Modeling Spatial and Temporal Dependencies of User Mobility in Wireless Mobile Networks

1 Modeling Spatial and T emporal Dependencie s of User Mobility in W ireless Mobile Netw orks W e i-Jen Hsu ∗ , Thrasyvoulos Spyropoulos † , K onstantinos Psoun is ‡ and Ahmed Helmy ∗ ∗ Dept. of Computer and Information Sc ience and Enginee ring, Uni versit y of Florida, Gaines ville, Florida † Computer Engineering an d Networks Lab, ETH Zurich ‡ Dept. of Electrical Eng ineering, Uni versity of Southern California, Los Angeles , Calif ornia Email: wjhsu@uﬂ.ed u, spyropoulos@tik.ee.ethz.c h, kpsoun is@usc.edu , he lmy@uﬂ.edu Abstract —Realistic mobility models are fundamental to ev al- uate the perform ance of protocols in mobile ad hoc networks. Unfortunately , there ar e no mobility models that capture the non- homogeneous behavio rs in both space and time commonly found in reality , while at th e same time bein g easy to use and analyze. Motiva ted by this, we propose a time-variant community mobility model, referred to as the TVC model, which realistically captures spatial and temporal corr elations. W e devise the communities that lead to skewed location visitin g preferences, and time periods that allow us to model time depend ent behaviors and periodic re-appearances of nodes at speciﬁc locations. T o demonstrate the power and ﬂexibility of th e TVC model, we use i t to generate syntheti c traces that match the ch aracteristics of a n umber of q ualitative ly dif ferent mobility t races, in cluding wireless LAN traces, vehicular mobility traces, and human encounter traces. More importantly , we show that, despite th e high lev el of realism achiev ed, our TVC model is still theoretically tractable. T o establi sh this, we derive a number of important quantities related to protocol performa nce, such as the av erage node degree, the hit ting time, and the meeting time, and provide examples of how to u tilize this theory to guide design d ecisions in routing p rotoco ls. I . I N T R O D U C T I O N Mobile a d hoc networks (MANETs) are self-organized, infrastructu re-less networks that could potentially supp ort many applications, such as vehicular networking ( V ANET) [4], wild-life tracking [19], and Internet provision to rural ar- eas [16], to n ame a few . Mob ility also enables message delivery in sparsely conn ected n etworks, gen erally known as delay tolerant network s ( DTNs). As the devices are easily portable and th e scenario s of deploymen t are inherently dy- namic, mo bility becomes one of the key chara cteristics in most of these networks. It has been shown th at mobility impacts MANETs in multiple ways, such as network cap acity [9], routing pe rforman ce [ 1], and cluster maintenanc e [2 4]. In short, the e valuation of protoc ols and services for MANETs seems to b e insepar able from the u nderlying mo bility models. It is, thus, of crucial importan ce to have suitab le mobility models as the fou ndation for the study o f ad ho c ne tworks. Ideally , a good mob ility model should achieve a number of go als: ( i) it sho uld ﬁrst captur e r ea listic mobility pattern s of scenarios in which one wants to ev entually operate the network; (ii) at the sam e time it is desirable that the model is mathematica lly tractable ; this is very im portant to allow researchers to der i ve perfor mance boun ds and und erstand th e limitations of various protoco ls un der the given scenario, as in [30], [31], [ 9], [6]; (iii) ﬁnally , it shou ld be ﬂexible enough to provid e qualitatively and qu antitativ ely different mobility characteristics by chan ging som e par ameters o f the mod el, yet in a re peatable and scalable manne r; design ing a new mob ility model f or each existing or new scenario is und esirable. Most existing mo bility models excel in o ne or, less often , two aspects o f the above req uirements, but none satisﬁes all of them at the same time. Our goal in this paper is, on on e hand, to improve the existing random mo bility models (e.g ., random walk, r andom d irection, etc.) an d synth etic mo bility models (e .g., [12], [11], [ 17]) on the fron t of realism , by considerin g empirically observed mobility ch aracteristics from the traces [14]. On th e oth er h and, the construction of th e model sh ould new model should be simple enoug h to allow in-depth th eor etical analysis , a nd be ﬂexible enou gh to hav e wider applicability than the mobility traces (which p rovide only a sing le snap shot of th e underlying mo bility pro cess) and cur rent trace-based mob ility models [ 33], [ 23], [22] whic h focus m ainly on matching mobility characteristics with a speciﬁc class of traces. The main c ontribution of this pap er is the pr oposal of a time-variant commu nity mobility mo del , refe rred to as the TVC m odel, which is realistic, ﬂexible, a nd ma thematically tractable . On e salient characteristic in the TVC mode l is location pr efer ence . Another important characteristic is the time-depen dent, periodical beh avior of n odes. T o o ur best knowledge, th is is the ﬁrst synthetic mo bility model that captures n on-ho mogeneo us behavior in both space and time . T o establish the ﬂexibility of ou r TVC mode l we show that we can m atch its two prominen t prope rties, loca tion visiting p r efer ence s and p eriodical re-appearance , w ith mu l- tiple WLAN traces, collected from environments such as university campu ses [10], [14] and corpora te buildings [ 2]. More inter estingly , althou gh we motiv ate the TV C mo del with the ob servations mad e on WLAN traces, ou r model is generic enoug h to have wider applicab ility . W e validate this claim b y examples of matching ou r T VC m odel with two add itional mobility traces: a vehicle mobility trace[36] and a hu man encoun ter tra ce[6]. In the latter case, we ar e e ven able to match our TVC mod el with some other mo bility characteristics not explicitly incorpo rated in o ur model by its construc tion, namely the in ter meetin g time and encoun ter duration between different users/devices. Finally , in add ition to th e improved realism, the T VC mod el can be ma thematically treated to derive analy tical expressions for importa nt qu antities o f interest, such as th e average no de de gr ee , th e hitting time and the meeting time . Th ese q uantities 2 are often fund amental to theoretically study issues such as routing perform ance, cap acity , connectivity , etc. W e show tha t our th eoretical der i vations ar e accu rate thro ugh simulation cases with a wide rang e of p arameter sets, an d addition ally provide examples o f how our theory cou ld be utilized in actual pr otocol design . T o our best knowledge, this is the ﬁrst synthetic mob ility m odel prop osed that m atches with tr aces from multiple scenarios, and has also been theoretically treated to the extent presented in th is pap er . W e make the co de of th e TVC model av ailable at [ 40]. The the paper is organized as f ollows: I n Section II we discuss related work. Our TVC model is then introdu ced in Section III. In Section IV , we show h ow to generate