

j = P tmove,j + P t

pause,j )K Transmission range of nodes

A(atj , b

tk)

The overlapped area betweenCommtj of nodea

andCommtk of nodeb

wt A specific relationship between a target coordinateand the communities in time periodt

Ωt The set of all possible relationships betweena target coordinate and the communities in time periodt

Ph(wt) Unit-time hitting probability

under the specific scenariowt

PH(wt) Hitting probability for a time periodtunder specific scenariowt

P tm Unit-time meeting probability in time periodt

P tM Meeting probability for a time periodt

a model that captures such spatial-temporal preferences ofhuman mobility in many contexts.

B. Construction of the Time-variant Community Model

In this section, we present the design of ourtime-variantcommunity (TVC) mobility model. We illustrate the model withan example in Fig. 2 and use this example to introduce thenotations we use (see Table I) in the rest of the paper.

First, to induceskewed location visiting preferences, wedefine somecommunities(or heavily-visited geographic areas).Take time period 1 (TP1) in Fig. 2 as an example, thecommunities are denoted asComm1

j and each of them isa square geographical area with edge lengthC1

j .1 A nodevisits these communities with differentprobabilities (detailsare given later) to capture its spatial preference in mobility.In the TVC model, the mobility process of a node consists

1For all parameters used in the paper, we follow the convention that thesubscript of a quantity represents its community index, andthe superscriptrepresents the time period index.

4

of epochsin these communities. When the node chooses tohave anepoch in community j (we say thatthe node isin state j during this epoch), it starts from the end pointof the previous epoch withinComm1

j and the epoch length(movement distance) is drawn from an exponential distributionwith averageLj, in the same order of the community edgelength. The node then picks a random speed uniformly in[vmin, vmax], and a direction (angle) uniformly in[0, 2π],and performs a random direction movement within the chosencommunity with the chosen epoch length2. The first differencebetween the TVC model and the standard Random Directionmodel is hence the spatial preference and location-dependentbehavior. Note that, a node can still roam around the wholesimulation area during some epochs, by assigning an additionalcommunity that corresponds to the whole simulation field (e.g.Comm1

3). We refer to such epochs asroaming epochs.We next explain how a node selects the next community

for a sequence of epochs. At the completion of an epoch, thenode remains stationary for a pause time uniformly chosenin [0, Dmax,j]. Then, depending on its current statei andtime period t, the node chooses the next epoch to be incommunityj with probabilitypti,j . This community selectionprocess is essentially a time-variant Markov chain that capturesthe spatial and temporal dependencies in nodal mobility andthus makes the community selection process in the TVCmodel non-i.i.d., an important feature absent in many syntheticmobility models even if they consider non-uniform mobilityfeatures. Now, if the end point of the previous epoch isin Commt

j (this can be the case when the node has twoconsecutive epochs inCommt

j , orCommtj containsCommt

i),the node starts the next epoch directly. If, on the other hand,the node is currently not inCommt

j , a transitional epochisinserted to bridge the two epochs in disjoint communities. Thenode selects a random coordinate point in the next community,moves directly towards this point on the shortest straight pathwith a random speed drawn from[vmin, vmax], and thencontinues with an epoch in the next community. Hence themovement trajectory of a node is always continuous in space.

We next introduce the structure in time. To capture time-dependent behavior, one creates multipletime periodswithdifferent community and parameter settings. As an example,there areV = 3 time periods with durationT 1, T 2, andT 3 in Fig. 2. These time periods follow aperiodic structure(e.g., a simple recurrent structure in Fig. 2 or the weeklyschedule in Fig. 3). This setup naturally captures thetemporalpreferences(e.g., go to work during the days and home duringthe nights) andperiodicity in human mobility. On the timeboundaries between time periods, each node continues withits ongoing epoch, and decides the next epoch according tothe new parameter settings in the new time period when itfinishes the current epoch.

As a final note, we choose to construct the TVC model withsimple building blocks introduced above due to its amenabilityto theoretical analysis [29] and flexibility. To further explainthe flexibility of our TVC model, we note that the number

2To avoid boundary effects, if the node hits the community boundary it isre-inserted from the other end of the area (i.e., ”torus” boundaries). Note thatwe could also choose random waypoint or random walk models for the typeof movement during each epoch.

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Fig. 3. An illustration of a simple weekly schedule, where weuse timeperiod1 (TP1) to capture weekday working hour, TP2 to capture night time,and TP3 to capture weekend day time.

of communities in each time period (denoted asSt) can bedifferent, and the communities can overlap (as in TP1 in Fig.2) or contain each other (as in TP2 in Fig. 2). Finally, thetime period structure, communities, and all other parameterscould be assigned differently foreach nodeto capture node-dependent mobility (e.g., people following different schedules,with different working places, etc.), while nodes can sharesome communities (i.e., the popular locations) as well. Thisconstruction allows for maximum flexibility when setting upthe simulations for nodes with heterogeneous behaviors3.

The benefit of using simple building blocks will becomeevident in Section V. At the same time, we will show nextthat these choices do not compromise our model’s ability toaccurately capture real life mobility scenarios.

IV. GENERATION OFMOBILITY SCENARIOS

The TVC model described in the previous section providesa general framework to model a wide range of mobilityscenarios. In this section, our aim is to demonstrate themodel’s flexibility and validate its realism by generating var-ious synthetic traces from the model, with matching mobilitycharacteristics to well-known, publicly-available traces (e.g.,WLANs, VANET, and human encounter traces). However, itis important to note that the use of such a model is not merelyto match it with any specific trace instance available; this isonly done for validation and calibration purposes. Rather,thegoal is to be able to reproduce a much larger range of realisticmobility instances than a single trace can provide4.

We first outline a general 3-step systematic process toconstruct specific mobility scenarios. Then, we demonstrateour success to generate matching mobility characteristicswiththree qualitatively different traces. All the parameter valueswe use in this section are also available in [40]5.

3When necessary, we use a pair of parentheses to include the node IDfor a particular parameter, e.g.,Ct

j(i) denotes the edge length of thej-thcommunity during time periodt for nodei.

4We have made our mobility trace generator available at [40].The toolprovides mobility traces in both ns-2[39] compatible format and time-location(i.e., (t, x, y)) format.

5Due to space limitations, we cannot list all parameters in this paper.