realistic mobility scen arios matched with various traces. T hen, in Section V, we present ou r theoretical framework and d eriv e generic expressions of various qu antities. Simulation validates the accuracy o f th ese expre ssions in Section VI. Addition ally , in Section VII, we motiv ate ou r theo retical f ramew ork f urther, by app lying our analysis to perfor mance pred ictions in proto- col design. Finally , we con clude the pap er in Section VII I. I I . R E L A T E D W O R K Mobility models hav e b een lo ng rec ognized as o ne of the fun damental com ponents that impacts the perf ormance of wireless ad hoc n etworks. A wid e variety of mo bility models are a vailable in the resear ch com munity ( see [5 ] fo r a g ood survey). Amon g all mo bility mo dels, the p opularity of random mo bility models (e.g ., rand om walk, rand om di- rection, and rand om waypoint) roo ts in its simplicity and mathematical tractability . A number of importan t prop erties for these mo dels have been studied, such as the station ary nod al distribution [3], the hitting and meeting times [29], and the meeting duration [ 18]. These quantities in turn enable routing protoco l analysis to produ ce perform ance boun ds [30], [31]. Howe ver , random mo bility mo dels are based on ov er-simpliﬁed assumptions, and as has been shown rece ntly an d we will also show in the p aper, the resultin g mobility char acteristics are very d ifferent from real-life scenar ios. Hence, it is deb atable whether th e ﬁndings un der these mo dels will direc tly tran slate into perfo rmance in r eal-world imp lementations of MANETs. More re cently , a n ar ray of synthetic mob ility mod els are propo sed to improve the re alism of the s imple random mobility models . Mor e complex rules are introduced to m ake the nodes follow a popu larity distribution wh en selecting th e next destination [ 12], stay on d esignated p aths fo r movements [17], or move as a group [ 11]. T hese rules enrich the scenarios covered b y the synth etic mobility models , but at th e same time make theo retical tre atment of these mo dels difﬁcult. In addition, most synthetic mob ility models are still limited to i.i.d. models, a nd the mob ility decisions are also independ ent of the curre nt location of no des and time of simula tion. A different app roach to mobility mo deling is by emp irical mobility trace collection . Along this line, researcher s h av e exploited existing wireless ne twork inf rastructure, such as wireless LANs (e.g., [2], [25 ], [10]) or cellular phon e n etworks (e.g., [ 7]), to track u ser mobility by mon itoring their lo cations. Such traces can be replay ed as input mobility p atterns for simulations of network protoco ls [13]. More recen tly , DTN- speciﬁc testbeds [6], [4], [19] aim at co llecting en counter ev ents between mobile nodes instead of the m obility p atterns. Some initial efforts to math ematically analyze these traces can be fou nd in [6], [2 0]. Y et, th e size o f the traces and the environments in which the experiments are pe rformed can not be adjusted at will b y th e resear chers. T o imp rove the ﬂexibility of traces, the ap proach of trace-based mo bility models have also been proposed [33], [23], [ 22]. These models discover the under lying mobility rules that lead t o the observed proper ties (such as th e du ration of stay at loca tions, the arrival patterns, etc.) in th e traces. Statistical an alysis is then used to determine proper par ameters o f th e m odel to ma tch it with the particular trace. The go al of this work is to combine the strengths of various approa ches to mo bility modeling and p ropose a r ealistic, ﬂ ex- ible, a nd mathematica lly tractable synthetic mobility model . Our work is partly motiv ated by se veral pro minent, comm on proper ties in multiple WLAN traces (e.g., traces a vailable from public archi ves [38], [37]) we observed in [14], based on which we constru ct the TVC m odel. This mod el extends the con cept of commu nities propo sed by us in [2 9] an d also introdu ces time-depen dent b ehavior . A pre liminary version o f th e mo del has been presented in [15]. I n this work we highligh t the ﬂexibility o f the TVC model by match ing the syn thetic traces with two additional, q ualitativ ely different traces to WLAN traces (i.e., vehicular an d human encounter traces, in section IV). W e a lso extend and present mor e generic theoretical results und er the scen ario with multip le commu nities (section V), an d d isplay its applicatio ns on protoco l p erforman ce prediction ( section VI I). W e differentiate our work from other trace-based mod - els [33], [23], [22] in several aspects. First, a mong a ll e fforts of providin g re alistic mobility mo dels, to ou r best k nowledge, this is the ﬁrst work to explicitly capture tim e-variant mobility characteristics. Alth ough capturing time- depend ent behavior is suggested in [22], it h as not been inco rporated in the particular pa per . Second , while previous works em phasize the capability to truth fully recrea te the mobility character istics observed from th e traces, we also striv e to ensure at the same time the mathematical tractability of the model. Our motiv ation is to facilitate the application of our mo del f or perfor mance pr ediction of various c ommunica tion pro tocols. Finally , m ost of the o ther trace- based mo dels h a ve not been shown as capab le to match mobility characteristics of a di verse set o f trac es, since th eir fo cus is mostly on one p articular tra ce or at most a single class of traces (e.g., WLAN tr ace). W e go beyond th at and re- produ ce matching m obility characteristics of se veral q ualitatively differ e nt tr aces, in cluding WLAN, vehicle, and hu man enco unter tr aces. As a ﬁnal n ote, in [26], the au thors assume the attra ction of a commu nity (i.e., a geo graphical area) to a mo bile nod e is der i ved fro m the num ber o f friends of this node cur rently residing in the commun ity . I n o ur pap er we assume that the nodes make movement decision s indep endently of the others (noneth eless, node sharing the sam e commun ity will exhibit mobility cor relation, captu ring the social f eature indirec tly). Mobility mo dels with inter-node depend ency require a solid understan ding of the socia l network stru cture, which is an importan t area unde r d e velopment. W e plan to work further in th is d irection in the futur e. 3 I I I . T I M E - V A R I A N T C O M M U N I T Y M O B I L I T Y M O D E L A. Mobility Characteristics Observed in WLAN T r aces The main objective o f this paper is to propo se a mobility model that captu res th e importan t mob ility ch aracteristics observed in daily life. T o better understand this mobility , we have cond ucted extensive analysis o f a number of w ireless LAN traces co llected by several research group s (e.g., traces av a ilable at [3 8] o r [37]). The rea son for this ch oice is that WLAN traces log inf ormation r egarding large n umbers of nodes, and thu s ar e re liable for statistical analysis. After analyzing a large numb er o f traces, we have observed two importan t pr operties that are common in all of them: (a) skewed location visiting pr efer ences and (b ) time-depe ndent mob ility behavio r [14]. More speciﬁcally , the location visiting pr efer enc e refers to the p ercentage of tim e a n ode spends at a g i ven acc ess point (AP). W e refer to th e coverage are a o f an access point as a loca tion . I n Fig. 1(a) , w e d raw the prob ability density function of the p ercentages of online time an average user spends at each loc ation, r anking the locations from the mo st fa vorite place to th e least fo r various traces. The d istribution appears highly skewed ; more than 95% of user’ s online time is spent at only top ﬁve APs. The time-dependen t mo bility behavio r r efers to the observation that nod es visit different locations, depe nding the time of th e day . In Fig. 1 (b) we p lot the pro bability of a no de re-appear ing at the same loca tion at some time in the futur e, as a fu nction o f the elapsed tim e. It is clear th at this probab ility displays some amount of periodicity , as the m obile nod es have stro nger tenden cy to re-app ear a t a previously visited location after a time gap of in teger multiples of days. A slightly higher peak on the 7th day , sug gesting a stronger weekly correlation in loca tion visiting pr eferences, could also be obser ved in some cu rves (e.g ., MIT). Unfortu nately , these two prominen t realistic mo bility c har- acteristics are not capture d by common ly u sed simple rand om models, as they do no t p ossess any space or time dependen t features in user mob ility . This is demonstrated in Fig. 1 by a straight line (unifo rm distribution) fo r the Random Direction model. The same could be obtained from Random W aypoint, Random walk, etc., or even more sophisticated models without spatial-tempor al preferen ces ( e.g., [ 11], [17]). T here ar e som e more rece nt mod els (e.g. , [29], [ 12], [3 3], [23]) th at aim at capturing spatial pr eference explicitly . As shown in Fig. 1(a) using th e simple commun ity m odel [ 29], with app ropriately assigned par ameters this m odel is able to capture th e skew ed location visiting p r efer ence , to some extent. Howe ver , time- depend ent behavior is not cap tured, and thus the periodica l r e-appearance pr operty cann ot be rep roduced , as shown by the ﬂat curve lab eled community mo del in Fig. 1 (b). It is o ur g oal to desig n a mob ility mo del that successfully captures th e skew ed lo cation p r e fer en ce an d time- depende ncy mobility pr operties observed in the traces in a n analytica lly tractable fashion . W e be lie ve tha t altho ugh the ab ove ob ser- vations are made based on WLAN trace s, th e two proper ties in qu estion are indeed prev alent in real-life mob ility . This belief is sup ported by typica l daily a cti vities o f hum ans: most of u s tend to spend most time at a h andful of f requently visited locations, and a recurrent daily or weekly schedule is an inseparab le p art o f o ur lives. It is essential to d esign 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1 21 41 61 81 101 AP sorted by total visit time F r a c t i o n o f o n l i n e t i m e USC MIT Dart-04 Dart-03 UCSD Random direction model Community model (a) Skewe d location visitin g preferen ces. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 R e - a p p e a r a n c e p r o b a b i l i t y Dart-04 Dart-03 USC MIT Random direction model Community model Time gap (days) (b) Periodical re-appea rance at the same location . Fig. 1. T wo important mobility features observed from WL AN traces. Labels of trac es used: MIT: trace from [2], Dart: trace from [10], UCSD: trace from [25], USC: trace from [14]. T ABLE I P A R A M E T E R S O F T H E T I M E - V A R I A N T C O M M U N I T Y M O B I L I T Y M O D E L 1 N Edge le ngth of simulation area V Number of time periods T t Duration of t -th time period S t Number of communities in time period t C t j Edge le ngth of community j in time period t C o mm t j The j -th community during time period t p t i,j The probability to choose community j when the previous community is i , during time period t π t j Stationary probability of a n epoch in community j during time period t v min , v max , v Minimum, maximum, and av erage speed 1 D max,j , D j Maximum and av erage paus e time after each epoch 1 L j A verage e poch length for community j P t move,j | P t pause,j Probability that a node is moving | pausing when being in c ommunity j during period t P t j Fraction of time the node is in state j ( P t j = P t move,j + P t pause,j ) K Transmission range of nodes A ( a t j , b t k ) The ov erlapped area between C omm t j of node a and C omm t k of node b w t A speciﬁc relationship between a target coordinate and the communities in time period t Ω t The set of a ll pos sible relationships between a target coordinate and the communities in time period t P h ( w t ) Unit-time hitting probability under the speciﬁc scenario w t P H ( w t ) Hitting probability for a time period t under speciﬁc scenario w t P t m Unit-time meeting probability in time period t P t M Meeting probability for a time period t a mod el that captu res such spatial-tempo ral preferences of human mobility in many co ntexts. B. Construction of th e T ime-variant Commun ity Mod el In this section, we present the d esign of ou r time-variant community (TVC) mobility model . W e illustrate the mo del with an example in Fig. 2 and use this example to intro duce the notations we use (see T able I) in th e rest o f the paper . First, to induce skewed location visiting pr efer ence s , we deﬁne some c ommunities (or heavily-visited geo graphic areas). T ake time perio d 1 ( TP1) in Fig. 2 as an examp le, the commun ities are deno ted as C omm 1 j and each of them is a square geogra phical area with edge length C 1 j . 1 A nod e visits these commu nities with different p r ob abilities ( details are given later ) to capture its spatial pref erence in mo bility . In th e T VC model, the mobility proc ess of a node consists 1 For all parameters used in the paper , we follo w the con ve ntion that the subscript of a quantit y represent s its community inde x, and the superscript represent s the time period index . 