5

STEP 1: Determine the Structure in Space and Time• (1.1 Number of communities) Each community in the TVCmodel corresponds to a location visited frequently by nodes(i.e., the most visited location in Fig. 1(a) corresponds tothemost popular community in the model, and so on). The numberof communities needed is thus determined by how closely onewants the mobility characteristics to match with the curvesin Fig. 1(a). Due to the nature of skewed location visitingpreference, in our experience, only two or three communitiesare needed to capture up to85% of the user online timespent at the most popular locations. Such a simple spatialstructure yields simple theoretical expressions. However, ifone wants the model to capture more details (e.g., for detailedsimulation), the user can instantiate as many communities asneeded to explicitly represent the less visited locations.• (1.2 Location of communities) If the map of the targetenvironment is available, one should observe the map andidentify the points of attraction in the given environment toassign the communities accordingly. The methods describedin [22] could be applied to help choosing the “hot spots” oncampus, by adding up the time users spend at each location ona 2-D map and identifying the peaks. Alternatively, if the mapis not available, one can instantiate communities atrandomlocations6. One way to do so is to simply divide the simulationarea into equal-sized grid cells, and assign randomly chosencells as communities.• (1.3 Time period structure) From the curves in Fig. 1(b),one observes there-appearanceperiodicity and decides on thetime period structure accordingly. Typically, human activitiesare bounded by daily and weekly schedules so a time periodstructure shown in Fig. 3 would suffice for most applications.If capturing finer behavior based on time-of-day is necessary,one could additionally split the day into time periods withdifferent mobile node behavior. We illustrate this in our thirdcase study, the human encounter trace.STEP 2: Assign Other Parameters After the space/timestructure is determined, one has to determine the remainingparameters for each community and time period. This includesπtj , Dt

j, and Ltj, which represent the stationary probability

(which is calculated after selecting properpti,j ’s that lead toa desired stationary distribution using simple Markov chaintheory), average pause time, and average epoch length, respec-tively, at communityj during time periodt. These parameterscan be determined by referring to the curves in Fig. 1. We givesome general rules of how the parameters change the curvesin Fig. 1 below. The detailed adjustments we make for eachspecific case studies will be discussed later.• The average epoch length in each community,Lt

j , shouldbe at least in the same order as the edge length of thecommunity,Ct

j . This is to ensure that the end point of theepoch becomes almost independent of its starting point, sincethe mixing time of the corresponding process becomes quitesmall. (The motivation for this requirement is to keep thetheoretical analysis tractable.)• The average duration the node stays in communityj isgiven byπt

j(Dtj +Lt

j/v). The ratio between the durations thenode stays in each community shapes the location visiting

6Concerning matching with the two mobility properties shownin Fig. 1,the actual locations of the communities do not make a difference.

preference curve in Fig. 1(a).• The highest peak of the re-appearance probability curve(on the7-th day under the weekly schedule) in Fig. 1(b) isdetermined by the weighted average probability of the nodeappearing in the same community during the same type oftime period. This value is

∑V

t=1T t

P

Vk=1

Tk

∑St

j=1(Ptj )

2, where

P tj denotes the fraction of time the node spends in community

j.STEP 3: Adjust User On-off Pattern (Optional) The mo-bility trace generated by the TVC model is an “always-on”mobility trajectory (i.e., the mobile nodes are always presentsomewhere in the simulation field). However, in some situa-tions some nodes might be absent occasionally. For example,in a WLAN setting, nodes (e.g. laptops) are often turned offwhen travelling from one location to another and the “off”time is often not negligible [14]. Thus one may need to makeoptional adjustments to turn nodes off in the generated trace,depending on the actual environment to match with. To addressthis we assign a probabilityPon,j as the probability for thenode to be “on” in communityj. In two of the case studies wepresent (WLAN and vehicular trace), we utilize this featureasthe nodes are not always-on in the actual traces.

Note that it is possible to automate part of the above com-munity and parameter selection. This can be done by feedingthe curves in Fig. 1 and the desired level of matching to aprogram that executes the above steps. Automatic generationof proper synthetic traces is a direction of our future work.

Next, we look into three specific case studies and applythe fore-mentioned procedure in each case, to display that theTVC model successfully produces synthetic mobility traceswith matching characteristics observed in the real traces.

A. WLAN Traces

In the first example, we show that the TVC model can re-create thelocation preferencesand re-appearance probabilitycurves observed in WLANs. We use the MIT WLAN trace(first presented in [2]) as the main example here7. We split theMIT trace into two halves and generate a matching synthetictrace with observed mobility characteristics from the firsthalf (the training data set). We then compare our synthetictrace with the mobility characteristics of the second half (thevalidation data set). Note that, the mobility characteristics aresimilar across the two halves (shown by the two very closethick black curves in Fig. 4). We generate two synthetic traceswith the TVC model, asimplifiedone and acomplexone, todisplay its flexibility to have different levels of matchingtothe WLAN trace.

The simplified model (shown by thin black curves) uses onlyone community and two time periods (for the day time andnight time), with parameters listed asModel-1in Table II. Thesimple model captures the major trends but still shows severalnoticeable differences: (a) the tail in themodel-simplifiedcurve in Fig. 4(a) is “flat” as opposed to the exponentiallydiminishing tail of theMIT curve. (b) the peaks in themodel-simplifiedcurve in Fig. 4(b) are of equal heights.

7We also achieve good matching with the USC[14] or the Dartmouth[10]traces, but do not show it here due to space limitations.

6

We can improve the matching between the synthetic traceand the real trace by adding complexity in both space and time,with the following detailed procedure. (STEP1): We divide thesimulation area into 10-by-10 grid cells. Since we want to havea close match with the curve in Fig. 4(a), we assign randomly15 of the cells as communities to each node (Intuitively, thisnumber corresponds to the number of distinct access pointsthat a person may connect to on a university campus over aperiod of one month.). For the time period structure we use thesimple weekly structure shown in Fig. 3, allocating8 hours forday time (TP1, TP3) and16 hours for night time (TP2), as thistrace is collected from a corporate environment. (STEP2 andSTEP3): In the actual WLAN trace the nodes are “on” only fora low percentage of time. We capture this phenomenon withan additional parameter,P t

on,j , the probability the node is “on”in statej. In WLAN, the nodes are typically “on” (i.e., appearat the current location) when they are not moving. Under thison-off pattern,P t

on,j = Dtj/(D

tj + Lt

j/v). We then considerthe on-off pattern and parameter assignment jointly. (1) Wefirst assign the sameDt

j , Ltj, P

ton,j to all communities, then

assignπtj with a value equal to the fraction of time spent at

the j-th location in Fig. 1(a). This assignment strategy makesthe node “on” for the same amount of time in each communityduring each visit, and the total time in each community(and hence the observed location visiting preference curve) istherefore determined by the value ofπt

j . (2) Due to the on-offpattern, the peak value in the re-appearance probability curvebecomes

∑Vt=1

T t

P

Vk=1

Tk

∑St

j=1(Ptj )

2(P ton,j)

2. To shape the re-

appearance probabilities, we adjust theDtj values, which, in

turn, adjust the values ofP ton,j and set the re-appearance

probabilities to the desirable values to match with the curve inFig. 1(b). Note that by adjusting theDt

j values in a consistentmanner among all communities we do not change the locationvisiting probability curve that has already been matched intheprevious step.

As it is evident from Fig. 4, this model, which is labeledModel-complex and corresponds to the red curves in theplot, yields synthetic traces whose characteristics matchveryclosely with those of the MIT trace.