4 of ep ochs in these commun ities. When the node ch ooses to have an epoch in commu nity j (we say that th e no de is in state j durin g th is epoch) , it starts fro m the end point of the pr e vious ep och within C omm 1 j and the ep och length (movement distance) is drawn from an exponential distribution with av erage L j , in the same or der of the co mmunity edg e length. The node then picks a random speed uniformly in [ v min , v max ] , and a directio n ( angle) unifor mly in [0 , 2 π ] , and perform s a rand om d irection movement within the ch osen commun ity with the c hosen epo ch length 2 . Th e ﬁrst difference between the TVC m odel and the standa rd Random Direction model is hence the sp atial pref erence and lo cation-dep endent behavior . No te that, a node can still roam a round the whole simulation area du ring so me epochs, by assignin g an additional commun ity that correspond s to the wh ole simulation ﬁeld ( e.g. C omm 1 3 ). W e ref er to such epo chs as r oaming epochs . W e next explain how a node selects the n ext com munity for a sequen ce o f epoch s. At the comp letion of an epoch, the node remains station ary fo r a p ause time unifo rmly chosen in [0 , D max,j ] . Then , depend ing on its cu rrent state i and time p eriod t , the nod e chooses th e next epoch to be in commun ity j with pr obability p t i,j . This commun ity selectio n process is essentially a time-variant Markov chain that captures the spatial and tempo ral depend encies in nodal mob ility a nd thus makes the commun ity selection process in th e TVC model non - i.i.d. , an important feature absent in many synthetic mobility models even if they consider no n-unifo rm mobility features. Now , if the end po int of the p revious epo ch is in C omm t j (this can be th e case wh en the n ode has two consecutive epoc hs in C omm t j , or C omm t j contains C omm t i ), the nod e starts the next ep och directly . If, on the o ther hand , the nod e is currently not in C omm t j , a transitional epo ch is inserted to bridge th e two ep ochs in disjoint com munities. The node selects a random coo rdinate point in th e next commu nity , moves directly towards this po int on the sho rtest straigh t path with a rando m speed drawn f rom [ v min , v max ] , and then continues with an epoch in th e next commu nity . Hence th e movement trajectory of a node is always co ntinuou s in spac e. W e next introd uce the structure in time. T o captu re tim e- depend ent behavior, o ne c reates mu ltiple time periods with different commun ity and par ameter settings. As an example, there are V = 3 time p eriods with dur ation T 1 , T 2 , and T 3 in Fig. 2. These time perio ds follow a p eriodic structure (e.g., a simple r ecurrent structure in Fig. 2 o r the weekly schedule in Fig. 3). Th is setup naturally captures the temporal pr efer en ces (e.g ., go to work du ring the days an d h ome dur ing the nights) and periodicity in human mobility . On the time bound aries between time periods, each nod e con tinues with its on going epoc h, a nd de cides the n ext epoch ac cording to the new p arameter settings in th e new time perio d when it ﬁnishes the curren t epoch. As a ﬁnal n ote, we choose to c onstruct the TVC m odel with simple building blocks introd uced above due to its am enability to theor etical analysis [29] an d ﬂexibility . T o further explain the ﬂexibility of our TVC mode l, we note that the numb er 2 T o avoid boundary ef fect s, if the node hits the community boundary it is re-insert ed from the other end of the area (i.e., ”torus” boundaries). Note that we could also choose random waypoint or random walk models for the type of m ov ement during each epoch. 7 LP H 5 H S H WL WLY HW LP H S H U LR G V WUX FWX U H  9  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 LP HS HU LR G   7 3  7 LP H S H UL R G   7 3   7 LP H S H UL R G   7 3   & R P P   & R P P   & R P P   & R P P   & R P P   & R P P   & R P P   & R P P   & R P P   1 &   &   &   &   &   &   &   &   &   7  7  7  6   6   6   Fig. 2. Illustration of a generic scenario of time-v ariant mobility model, with three time peri ods and dif ferent numbers of communit ies in each time period. 7 LP H 7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  73  73  7 3  7 3  7 3  7 3  : H H N G D \ V : H H N H Q G 7 LP H 7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  73  73  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  7 3  : H H N G D \ V : H H N H Q G Fig. 3. An illustrati on 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 capt ure weekend day time. of comm unities in each time perio d ( denoted as S t ) can be different, and the commu nities can overlap (as in TP1 in Fig . 2) or co ntain each oth er (as in TP2 in Fig. 2). Finally , the time p eriod structur e, comm unities, and all o ther par ameters could be assigned differently for each n ode to c apture node- depend ent mobility (e.g., peo ple following different schedules, with d ifferent working p laces, etc.), while nod es can share some commun ities (i.e., the po pular lo cations) as well. This construction allows for maximu m ﬂexibility when setting up the simu lations for nodes with heter ogeneou s behaviors 3 . The be neﬁt of using simp le building blo cks will become evident in Section V. At the sam e time, we will show next that these choices d o not comp romise our mo del’ s ability to accurately capture real life mob ility scenarios. I V . G E N E R A T I O N O F M O B I L I T Y S C E N A R I O S The TVC m odel de scribed in the previous section provid es a g eneral f ramework to model a wid e r ange o f mo bility scenarios. In this section , our aim is to demonstrate the model’ s ﬂexibility and validate its realism by ge nerating var - ious sy nthetic traces f rom the model, with matching mo bility characteristics to well-kn own, pub licly-av ailable trac es (e. g., WLANs, V ANET , and human enco unter traces). Howe ver , it is im portant to note that the u se o f such a mo del is not merely to match it with any speciﬁc trace in stance av ailable; th is is only d one for validation and calibratio n p urposes. Rath er , the goal is to be able to reprod uce a much larger ran ge o f re alistic mobility in stances than a single trace can pr ovide 4 . W e ﬁrst outlin e a g eneral 3 -step systematic process to construct speciﬁc mob ility scenario s. Then, we dem onstrate our success to gene rate matching mobility char acteristics with three q ualitativ ely different tr aces. All the parameter values we use in this section are also available in [40] 5 . 3 When necessary , we use a pair of parenthese s to includ e the node ID for a partic ular paramete r , e.g., C t j ( i ) denotes the edge length of the j -th community during time period t for node i . 4 W e have made our mobility trace generat or av aila ble at [40]. The tool provi des mobility traces in both ns-2[39] compatible format and time-locatio n (i.e., ( t, x, y ) ) format. 5 Due to space limitat ions, we cannot list all parameters in this paper . 5 STEP 1: De termine the Structure in Space and Time • (1.1 Numb er of commu nities) Each com munity in the TVC model correspo nds to a lo cation visited f requently by nodes (i.e., the most v isited location in Fig. 1(a) cor responds to the most popular c ommunity in the mo del, and so on). The number of communities n eeded is thus determin ed by how closely one wants th e mo bility cha racteristics to ma tch with th e cu rves in Fig. 1(a). Due to the natur e o f ske wed locatio n v isiting preferen ce, in o ur experience, only two or three commu nities are ne eded to capture u p to 85% o f th e user online time spent at the most popular locations. Such a simple spatial structure yields simp le theoretical expressions. Howe ver, if one wants the model to capture mor e details (e.g., for detailed simulation), the user can instantiate as many co mmunities as needed to explicitly rep resent the less visited lo cations. • (1.2 Loc ation of commun ities) If the map of the target en vironmen t is av ailable, one should observe the map and identify th e poin ts o f attraction in the giv en environment to assign the commu nities according ly . The m ethods d escribed in [2 2] c ould be applied to help ch oosing the “hot spots” o n campus, b y ad ding up th e time users spe nd at each lo cation on a 2-D map and identifyin g the peaks. Alternatively , if the map is not a vailable, on e can instan tiate comm unities at random locations 6 . One way to do so is to simply divide the simulation area into equal-sized gr id cells, and assign random ly chosen cells as commun ities. • (1.3 T ime period stru cture) Fr om the cu rves in Fig. 1(b ), one ob serves the re-appearance periodicity an d de cides on the time pe riod structu re accord ingly . T ypically , hum an ac ti vities are b ounded by d aily an d weekly sche dules so a time period structure shown in Fig. 3 would sufﬁce for mo st app lications. If captu ring ﬁner b ehavior b ased on time-of -day is necessary , one co uld addition ally split the d ay into time pe riods with different mobile node behavior . W e illustrate this in ou r third case study , the hum an enco unter trace. STEP 2: Assign Other Paramet ers After the space/time structure is d etermined, one h as to determ ine the r emaining parameters for each co mmunity and time period. This in cludes π t j , D t j , and L t j , which represent the station ary probability (which is calcu lated after selecting pro per p t i,j ’ s that lead to a d esired stationary distribution using simple Markov ch ain theory) , a verage pause time, an d average epoch length , respec- ti vely , a t commu nity j durin g time p eriod t . These par ameters can be d etermined by ref erring to the cur ves in Fig. 1. W e give some general rules of how the parameter s chang e the cu rves in Fig. 1 below . The detailed ad justments we make fo r each speciﬁc case studies will be discussed later . • Th e av erage epoch len gth in each commu nity , L t j , shou ld be at least in the same ord er as th e edge length of the commun ity , C t j . This is to e nsure that the end po int of the epoch becomes almo st in depend ent of its starting p oint, since the mixin g time of the correspo nding pro cess becomes quite small. (The mo ti vation for this req uirement is to keep the theoretical a nalysis tractable.) • The av erage duration the node stays in comm unity j is giv en by π t j ( D t j + L t j / v ) . The ratio be tween the dur ations the node stays in each commun ity shapes the location v isiting 6 Concerni ng matching with the two mobility properties s ho wn in Fig. 1 , the actual locati ons of the communities do not make a differen ce. preferen ce c urve in Fig. 1(a) . • The high est peak of the re-ap pearance probability curve (on the 7 -th da y under the weekly sched ule) in Fig. 1(b) is determined by the we ighted av erage probab ility of the nod e appearin g in th e same community du ring the same typ e o f time pe riod. This value is P V t =1 T t P V k =1 T k P S t j =1 ( P t j ) 2 , whe re P t j denotes the f raction o f time the nod e spend s in com munity j . STEP 3: Adj ust User On-o ff Pattern (O ptional) The mo- bility trace g enerated b y the TVC m odel is an “always-on” mobility tra jectory (i.e. , the mobile nodes ar e always p resent somewhere in the simulatio n ﬁeld). Howe ver , in some situa- tions so me nodes might be absent occa sionally . For exam ple, in a WLAN setting , no des (e. g. lap tops) ar e ofte n tur ned off when travelling from on e lo cation to anoth er and the “off ” time is often n ot negligible [1 4]. Thus on e ma y need to make optiona l adjustments to turn nodes off in the gener ated trace, depend ing on the actual en vironmen t to match with. T o address this we assign a probab ility P on,j as the probab ility fo r th e node to be “o n” in commun ity j . In two o f the case studies w e present (WLAN an d vehicu lar trace), we utilize this f eature as the n odes are not always-on in th e actua l tr aces. Note th at it is po ssible to autom ate p art o f the above com- munity and pa rameter selection. This can be done by fe eding the curves in Fig. 1 an d the desired lev el o f match ing to a progr am th at executes the above steps. Autom atic generatio n of proper synthetic traces is a dir ection of ou r f uture work. Next, we look into th ree speciﬁc case studies and app ly the fo re-mention ed p rocedur e in each case, to display that the TVC mo del successfully prod uces synthetic mob ility trac es with m atching character istics o bserved in th e real traces. A. WLAN T races In the ﬁrst example, we show that the T VC mode l can re- create the locatio n pr efer en ces and r e-appearance pr o bability curves ob served in WLANs. W e use th e MIT WLAN tr ace (ﬁrst pr esented in [2]) as the main examp le her e 7 . W e split th e MIT trace in to two halves and gen erate a matching syn thetic trace with o bserved mob ility characteristics fro m the ﬁrst half (the trainin g data set). W e th en co mpare our sy nthetic trace with the mobility characteristics of the second half (th e validation data set). No te that, the mo bility characteristics are similar across th e two halves (shown by the two very close thick black cur ves in Fig. 4). W e gen erate two synth etic traces with the TVC mod el, a simpliﬁe d one an d a co mplex one, to display its ﬂexibility to have different levels of matching to the WLAN trace. The s impliﬁed model (sh own by thin b lack curves) uses only one commun ity and two time p eriods (for the day time and night time) , with parameters listed as Mod el-1 in T able I I. Th e simple model capture s the major trends but still shows several noticeable d ifferences: (a) the tail in the model-simpliﬁed curve in Fig. 4( a) is “ﬂat” as opposed to th e exponen tially diminishing tail of the MIT curve. (b) the peaks in th e model- simpliﬁed curve in Fig . 4( b) are o f eq ual h eights. 7 W e also achie ve good matching with the USC[14] or the Dartmouth[10] traces, but do not sho w it here due to space limitations. 