B. Vehicle Mobility Traces

In this example we display thatskewed location visitingpreferencesand periodical re-appearanceare also prominentmobility properties in vehicle mobility traces. We obtain avehicle movement trace from [36], a website that tracks par-ticipating taxis in the greater San Francisco area. We processa 40-day trace obtained between Sep. 22, 2006 and Nov. 1,2006 for 549 taxis to obtain their mobility characteristics. Theresults are shown in Fig. 5 with the labelVehicle-trace. It isinteresting to observe that the trend of vehicular movementsis very similar to that of WLAN users in terms of these twoproperties.

We use30 communities and the weekly time schedule in(STEP1). We need more communities for this trace as thetaxis are more mobile and visit more places than people onuniversity campuses. From the actual trace, we discover thatthe taxis are offline (i.e., not reporting their locations) when notin operation. Hence we assume that the nodes are “on” only

Fra

ctio

n o

f v

isit

tim

e

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1 11 21 31 41

MIT

Model-simplified

Model-complex

AP sorted by total visit time

(a) Skewed location visiting preferences.

Time gap (days)

Re-

appea

rance

pro

bab

ilit

y

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8

MIT

Model-simplified

Model-complex

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

Fig. 4. Matching mobility characteristics of the synthetictraces to the MITWLAN trace.

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1 11 21 31 41 51

Fra

ctio

n o

f vis

it t

ime

Vehicle-trace

Model

Location sorted by visit time

(a) Location visiting preferences.

Time gap (days)

Re-

app

eara

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pro

bab

ilit

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0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8

Vehicle-trace

Model

(b) Periodical re-appearance.

Fig. 5. Matching mobility characteristics of the synthetictrace to the vehiclemobility trace.

when they are moving. The pause times between epochs areconsidered as breaks in taxi operation. Therefore in (STEP3),P ton,j = (Lt

j/v)/(Dtj+Lt

j/v), and we adjust the parameters ina similar way as described in the previous section. The curvesin Fig. 5 with labelModelmatch with the curves withVehicle-tracelabel well. As a final note, although vehicular movementsare generally constrained by streets and our TVC model doesnot capture such microscopic behaviors, designated paths andother constraints could still be added in the model’s map(for vehicular or human mobility) without losing its basicproperties. We defer this for future work.

C. Human Encounter Traces

In this example, we show that the TVC model is genericenough to mimic the encounter properties of mobile humannetworks observed in an experiment performed at INFOCOM2005 [6]. In this experiment, wireless devices were distributedto 41 participants at the conference to log encounters betweennodes (i.e., coming within Bluetooth communication range)asthey moved around the premises of the conference area. Theinter-meeting time and the encounter duration distributions ofall 820 pairs of users obtained from this trace are shown inFig. 6 with labelCambridge-INFOCOM-trace.

To mimic such behaviors using our TVC model, we observethe conference schedule at INFOCOM, and set up a daily

7

0.00001

0.0001

0.001

0.01

0.1

1

10 100 1000 10000 100000 1000000Inter-meeting time (s)

Cambridge-

INFOCOM-trace

Model

Pro

b(I

nte

r-m

eeti

ng

tim

e >

X)

(a) Inter-meeting Time.0.00001

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000 100000

Cambridge-

INFOCOM-trace

Model

Pro

b (

Mee

tin

g d

ura

tio

n >

X)

Meeting duration (s)

(b) Encounter duration.

Fig. 6. Matching inter-meeting timeand encounter durationdistributionswith the encounter trace.

recurrent schedule with five different types of time periods(STEP1): technical sessions, coffee breaks, breakfast/lunchtime, evening, and late night (see [40] for the detailed pa-rameters). For each time period we set up communities as theconference rooms, the dining room, etc. We also generate acommunity that is far away from the rest of the communitiesfor each node and make the node sometimes isolated in thiscommunity to capture the behavior of patrons skipping partof the conference. In (STEP2), we use the theory presentedin section V to adjust the parameters and shape the inter-meeting time and encounter duration curves. For example, astronger tendency for nodes to choose roaming epochs (settinglarger πt

r) would increase the meeting probability (see, e.g.,Eq. (18)), hence reducing inter-meeting times. Finally, sincethe devices used to collect the encounter traces are always-on,we do not apply any changes to the synthetic trace (STEP3).We randomly generate820 pairs of users and show theircorresponding distributions of the inter-meeting time andtheencounter duration in Fig. 6 with labelModel. It is clear thatour TVC model has the capability to reproduce the observeddistributions, even if it is not constructed explicitly to do so.This displays its success in capturing the decisive factorsoftypical human mobility.

It is clear from the cases studied here that the TVCmodel is flexible to capture mobility characteristics fromvarious environments well. In addition, with the respectiveconfiguration, it is possible to generate synthetic traces withmuch larger scale (i.e., more nodes) than the empirical oneswhile maintaining the same mobility characteristics. It isalsopossible to generate multiple instances of the synthetic traceswith the same mobility characteristics to complement theoriginal, empirically collected trace.

V. THEORETICAL ANALYSIS OF THE TVC MODEL

So far, we have established the flexibility of the TVC modelin terms of its ability to reproduce the properties observedinqualitatively different mobility traces. Yet, one of the biggestadvantages of our model is that, in addition to the realism, itis also analytically tractable with respect to some importantquantities which determine protocol performance. In the restof this paper, we focus on demonstrating this last point.

We start here by deriving the theoretic expressions ofvarious properties of the proposed mobility model assumingthe nodes are always “on”. The properties of interest aredefined below.• The average node degreeis the average number of nodesresiding within the communication range of a given node. Thisis a quantity of interest due to its implication on the success

rate of various tasks (e.g. geographic routing [28]) in mobilead hoc networks.• Thehitting time is the time it takes a node, starting from thestationary distribution, to move within transmission range ofa fixed, randomly chosen target coordinate in the simulationfield.• The meeting timeis the time until two mobile nodes,both starting from the stationary distribution, move into thetransmission range of each other. Thehitting and meetingtimes are of interest due to their close relationship to theperformance of DTN routing protocols.

We note that a preliminary version of some of the theoreticalderivations presented here appear under a special case of ourTVC model in [15] (that model included one community andtwo time periods only). Here, we generalize all derivationsfor any community and time-period structure. We start witha useful lemma that calculates the probability of a node toreside in a particular state.

Lemma 5.1: The probability that a node moves, pauses(after the completion of an epoch) in statej, or performs atransitional epoch at any given time instant during time periodt, respectively, is:

P tmove,j = πt

j(Ltj/v

tj)/Ψ, (1)

P tpause,j = πt

jDtj/Ψ, (2)

P ttr =

St

∑

k=1

πtk

∑

∀n

ptk,nLtr(k,n)/vtk/Ψ. (3)

whereΨ =∑St

k=1 πtk(L

tk/v

tk +Dt

k +∑

∀n ptk,nLtr(k,n)/v

tk)

andLtr(k,n) is the average length of a transitional epoch fromcommunityk to communityn.