6 W e can improve the matching between th e synthetic trace and the real trac e by adding com plexity in both space and time, with the fo llowing detailed procedu re. (STEP1): W e divide the simulation area into 10-by-10 grid cells. Since we w ant to ha ve a close m atch with the cu rve in Fig. 4(a), we assign ran domly 15 of the cells as commun ities to each n ode (I ntuitiv ely , this number correspo nds to th e num ber of distinct access points that a person may co nnect to on a university campus over a period of on e month.). For the time period struc ture we use the simple weekly structu re shown in Fig. 3, alloc ating 8 hours for day time (TP1, TP3) and 1 6 hours fo r night tim e (TP2), as th is trace is collected from a corpo rate environmen t. (STEP2 an d STEP3): In the actual WLAN trace the nod es are “on” only for a low per centage of tim e. W e capture this ph enomeno n with an additional parameter, P t on,j , the prob ability the nod e is “on” in state j . In WLAN, the n odes ar e typ ically “o n” (i. e., ap pear at the cu rrent loc ation) wh en they ar e n ot m oving. Un der this on-off pattern , P t on,j = D t j / ( D t j + L t j / v ) . W e then consider the on-off pattern and parame ter assignmen t jointly . ( 1) W e ﬁrst assign the same D t j , L t j , P t on,j to all commu nities, then assign π t j with a value e qual to the frac tion o f time spent at the j -th locatio n in Fig. 1(a). This assignmen t strategy makes the n ode “on” fo r the same amount of tim e in ea ch commu nity during each visit, and the total tim e in each commun ity (and henc e the ob served locatio n v isiting preferen ce curve) is therefor e deter mined by the value of π t j . (2) Due to the on -off pattern, the peak value in th e re- appearan ce probab ility cu rve becomes P V t =1 T t P V k =1 T k P S t j =1 ( P t j ) 2 ( P t on,j ) 2 . T o shape the re- appearan ce pr obabilities, we adjust the D t j values, which, in turn, adjust the values of P t on,j and set the r e-appear ance probab ilities to the d esirable values to match with the curve in Fig. 1(b). No te that b y adju sting th e D t j values in a co nsistent manner among all co mmunities we do not change the location visiting probability cu rve that h as a lready be en ma tched in the previous step. As it is evident f rom Fig. 4 , th is mod el, wh ich is labeled Model-co mplex and co rrespond s to the red curves in the plot, yields synthetic traces wh ose cha racteristics match very closely with those of the MIT trace. B. V ehicle Mo bility T races In this example we display that skewed location visiting pr efer en ces a nd p eriodical r e- appearance are also pro minent mobility properties in vehicle mob ility traces. W e obtain a vehicle movement trace from [ 36], a web site that tracks par- ticipating taxis in the gre ater San Fran cisco are a. W e process a 40- day trac e obtained between Sep. 22, 2 006 and Nov . 1, 2006 for 549 taxis to obtain the ir mobility ch aracteristics. T he results are shown in Fig. 5 with the label V eh icle-trace . It is interesting to ob serve th at the trend o f vehicu lar movements is very similar to that of WLAN u sers in terms of th ese two proper ties. W e use 3 0 commu nities an d the weekly time schedule in (STEP1). W e need m ore co mmunities fo r th is tr ace as the taxis are more mob ile an d visit more places than peop le o n university campuses. From th e actual trace, we discover that the taxis ar e o fﬂine (i.e., not r eporting their location s) wh en not in operation . Hence we assume that the nodes are “on” only F r a c t i o n o f v i s i t t i m e 1.E-0 6 1.E-0 5 1.E-0 4 1.E-0 3 1.E-0 2 1.E-0 1 1.E+ 00 1 11 21 31 41 MIT Mode l-sim plifie d Mode l-c om plex AP sorted by total visit time (a) Skewe d locati on visiting preferenc es. Time gap (days) R e - a p p e a r a n c e p r o b a b i l i t y 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 MIT Mod el-sim pli fied Mod el-c om plex (b) Periodical re-appearan ce at the same location. Fig. 4. Matching mobilit y characterist ics of the synthetic trace s to the MIT WLAN 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 Fraction of visit time Vehicle-trace Model Location sorted by visit time (a) Location visiting prefere nces. Time gap (days) R e - a p p e a r a n c e p r o b a b i l i t y 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-appea rance. Fig. 5. Matching mobilit y characterist ics of the synthetic trac e to the vehicle mobility trace. when they are moving. The pause times b etween epoch s ar e considered as b reaks in taxi operation . Ther efore in (STEP3), P t on,j = ( L t j / v ) / ( D t j + L t j / v ) , an d we a djust the par ameters in a similar way as d escribed in the pr evious section . Th e curves in Fig. 5 with label Mod el match with th e cur ves with V ehicle- trace label well. As a ﬁnal note, although v ehicular movements are gen erally con strained b y streets and our TVC model do es not capture such microsco pic behaviors, designated paths and other constraints could still be add ed in the m odel’ s map (for vehicula r or human mobility) witho ut losing its basic proper ties. W e defer this for futu re work. C. Human En counter T r aces In this examp le, we sh ow th at the TVC model is g eneric enoug h to mimic the en counter pro perties o f mo bile human networks ob served in an experiment perf ormed at INFOCOM 2005 [6]. In th is exp eriment, wireless devices wer e d istributed to 41 particip ants at the conf erence to log enco unters betwee n nodes (i. e., co ming within Bluetooth co mmunicatio n range) as they moved aroun d th e pre mises of the co nferenc e area. The inter-meeting time and the encou nter du ration d istributions of all 820 p airs of users obtained from this trace a re shown in Fig. 6 with label Cambridge-INFOCOM-trace . T o mimic such b ehaviors using our TVC model, we o bserve the conferenc e schedule at INFOCOM, an d set up a daily 7 0.000 01 0.000 1 0.001 0.01 0.1 1 10 10 0 1000 10000 100 000 1000 000 Inter-meeting time (s) Cambridge- INFOCOM-trace Model P r o b ( I n t e r - m e e t i n g t i m e > X ) (a) Inter-meeti ng Time. 0.00001 0.0001 0.001 0.01 0.1 1 1 10 100 1000 10000 100000 Cambridge- INFOCOM-trace Model Prob (Meeting duration > X) Meeting duration (s) (b) Encounter duration. Fig. 6. Match ing inter-mee ting time and encounter duration distrib utions with the encounter trace . recurren t schedule with ﬁve different typ es of time pe riods (STEP1): tech nical sessions, coffee breaks, b reakfast/lunch time, evening, and late night (see [40] for the d etailed pa- rameters). For ea ch time perio d we set up comm unities as the confere nce roo ms, th e dining room , etc. W e also gen erate a commun ity th at is far away from the rest of the commun ities for each no de and make the no de sometimes isolated in this commun ity to captur e the behavior of pa trons skippin g part of the conferen ce. In (STEP2), we use the theory pr esented in section V to adjust the parameters and sha pe the inter - meeting tim e and encounter d uration cur ves. For examp le, a stronger tend ency for nod es to ch oose roam ing epochs (setting larger π t r ) would in crease the m eeting probab ility (see, e.g., Eq. (18)), hence red ucing inter-meeting times. Finally , since the devices u sed to collect the enco unter trace s ar e alw ays-on, we do no t apply any chan ges to the synthetic tra ce (STEP3). W e r andomly gener ate 8 20 pairs of users an d show their correspo nding d istributions of the inter-meeting time and the encoun ter d uration in Fig. 6 with label Model . It is clear tha t our TVC model h as the cap ability to reprodu ce the observed distributions, ev en if it is n ot con structed explicitly to d o so. This d isplays its success in captur ing the decisiv e factors o f typical human mobility . It is clear from the ca ses studied here that the TVC model is ﬂexible to c apture mobility char acteristics from various environments well. In addition , with th e respecti ve conﬁgur ation, it is po ssible to generate synthetic traces with