Proof: The probability for a node to be in statej (πtj)

can be easily derived with Markov chain theory from thestate transition probabilities (pti,j). The above result followsfrom the ratio of the average durations of the moving part(Lt

j/vtj) and the pause part (Dt

j) of regular epochs, and the

transitional epochs (Ltr(k,n)/vtk), weighted by the probabilities

of the states. The expected length of the transitional epochs,Ltr(k,n), can be calculated as follows. Note that if communityn contains communityk, no transitional epoch is needed (i.e.,Ltr(k,n) = 0). The transitional epoch is thus needed for aroaming node to go back to a smaller community, and as theprevious roaming epoch ends at a random location in the wholesimulation field, by symmetry, the expected length of thetransitional epoch is the average length to move to the centerof the simulation field from a random point in the simulationfield. Numerical analysis concludesLtr = 0.3826N in thiscase.

Note that the above stationary probabilities can be calculatedfor each time period and node separately. We useP t

j (i) todenote the probability that nodei is in statej during timeperiodt (i.e., P t

j (i) = P tmove,j(i) + P t

pause,j(i)).

A. Derivation of the Average Node Degree

The average node degree of a node is defined as the expectednumber of nodes falling within its communication range. Each

8

node contributes to the average node degree independently,asnodes make independent movement decisions.

Lemma 5.2: Consider a pair of nodes,a and b. Assumefurther that, in time periodt, communityj of nodea andcommunityk of nodeb overlap with each other for an areaA(atj , b

tk). Then, the contribution of nodeb to the average node

degree of nodea, whena resides in itsj-th community andbresides in itsk-th community, is given by

πK2

Ctj

2(a)

A(atj , btk)

Ctk

2(b)

, (4)

whereK is the communication range of the nodes.Proof: Since nodes follow random direction movement

in each epoch, they are uniformly distributed within eachcommunity (i.e., they are at any point within the communityequally likely). The probability for nodeb to fall in the j-thcommunity of nodea is simply the ratio of the overlapped areaover the size of thek-th community of nodeb. Nodea coversany given point in its community equal-likely, hence givennodeb is in the overlapped area, it is within the communicationrange of nodea with probabilityπK2/Ct

j

2(a).

Following the same principle in Lemma 5.2, we include allcommunity pairs and arrive at the following Theorem.

Theorem 5.3: The average node degree of a given nodeais

∑

∀Commtj(a)

P tj (a)

∑

∀b

∑

∀Commtk(b)

P tj (b)

πK2

Ctj

2(a)

A(atj , btk)

Ctk

2(b)

. (5)

Proof: Eq. (5) is simply a weighted average of thenode degree of nodea conditioning on its states. For eachstate with probabilityP t

j (a), the expected node degree is asum over all other nodes’ probability of being within thecommunication range of nodea, again conditioning on allpossible states. Transitional epochs are treated the same wayas roaming epochs here. That is, when considering a nodein the transitional state with probabilityP t

tr, it has equivalentcontribution to the node degree as when it is in the roamingstate (i.e., the node appears uniformly in the simulation fieldduring transitional epochs since the it moves from anywhereinthe simulation field back to the local community.). Hence, withprobabilityP t

tr+P troam

8, the node has an effective communitysize of the simulation field,N .

Corollary 5.4: In the special case when all nodes choosetheir communities uniformly at random among the simulationfield, Eq. (5) degenerates to

∑

∀bπK2

N2 .Proof: This result follows from the fact that a randomly

chosen community is anywhere in the simulation field equallylikely.

B. Derivation of the Hitting Time

In the calculation of the average node degree, the depen-dence between consecutive epochs did not affect the deriva-tion. In fact, only the stationary occupancy probabilitiesπt

j

andP tj (i.e. the probability of being found in communityj)

8If the node has no roaming state in this time period, then we consideronly P t

tr.

are needed, since we were looking only at a randomsnapshotof the model. In the case of hitting and meeting times, we areinterested in counting the number of epochs until a given targetcoordinate is found (“hit”). Our approach is to try to calculatethe “hit probability” for a given epoch, and then count thenumber of such epochs needed on average until the destinationpoint is hit. If these probabilities were independent, thenonecould use a simple geometric distribution to derive the result.However, (1) consecutive epochs are strongly related, as theending point of one epoch is, naturally, the beginning pointof the next. This introduces aseemingdependency betweenthe hit probabilities of consecutive epochs, complicatingthederivation. What is more, (2) thetransitionbetween communi-ties (and epochs performed in each) are governed by the TVCmodel’s Markov chain and the respective community transitionprobabilitiespti,j . Thus, looking only at the stationary probabil-ities for “choosing” the next communityj (as in the previoussection) no longer suffices. Finally, (3) thetransitional epochsthemselves introduce further complications, as they cannot, inthis case, be handled as regular in-community or even roamingepochs.

The above three observations introduce dependencies that,at first glance, complicate our task. Nevertheless, we willshow how these dependencies can be “washed out” under a(minimally restricting) set of assumptions, and that stationaryprobabilities still suffice to derive a simple formula for therespective hitting time that holdsin the limit. The basis of ourargument is found in the proof of Lemma 5.7, upon whichthe rest of results in this section depend (In a nutshell, thefast mixing of the mobility process takes care of (1), the largenumber of epochs required to hit a target takes care of (2) inthe limit, and the dominance of local and roaming epochs overtransitional epochs takes care of (3).). In Section VI, we showthat the accuracy of our theory is not compromised by theseassumptions and that our derivations introduce little error inmost practical scenarios considered.

The sketch of the derivation of the hitting time is as follows:(i) We first condition on the relative location of the targetcoordinate with respect to a node’s communities (Lemma 5.5).We identify all possible sub-cases (i.e. whether the targetisinside or outside one or more of the node’s communities).A target inside a community is, naturally, expected to befound faster than a target outside all communities. Usingsimple geometric arguments, we calculate the probability ofeach of these sub-cases (Lemma 5.6) and take the weightedaverage of all sub-cases and the respective hitting time (tobecalculated per sub-case). (ii) For a given sub-case, we derivethe expected number of epochs (and the expected number oftime units) until the target is found (Lemma 5.7). (iii) Finally,we introduce the time-period factor, and account for the totalnumber of time periods needed to hit the target (Theorem 5.9).

The most influential factor for the hitting time is whether thetarget coordinate is chosen inside the node’s communities.Wedenote the possible relationships between the target locationand the set up of communities during time periodt as the setΩt. Note that the cardinality of setΩt is at most2S

t

(i.e. foreach of theSt communities, the target coordinate is either inor out of it).

Lemma 5.5: By the law of total probability, the average

9

hitting time can be written as

HT =∑

w1∈Ω1,...,wV ∈ΩV

P (w1, ..., wV )HT (w1, ..., wV ), (6)

wherew1, w2, ..., wV denote one particular relationship (i.e. acombination ofout, inS

t

) between the target coordinate andthe community set up during time period1, 2, ..., V , respec-tively. FunctionsP (·) and HT (·) denote the correspondingprobability for this scenario and the conditional hitting timeunder this scenario, respectively. Note that each sub-casew1, w2, ..., wV is disjoint from all other sub-cases.

To evaluate Eq. (6), we need to calculateP (w1, ..., wV ) andHT (w1, ..., wV ) for each possible sub-case(w1, ..., wV ).

Lemma 5.6: If the target coordinate is chosen independentof the communities and the communities in each time periodare chosen independently from other periods, then

P (w1, ..., wV ) = ΠVt=1P (wt), (7)

where P (wt) = A(wt)/N2, i.e., the probability of a sub-casewt is proportional to the areaA(wt) that correspondsto the specific scenariowt, which is a series of conditionsof the following type: (target ∈ commt

1, target /∈commt

2, ..., target ∈ commtS).

Proof: The result follows from simple geometric argu-ments.

The first step for calculatingHT (w1, ..., wV ) is to derivethe unit-time hitting probability in time periodt under targetcoordinate-community relationshipwt, denoted asP t

h(wt).

Lemma 5.7: For a given time periodt and a specificscenariowt,

P th(w

t) =

St

∑

j=1

I(target ∈ Commtj |w

t)P tmove,j2Kvtj/C

tj

2,

(8)whereI(·) is the indicator function.

Proof: In order to calculate the expected hitting time,let us first count the total number of epochs needed. Let usassume thatNe epochs are needed in total, and let us denote asepochEkm

(m) them-th epoch in sequence (that is occurringin communitykm). Let further,P (k1, k2, . . . , km) denote theprobability of the specific sequence of epochs occurring. Then,the probability that the target has not been found aftern epochsis

P (Ne > n) =X

k1,k2,...,kn

P (k1, k2, . . . , kn) · P (Ek1(1) = miss,

Ek2(2) = miss, . . . , Ekn(n) = miss). (9)

In order to simplify the above equation, we need to dealwith the inherent dependencies introduced by the transition ofepochs. First, since node movement is continuous, the end ofone epochEj(m), performed in communityj, is the beginningof the next,Ej(m+ 1), if performed in the same community9. Nevertheless, as explained in Section III-B, the expected“length” of an epochLt

j performed in communityj is in theorder of the square root of the community sizeCt

j . This is

9For the moment, we will ignore transitional epochs, and assume thatall epochs are performed inside some community; we deal withtransitionalepochs later.

sufficient for the node to “mix” in the community after justone epoch [29]. Consequently, we can write

P (Ne > n) =X

k1,k2,...,kn

P (k1, k2, . . . , kn) ·

nY

i=1

P (Eki= miss). (10)

An additional dependency arises from the transitionsbetween communities and the calculation of termP (k1, k2, . . . , kn). If epoch Ekm

(m) is performed incommunity km the next epochEkm+1

(m + 1) will beperformed in communitykm+1 with probability pkm,km+1

(the transition probability in the Markov Chain governing thecommunity transitions in the TVC model). Let us assumethat Ne denotes again the total epochs needed (of any type)to hit the target. Further, let there belj epochs of typej(i.e. performed in communityj) in the above mix ofNe

total epochs. WhenNe → ∞, then li → πiNe, that is, thetotal number of epochs in communityi dependsonly on thestationary probability of communityi, πi. Thus,

P (k1, k2, . . . , kn) = πk1· πk2

. . . πkn. (11)

Consequently, Eq.(10) becomes

P (Ne > n) =n∏

i=1

πki· P (Eki

= miss). (12)

This implies that,in the limit10, the total number of epochsneeded to hit the target can be approximated by a geometricdistribution, where the “average” epoch has a hit probabilityof

St

∑

i

πi · P (Ei = hit) (13)

As the final step, we need to calculate the probability of agiven epoch in communityj to hit the target. Instead of usingthis per epoch hit probability, we revert now to what we callthe unit-stephit probabilities,Ph. The unit-step probability isthe probability of encountering the target exactly within thenext time-unit (rather than within the duration of a wholeepoch). This discrete approximation provides an equivalentformulation to the above continuous case (see [29]), howeverit is more convenient to manipulate in the case of time-period boundaries and meeting times calculated later. (Notethat this approximation is again only possible when the averageepoch length is in the order of the respective community size,ensuring mixing after one epoch.)

Note that the hitting event can only occur when the node isphysically moving inside the community where the target islocated11. Whether the target is located inside communityj isdenoted using the indicator functionI(target ∈ Commt

j |wt).

If the target is outside the community, then this probabilityof hit is zero. If the target lies within communityj, then

10In practice, the requirement is that a large number of epochsis needed onaverage until the target is hit. In the sparse networks we’reinterested in, thisis a reasonable assumption, and as we shall show in Section VIthe achievedaccuracy is indeed high.

11We neglect the small probability that the target is chosen out of thecommunity but close to it, and make the contributions from epochs in statej zero if the chosen target coordinate is not in communityj.

10

when a node moves with average speedvj , on average itcovers a new area of2Kvj in unit time. Since a nodefollowing random direction movements visits the area it movesabout with equal probability, and the target coordinate ischosen at random, it falls in this newly covered area withprobability 2Kvj/C

tj

2 [29]. Hence the contribution to theunit-time hitting probability by movements made in statejis P t

move,j2Kvtj/Ctj

2. Thus, in Eq.(13),πj is replaced byI(target ∈ Commt

j |wt)P t

move,j in the unit-step case, and

P (Ei = hit) by 2Kvtj/Ctj

2.As a final remark, the contribution of transitional epochs

to the unit-time hitting probability is not equivalent to otherepochs (due to the dependency of end-points on local com-munities, which introduces bias after communities have beenchosen). Nevertheless, in a normal mobility scenario, we ex-pect a node to spend the majority of its time within one of thecommunities rather than in transitional epochs. Specifically,we will assume that community transition probabilities exhibita strong positive correlation, that is, if a node resides incommunityj, it has a higher probability of staying within thiscommunity for the next epoch, rather than leaving. In this case,the total contribution of transitional epochs is small, andcanbe safely ignored in order to not complicate our analysis. Theabove is a reasonable assumption for many target scenarios wecan imagine; simulation results show further that the time anode resides in transitional state is indeed less than10% in thescenarios considered, not significantly affecting the accuracyof the above expression.

Given the fore-mentioned assumptions about unit-step hit-ting probabilities, the corollary below follows.

Corollary 5.8: The probability for at least one hitting eventto occur during time periodt under scenariowt is

P tH(wt) = 1− (1− P t

h(wt))T

t

. (14)

Finally, using the law of total probability, we derive theconditional hitting time under a specific target-communityrelationship,HT (w1, ..., wV ).