much la rger scale (i.e., more no des) than the empir ical ones while maintain ing the same mobility chara cteristics. It is a lso possible to g enerate m ultiple instances of the synthetic traces with the same mob ility characteristics to comp lement the original, empirically collected trace. V . T H E O R E T I C A L A NA L Y S I S O F T H E T V C M O D E L So far, we have established the ﬂexibility of th e TVC model in terms of its ability to re produc e the pro perties ob served in qualitatively different mob ility trace s. Y et, one of the b iggest advantages of o ur m odel is that, in add ition to the realism, it is also analytically tr actable with respect to some importan t quantities which determine protoc ol per formanc e. In the r est of this paper, we focus on demonstra ting this last point. W e start h ere by d eriving the theoretic expre ssions of various pro perties of the pr oposed mobility model assuming the nod es are always “on ”. T he pro perties of interest are deﬁned b elow . • The average no de degr ee is the average nu mber of nodes residing within th e commun ication range of a giv en node. This is a qu antity of interest due to its implicatio n on the succe ss rate o f various tasks (e.g . geogr aphic ro uting [28]) in mobile ad h oc networks. • T he h itting time is the time it takes a no de, startin g f rom the stationary distribution, to move w ithin transmission ra nge of a ﬁxed, random ly cho sen target coo rdinate in the simulatio n ﬁeld. • Th e meeting time is the time until two mobile no des, both starting f rom th e stationary distribution, move into the transmission rang e of each oth er . T he hitting an d meeting times are of interest du e to their close relationship to th e perfor mance o f DTN routing protoc ols. W e no te that a preliminary v ersion of some o f the theoretical deriv ations presented h ere ap pear under a spec ial case of ou r TVC mo del in [1 5] (that mo del includ ed one co mmunity and two time p eriods only) . Here, we g eneralize a ll deriv ations for any commu nity a nd time-per iod structu re. W e start with a useful lemma th at calculates th e prob ability o f a no de to reside in a particu lar state. Lemma 5 .1: The pr o bability that a no de moves, pauses (after th e co mpletion of an epoch) in state j , or performs a transitional epoch at any given time instan t during time period t , r espectively , is: P t mov e,j = π t j ( L t j / v t j ) / Ψ , (1) P t pause,j = π t j D t j / Ψ , (2) P t tr = S t X k =1 π t k X ∀ n p t k,n L tr ( k,n ) /v t k / Ψ . (3) wher e Ψ = P S t k =1 π t k ( L t k /v t k + D t k + P ∀ n p t k,n L tr ( k,n ) /v t k ) and L tr ( k,n ) is th e average leng th of a transitional ep och fr om community k to commun ity n . Pr oo f: The p robability for a node to be in state j ( π t j ) can be easily derived with Markov chain theory fr om the state transition pr obabilities ( p t i,j ). The ab ove resu lt follows from the ratio of th e average duration s of the moving part ( L t j / v t j ) and the pa use part ( D t j ) of regular epoch s, and the transitional epochs ( L tr ( k,n ) /v t k ), weighted by the probabilities of the states. Th e expected length of the transitional epo chs, L tr ( k,n ) , can be ca lculated as follows. No te th at if comm unity n conta ins c ommunity k , no transitional epoc h is needed (i.e. , L tr ( k,n ) = 0 ) . The tr ansitional epoch is thus need ed for a roaming node to go back to a smaller community , and as the previous roam ing ep och ends at a random lo cation in the whole simulation ﬁeld , by sym metry , the expected length o f the transitional ep och is the average len gth to move to the center of the simulation ﬁeld from a ran dom poin t in th e simu lation ﬁeld. Numerical ana lysis conclud es L tr = 0 . 3 8 26 N in th is case. Note that the above stationary probabilities can be calculated for ea ch time p eriod and node sep arately . W e u se P t j ( i ) to denote the prob ability that node i is in state j d uring time period t (i.e., P t j ( i ) = P t mov e,j ( i ) + P t pause,j ( i ) ). A. Derivation o f the A verage No de Degr ee The a verag e n ode d egree of a node is deﬁned as the expected number o f no des falling with in its commun ication range. Each 8 node con tributes to the av erage nod e degree in depend ently , as nodes make independ ent movement decisions. Lemma 5 .2: Consider a pair of nodes, a and b . Assume further th at, in time period t , community j of n ode a and community k of node b overlap with each othe r for an area A ( a t j , b t k ) . Then, the contrib ution of n ode b to the averag e node de gr ee of n ode a , when a resides in its j -th community and b r esides in its k -th commun ity , is g iven by π K 2 C t j 2 ( a ) A ( a t j , b t k ) C t k 2 ( b ) , (4) wher e K is the communic ation range of th e no des. Pr oo f: Since nod es f ollow rando m dir ection m ovement in each epoch, they ar e unifo rmly distributed within each commun ity ( i.e., they are at any point within th e commun ity equally likely). The prob ability f or n ode b to fall in the j -th commun ity of node a is simply the ratio of the overlapped a rea over th e size of the k -th commun ity o f nod e b . Node a covers any g i ven point in its co mmunity equal- likely , hence given node b is in the overlapp ed area, it is within the commun ication range o f nod e a with pro bability π K 2 /C t j 2 ( a ) . Follo wing the same princip le in Lem ma 5.2, we includ e all commun ity p airs and arrive at th e following Theorem . Theor em 5. 3: The average n ode de gr ee of a g iven n ode a is X ∀ C omm t j ( a ) P t j ( a ) X ∀ b X ∀ C omm t k ( b ) P t j ( b ) π K 2 C t j 2 ( a ) A ( a t j , b t k ) C t k 2 ( b ) . (5) Pr oo f: Eq. (5) is simp ly a weighted average of the node degree of nod e a conditioning on its states. For each state with pr obability P t j ( a ) , the expec ted node degree is a sum over all other nodes’ prob ability of being within the commun ication range of node a , again condition ing on all possible states. T ransitional epoc hs are treated the same way as ro aming ep ochs her e. T hat is, when considerin g a node in th e transitional state with pro bability P t tr , it h as equiv alent contribution to the no de d egree as when it is in the roam ing state ( i.e., th e no de appear s unifo rmly in the simulation ﬁeld during transitional epo chs since the it moves from anywhere in the simulation ﬁeld back to the local community . ). Henc e, with probab ility P t tr + P t r oam 8 , the no de has an effectiv e community size of the simulation ﬁeld, N . Cor ollary 5.4: In the special case when all nod es choose their commun ities uniformly at random amo ng the simula tion ﬁeld, Eq. (5) degener ates to P ∀ b π K 2 N 2 . Pr oo f: This result fo llows from the fact that a randomly chosen commu nity is anywhere in the simulation ﬁeld eq ually likely . B. Derivation o f the Hitting T ime In th e calculation of the av erage node d egree, th e depe n- dence between consecutive ep ochs did no t affect th e de riv a- tion. In fact, only the station ary occupa ncy probab