Theorem 5.9:

HT (w1, ..., wV ) =

V∑

t=1

HT (w1, ..., wV |first hit in period t)·

P (w1, ..., wV , f irst hit in period t),(15)

where the probability for the first hitting event to happen intime periodt is

P (w1, ..., wV , f irst hit in period t)

=Πt−1

i=1(1− P iH(wi)) · P t

H(wt)

P,

(16)

and the hitting time under this specific condition is

HT (w1, ..., wV |first hit in period t)

=

V∑

i=1

T i · (1

P− 1) +

t−1∑

i=1

T i +1

P th(w

t),

(17)

whereP = 1 − ΠVt=1(1 − P t

H(wt)) is the hitting probabilityfor one full cycle of time periods.

Proof: Eq. (16) holds as each cycle of time periodsfollows the same repetitive structure, and for the first hittingevent to occur in time periodt it must not occur in time period1, ..., (t − 1). The first term in Eq. (17) corresponds to theexpected duration of full time period cycles until the hittingevent occurs. Since for each cycle the success probability ofhitting the target isP , in expectation it takes1/P cycles tohit the target, and there are1/P − 1 full cycles. The secondterm in Eq. (17) is the sum of duration of time periods beforethe time periodt in which the hitting event occurs in the lastcycle. Finally, the third term is the fraction of the last timeperiod before the hitting event occurs. Note that the last part isan approximation which holds if the time periods we considerare much longer than unit-time.

C. Derivation of the Meeting Time

The procedures of the derivation of the meeting time issimilar to that of the hitting time detailed in the last section.In short, we derive the unit-step (or unit-time) meeting prob-ability, Pm, and the meeting probability for each type of timeperiod,PM , and put them together to get the overall meetingtime in a similar fashion as in Theorem 5.9.

Similar to Lemma 5.7, we add up the contributions to themeeting probability from all community pairs from nodeaandb in the following Lemma.

Lemma 5.10: Let communityj of nodea and communitykof nodeb overlap with each other for an areaA(atj , b

tk) in time

period t. Then, the conditional unit-time meeting probabilityin time periodt when nodea and b are in its communityjand k, respectively, is

P tm(atj , b

tk) =

P tmove,j(a)P

tmove,k(b)v

2Kv

A(atj , btk)

A(atj , btk)

Ctj

2(a)

A(atj , btk)

Ctk

2(b)

+ P tmove,j(a)P

tstop,k(b)

2Kv

A(atj , btk)

A(atj , btk)

Ctj

2(a)

A(atj , btk)

Ctk

2(b)

+ P tstop,j(a)P

tmove,k(b)

2Kv

A(atj , btk)

A(atj , btk)

Ctj

2(a)

A(atj , btk)

Ctk

2(b)

.

(18)

Proof: Equation (18) consists of two parts:(I) Both of the nodes are moving within the overlapped area.

This adds the first term in Eq. (18) to the meeting probability.

The two ratios,A(at

j ,btk)

Ctj2(a)

andA(at

j ,btk)

Ctk2(b)

, capture the probabilities

that the nodes are in the overlapped area of the communities.The contribution to the unit-time meeting probability is theproduct of probabilities of both nodes moving within theoverlapped area and the term2Kv

A(atj,bt

k) , which reflects the

covered area in unit time. We use the fact that when bothnodes move according to the random direction model, one cancalculate the effective (extra) area covered by assuming thatone node is static, and the other is moving with the (higher)relative speedbetween the two. This difference is capture withthe multiplicative factorv [29].

(II) One node is moving in the overlapped area, and theother one pauses within the area. This adds the remaining twoterms in Eq. (18) to the unit-time meeting probability. These

11

terms follow similar rationale as the previous one, with thedifference that now only one node is moving. The second termcorresponds to the case when nodea moves (andb is static),and the third term corresponds to the case when nodeb moves(anda is static).

The derivation of the unit-time meeting probability betweennodesa andb for time periodt includes all possible scenariosof community overlap. If nodea has St(a) communitiesand nodeb has St(b) communities, there can be at mostSt(a)St(b) community-overlapping scenarios in time periodt. For similar reason detailed in the proof of Lemma 5.7, weneglect the contribution of transitional epochs to the unit-timemeeting probability.

Note that (18) is the general form of Equation (13) and (14)in [15]. If we assume perfect overlap and a single communityfrom both nodes, we arrive at (14). If we assume no over-lap, we result in (13). Also note in the general expressionspresented in this paper, the whole simulation area is alsoconsidered as a community. Therefore we do not have toinclude a separate term to capture the roaming epochs.

Corollary 5.11: The probability for at least one meetingevent to occur during time periodt is

P tM =1−

∑

∀(j,k)

Pov(atj , b

tk) · (1− P t

m(atj , btk))

T t

, (19)

wherePov(atj , b

tk) is the probability that the communityj of

nodea overlaps with communityk of nodeb. This quantityis simply 1 if the communities have fixed assignments andA(atj , b

tk) 6= 0. If the communities are chosen randomly, this

probability can be derived by Lemma 4.5 in [15]. Due to spaceconstraint, the Lemma is not reproduced here.

Finally, similarly to Theorem 5.9, the expected meeting timecan be calculated using the results in the Lemmas in thissection.

Theorem 5.12: The expected meeting time is

MT =V∑

t=1

MT (meet in period t)P (meet in period t).

(20)Where the quantities in the above equation are calculated by

P (meet in period t) =Πt−1

i=1(1− P iM ) · P t

M

Q, (21)

MT (meet in period t) =V∑

i=1

T i · (1

Q− 1) +

t−1∑

i=1

T i +1

P tm

,

(22)whereQ = 1 −ΠV

i=1(1 − P iM ) is the meeting probability for

one full cycle of time periods.

Proof: The proof is parallel to that of Theorem 5.9 andis omitted due to space limitations (see [40] for details).

As a final note, we can easily modify the above theoryto account for potential “off” periods (e.g. by introducinga per step or per epoch “off” probability, and a respectivemultiplicative factor). Due to space limitations, we do notinclude here these modifications.

VI. VALIDATION OF THE THEORY WITH SIMULATIONS

In this section, we compare the theoretical derivations of theprevious section against the corresponding simulation results,for various parameter settings. Through extensive simulationswith multiple scenarios and parameter settings, we establishthe accuracy of the theoretical framework. Due to spacelimitations, we can only show some examples of the simulationresults we have. More complete results can be found at [40].

We summarize the parameters for the tested scenarios inTable II. We use two different setup of the TVC model forthe simulation cases. The parameters listed in Table II are forthe simple models (Model 1 − 4), where we have two timeperiods with two communities in each time period (one of thecommunities is the whole simulation field). We also simulatefor more generic setup of the TVC model (Model 5−7, referto [40] for its parameters), where we have three communities(one of them is the simulation field) in each time period. Forthe generic models, we have experimented with two ways ofcommunity placement: in a tiered fashion (as drawn in TP2 inFig. 2), or in a random fashion. Our discrete-time simulatoriswritten in C++. More details about the simulator, as well asthe source code, can be found at [40].

A. The Average Node Degree

For the average node degree, we create simulation scenarioswith 50 nodes in the simulation area, and calculate the averagenode degree of each node by taking the time average acrosssnapshots taken every second during the simulation, and thenaverage across all nodes. All the simulation runs last for60000seconds in this subsection.

As we show in Corollary 5.4, when the communities arerandomly chosen, the average node degree turns out to be theaverage number of nodes falling in the communication rangeof a given node,as if all nodes are uniformly distributed.Hence the average node degree does not depend on the exactchoices of community setup (i.e. single, multiple, or multi-tiercommunities) or other parameters. In Fig. 7 (a), we see thesimulation curves follow the prediction of the theory well.

To make the scenario a bit more realistic, we simulatesome more scenarios when the communities are fixed. Amongthe 50 nodes, we make25 of them pick the communitycentered at(300, 300) and the other25 pick the communitycentered at(700, 700). We simulate scenarios for all sevensets of parameters, and show some example results in Fig.7 (b). In the simulations, when the communication rangesare small as compared to the edge of the communities, therelative errors are low, indicating a good match between thetheory and the simulation. However, as the communicationrange increases, the area covered by the communication diskbecomes comparable to the size of the community and Eq. (4)is no longer accurate since the communication disk extendsout of the overlapped area in most cases. That is the reasonfor the discrepancies between the theory and simulation.BesidesModel-3, we observe at most20% of error when thecommunication disk is less than20% the size of the inner-most community, indicating that our theory is valid when thecommunication range is relatively small.

12

TABLE IIPARAMETERS FOR THE SCENARIOS IN THE SIMULATION

We use the same movement speed for all node:vmax = 15 andvmin = 5 in all scenarios. In all cases we use two time periods and theyare named as time period1 and2 forconsistency. We only list the parameters for the simple models (Model 1 − 4) here. Please refer to [40] for the details of the generic models (Model 5 − 7).

Model name Description N C1l C2

l Dmax,l Dmax,r Ll Lr π1l π1

r π2l π2

r T 1 T 2

Model 1 Match with the MIT trace 1000 100 100 100 50 80 520 0.714 0.286 0.8 0.2 5760 2880Model 2 Highly attractive communities 1000 200 50 100 200 52 520 0.667 0.333 0.889 0.111 3000 2000Model 3 Not attractive communities 1000 100 100 50 200 80 800 0.5 0.5 0.667 0.333 2000 1000Model 4 Large-size communities 1000 200 250 50 100 200 800 0.7 0.3 0.889 0.111 2000 1000

Communication Range (K)

Av

erag

e N

od

e D

egre

e

0

0.5

1

1.5

2

0 20 40 60 80 100

Theory Model1 Model5

(a) Randomly placed community.

Communication Range (K)

Aver

age

Node

Deg

ree

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50

Model5-simModel5-theoryModel7-simModel7-theoryModel3-simModel3-theory

(b) Fixed communities.

Fig. 7. Examples of simulation results (the average node degree).

B. The Hitting Time and the Meeting Time

We perform simulations for the hitting and the meetingtimes for50, 000 independent iterations for each scenario, andcompare the average results with the theoretical values derivedfrom the corresponding equations (i.e. (6) and (20)). To findout the hitting or the meeting time, we move the nodes inthe simulator indefinitely until they hit the target or meet witheach other, respectively.

Again we show some example results in Fig. 8. For allthe scenarios (including the ones not listed here), the relativeerrors are within acceptable range. The absolute values forthe error are within15% for the hitting time and within20% for the meeting time. For more than70% of the testedscenarios, the error is below10% (refer to [40] for otherfigures). These results display the accuracy of our theory undera wide range of parameter settings. The errors between thetheoretical and simulation results are mainly due to someof the approximations we made in the various derivations.For example, the approximation of the hitting and meetingprocesses with discrete, unit-time Bernoulli trials is valid onlyfor the epochs that are long enough (in the order of communitysize) and if there are a lot of epochs. Furthermore, there existsome border effects – when a node is close to the border ofa community, it could also “see” some other nodes outsideof the community if its transmission range is large enough.However, we have chosen to ignore such occurrences to keepour analysis simpler. Nevertheless, as shown in the figures,the errors are always within acceptable ranges, justifyingoursimplifying assumptions.

VII. U SING THEORY FORPERFORMANCEPREDICTION

Although the various theoretical quantities derived for theTVC 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. Weillustrate this point with two examples in this section.

Hit

tin

g T

ime

(s)

Communication range (K)

0

10000

20000

30000

40000

50000

10 20 30 40 50 60 70

Model1-simModel1-theoryModel2-simModel2-theoryModel6(multi_tier)-simModel6(multi_tier)-theory

(a) Hitting Time.

Mee

tin

g T

ime

(s)

Communication range (K)

0

3000

6000

9000

12000

15000

10 20 30 40 50 60 70

Model5(tiered_comm)-simModel5(tiered_comm)-theoryModel7(multi_comm)-simModel7(multi_comm)-theoryModel3-simModel3-theory

(b) Meeting Time.

Fig. 8. Examples of simulation results.

A. Estimation of the Number of Nodes Needed for GeographicRouting

It has been shown in geographic routing that the averagenode degree determines the success rate of messages deliv-ered [28]. Thus, using the results of Section V-A we canestimate the number of nodes (as a function of the averagenode degree) needed to achieve a target performance forgeographic routing, for a given scenario.

We consider the same setup as in Section VI-A, wherehalf 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 routingmessages across one of the communities, from coordinate(250, 250) to coordinate(350, 350) with simple geographicrouting (i.e., greedy forwarding only, without face routing[21]). Using simulations we obtain the success rate of geo-graphic routing under various communication ranges when200nodes move according to the mobility parameters ofModel-1 (Table II). Results are shown in Fig. 9 (each point is thepercentage of success out of2000 trials). If we assume themobility model is different, sayModel-3, we would like toknow how many nodes we need to achieve similar perfor-mance. Using Eq. (5) we find that760 nodes are needed tocreate a similar average node degree forModel-3. To validatethis, we also simulate geographic routing for a scenario where760 nodes followModel-3. Comparing the resulting messagedelivery ratio for this scenario to the original scenario (200nodes withModel-1) in Fig. 9, we see that similar successrates are achieved under the same transmission range, whichconfirms the accuracy of our analysis.

B. Predicting Message Delivery Delay with Epidemic Routing

Epidemic routing is a simple and popular protocol thathas been proposed for networks where nodal connectivity isintermittent (i.e., in Delay Tolerant Networks) [34]. It has beenshown that message propagation under epidemic routing canbe modeled with sufficient accuracy using a simple fluid-based

13

0

0.2

0.4

0.6

0.8

1

10 15 20 25 30 35 40

Model1_200nodes

Model3_760nodes

Nodal communication range

Su

cces

s ra

te

Fig. 9. Geographic routing successrate under different mobility parame-ter sets and node numbers.

Simulation timeNo

des

rec

eiv

ed t

he

mes

sag

e

0

10

20

30

40

50

60

0 500 1000 1500 2000

Theory (SI model)

Simulation

Fig. 10. Packet propagation with epi-demic routing withtwo node groupswith different communities.

model [35]. (Note that its performance has also been analyzedusing Markov Chain [8] and Random Walk [30] models.) Thisfluid model has been borrowed from the Mathematical Biologycommunity, and is usually referred to as the SI (Susceptible-Infected) epidemic model. The gist of the SI model is thatthe rate by which the number of “infected” nodes increases(“infected” nodes here are nodes who have received a copyof the message) can be approximated by the product of threequantities: the number of already infected nodes, the numberof susceptible (not yet infected) nodes, and the pair-wisecontact rate,β (assuming nodes meet independently – thiscontact rate is equivalent to the unit-step meeting probabilitiescalculated in (18)). Thus, one could plug-in these meetingprobabilities into the SI model equations and calculate thedelay for epidemic routing. Yet, in the TVC model (and oftenin real life) there are multiple groups of nodes with differentcommunities, and thus different pair-wise contact rates thatdepend on the community setup. For example, nodes with thesame or overlapping communities tend to meet much moreoften than nodes in far away communities. For this reason,we extend the basic SI model to a more general scenario.

We consider the following setup in the case study: We useModel-3 (Table II) for the mobility parameters. A total ofM = 50 nodes are divided into two groups of25 nodes each.One group has its community centered at(300, 300) and theother at(700, 700). One packet starts from a randomly pickedsource node and spreads to all other nodes in the network. Thepropagation of the message can be described by the followingequations:

dI1(t)dt

= βovI1(t)S1(t) + βno ovI2(t)S1(t)dI2(t)dt

= βovI2(t)S2(t) + βno ovI1(t)S2(t)S1(t) + I1(t) = M/2S2(t) + I2(t) = M/2.

(23)

whereSx(t) and Ix(t) denote the number of susceptible andinfected nodes at timet in groupx, respectively. Parametersβov and βno ov represent the pair-wise unit-time meetingprobability when the communities are overlapped (i.e., fornodes in the same group) and not overlapped (i.e., nodesin different groups), respectively. We use Eq. (18) to obtainthese quantities. This model is an extension from the standardSI model [35] and similar extensions can be made for morethan two groups [32]. The first equation governs the changeof infected nodes in the first group. Notice that the infectionto susceptible nodes in the group (S1(t)) can come from theinfected nodes in the same group (I1(t)) or the other group(I2(t)). We can solve the system of equations in (23) to get

the evolution of the total infected nodes in the network. Ascan be seen in Fig. 10, thetheory curve closely follows thetrend in thesimulationcurve. This indicates first that scenariosgenerated by our mobility model are still amenable to fluidmodel based mathematical analysis (SI), despite the increasedcomplexity introduced by the concept of communities. It alsoshows that results produced thus can be used by a systemdesigner to predict how fast messages propagate in a givennetwork environment. This might, for example, determineif extra nodes are needed in a wireless content distributionnetwork to speed up message dissemination.

As a final note, in addition to epidemic routing, the theoreti-cal results for the hitting and meeting times could be applied topredict the delay of various other DTN routing protocols (seee.g. [29], [30], [35]), for a large range of mobility scenariosthat can be captured by the TVC model.

VIII. C ONCLUSION AND FUTURE WORK

We have proposed atime-variant community mobility modelfor wireless mobile networks. Our model preserves commonmobility characteristics, namelyskewed location visiting pref-erencesand periodical re-appearanceobserved in empiricalmobility traces. We have tuned the TVC model to matchwith the mobility characteristics of various traces (WLANtraces, a vehicle mobility trace, and an encounter trace ofmoving human beings), displaying its flexibility and generality.A mobility trace generator of our model is available at[40]. In addition to providing realistic mobility patterns, theTVC model can be mathematically analyzed to derive severalquantities of interest: theaverage node degree, thehitting timeand themeeting time. Through extensive simulations, we haveverified the accuracy of our theory.

In the future we would like to further analyze the perfor-mance of various routing protocols (e.g., [30], [31]) underthetime-variant community mobility model. We also would liketo construct a systematic way to automatically generate theconfiguration files, such that the communities and time periodsof nodes are set to capture the inter-node encounter propertieswe observe in various traces (for example, the Small Worldencounter patterns observed in WLAN traces [13]).

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available at http://nile.cise.ufl.edu/˜weijenhs/TVCmodel

Wei-jen Hsu was born in Taipei, Taiwan, in March1977. He received the B.S. degree in ElectricalEngineering and the M.S. degree in Communica-tion Engineering, respectively, from National TaiwanUniversity, in June 1999 and June 2001. He receivedthe Engineer degree in Electrical Engineering fromUniversity of Southern California, in August 2006.He is currently a Ph.D. student in the CISE De-partment, University of Florida. His main researchinterest involves the utilization of realistic measure-ment data in various tasks in computer networks,

including user modeling and behavior-aware protocol design.

Thrasyvoulos Spyropoulos was born in Athens,Greece, in July 1976. He received the Diplomain Electrical and Computer Engineering from theNational Technical University of Athens, Greece, inFebruary 2000. In May 2006, he received the Ph.Ddegree in Electrical Engineering from the Universityof Southern California. In 2006-07, he was a post-doctoral researcher at INRIA, Sophia-Antipolis. Heis currently a senior researcher with the Swiss Fed-eral Institute of Technology (ETH), Zurich.

Konstantinos PsounisKonstantinos Psounis is anassistant professor of EE and CS at the Universityof Southern California. He received his first de-gree from National Technical University of Athens,Greece, in 1997, and the M.S. and Ph.D. degreesfrom Stanford University in 1999 and 2002 re-spectively. Konstantinos models and analyzes theperformance of a variety of networks, and designsmethods and algorithms to solve problems relatedto such systems. He is the author of more than 40research papers, has received faculty awards from

NSF and the Zumberge foundation, and has been a Stanford graduate fellowthroughout his graduate studies.

Ahmed Helmy Dr. Ahmed Helmy received hisPh.D. in Computer Science (1999), M.S. in Elec-trical Engineering (EE) (1995) from the Universityof Southern California (USC). He is Associate Pro-fessor and director of the wireless networking lab atthe CISE Dept, University of Florida. From 1999 to2006, he was faculty with EE-USC. He was a keyresearcher in the network simulator (NS-2) and theprotocol independent multicast (PIM-SM) projectsat USC/ISI. In 2002, he received the NSF CAREERAward. His interests include network protocol design

and analysis for mobile ad hoc and sensor networks, and mobility modeling.