ilities π t j and P t j (i.e. the probab ility of being fou nd in co mmunity j ) 8 If the node has no roaming state in this time period, then we consider only P t tr . are nee ded, since we wer e lookin g only at a rando m snapshot of the mode l. In the case of hitting and meeting tim es, we are interested in counting the number of epochs until a given target coordin ate is foun d (“h it”). Our app roach is to try to calculate the “hit pro bability” for a given epo ch, and then count the number of such ep ochs needed on average until the de stination point is hit. I f these pro babilities we re ind ependen t, then one could use a simple geometric distribution to derive the re sult. Howe ver , (1) consecutive ep ochs a re stro ngly related, as the ending poin t of o ne epoch is, naturally , the beginning point of th e next. This introdu ces a seeming d ependen cy between the hit p robabilities of con secutiv e epochs, comp licating the deriv ation. What is mo re, (2) the transition betwee n commu ni- ties (and ep ochs p erform ed in each) are g overned by the TVC model’ s Mar kov chain and the respective comm unity transition probab ilities p t i,j . Thus, l ooking only at the stationary probab il- ities for “ch oosing” the next co mmunity j (as in the p revious section) no longe r sufﬁces. Finally , (3) the transitional epochs themselves introduce f urther complicatio ns, as they can not, in this case, b e handled as r egular in-community or even roaming epochs. The above three observations in troduce depend encies th at, at ﬁrst g lance, complicate ou r task . Nevertheless, we will show how th ese depen dencies can be “washed out” und er a (minimally restrictin g) set of assumption s, and th at stationary probab ilities still sufﬁce to de riv e a simple fo rmula for the respective hitting time that holds in the limit . The basis of o ur argument is fo und in the proo f of Lem ma 5.7, upon which the rest of results in this section depend (In a nu tshell, the fast mixing of th e mobility pro cess takes c are of (1), th e large number o f ep ochs required to hit a target takes car e of (2) in the limit, and the dominance of loc al and roaming epo chs over transitional epochs takes care of (3 ).). In Section VI, we show that the acc uracy of our theor y is not comp romised by these assumptions a nd that o ur derivations introduce little err or in most p ractical scenarios considered . The sketch of the der i vation of the hitting time is as f ollows: (i) W e ﬁrst co ndition on the r elati ve location of the target coordin ate with respect to a node’ s communities (Lemma 5.5). W e identify all po ssible sub-cases (i.e. whether the target is inside or outside one or m ore of the no de’ s com munities). A target inside a com munity is, naturally , expected to be found faster than a target outside all commun ities. Using simple geome tric arguments, we calculate th e prob ability of each of these sub-cases (Lemma 5.6) and take the weig hted av erage of all sub-cases and the resp ecti ve hitting tim e (to be calculated p er sub-case). (ii) For a given sub-ca se, we derive the expec ted numb er o f epo chs (an d the expected num ber of time un its) until th e target is fo und (Lemma 5 .7). (iii) Fin ally , we intr oduce th e time- period factor, an d acc ount fo r the total number o f time p eriods needed to hit the target (Th eorem 5.9). The most inﬂuential f actor for the h itting time is whether th e target coo rdinate is c hosen inside th e node’ s c ommunities. W e denote the possible relationships between the target lo cation and the set u p of comm unities during time pe riod t as the set Ω t . No te that the cardin ality o f set Ω t is a t most 2 S t (i.e. fo r each of the S t commun ities, the target coord inate is e ither in or out of it). Lemma 5 .5: By th e law of total pr o bability , the a verag e 9 hitting time can be written as H T = X w 1 ∈ Ω 1 ,...,w V ∈ Ω V P ( w 1 , ..., w V ) H T ( w 1 , ..., w V ) , (6) wher e w 1 , w 2 , ..., w V denote one pa rticular r elatio nship (i.e. a combinatio n of { out, in } S t ) between the tar get coor dinate and the commun ity set u p during time p eriod 1 , 2 , ..., V , r espec- tively . Fu nctions P ( · ) an d H T ( · ) de note th e corr esponding pr o bability for this scenario and th e conditio nal h itting time under this scenario, r espectively . Note that each sub -case { w 1 , w 2 , ..., w V } is disjoint fr om a ll other sub-cases. T o e valuate Eq. (6), we n eed to calcu late P ( w 1 , ..., w V ) and H T ( w 1 , ..., w V ) for each po ssible sub- case ( w 1 , ..., w V ) . Lemma 5 .6: If the tar get coordinate is chosen indepen dent of the co mmunities a nd th e communities in each time p eriod ar e chosen indep endently fr o m other periods, then P ( w 1 , ..., w V ) = Π V t =1 P ( w t ) , (7) wher e P ( w t ) = A ( w t ) / N 2 , i.e., the pr ob ability of a sub- case w t is pr oportional to the a r ea A ( w t ) that corr esponds to th e speciﬁc scenario w t , which is a series of con ditions of the fo llowing type : ( { targ et ∈ comm t 1 } , { tar g e t / ∈ comm t 2 } , ..., { targ et ∈ comm t S } ). Pr oo f: The result fo llows from sim ple geom etric argu- ments. The ﬁrst step fo r calcu lating H T ( w 1 , ..., w V ) is to derive the u nit-time hitting probab ility in time p eriod t under target coordin ate-commu nity relationship w t , denote d as P t h ( w t ) . Lemma 5 .7: F or a g iven time p eriod t an d a speciﬁc scenario w t , P t h ( w t ) = S t X j =1 I ( tar g et ∈ C omm t j | w t ) P t mov e,j 2 K v t j /C t j 2 , (8) wher e I ( · ) is the ind icator functio n. Pr oo f: I n o rder to calculate the expected hitting time, let us ﬁrst coun t the to tal number of epo chs ne eded. Le t us assume that N e epochs are needed in total, and let u s denote as epoch E k m ( m ) the m -th epoch in sequ ence (th at is o ccurring in co mmunity k m ). L et fu rther, P ( k 1 , k 2 , . . . , k m ) denote th e probab ility of the speciﬁc sequence of epochs occur ring. Then, the probability that the target has not been fo und a fter n ep ochs is P ( N e > n ) = X k 1 ,k 2 ,...,k n P ( k 1 , k 2 , . . . , k n ) · P ( E k 1 (1) = miss , E k 2 (2) = miss , . . . , E k n ( n ) = miss) . (9) In o rder to simplify th e a bove eq uation, we ne ed to dea l with the inh erent depen dencies intro duced by the tran sition of epochs. First, since n ode movement is continu ous, the e nd of one epoch E j ( m ) , perfor med in co mmunity j , is the beginning of the next, E j ( m + 1) , if perfo rmed in the same c ommunity 9 . Nevertheless, as explained in Section III-B, the expected “length” of an ep och L t j perfor med in comm unity j is in the order of the square root of th e co mmunity size C t j . This is 9 For the moment, we will ignore transit ional epochs, and assume that all epochs are performed inside some community; we deal with transitional epochs later . sufﬁcient for the node to “mix ” in th e com munity after ju st one epoch [29]. Consequen tly , we can write P ( N e > n ) = X k 1 ,k 2 ,...,k n P ( k 1 , k 2 , . . . , k n ) · n Y i =1 P ( E k i = miss ) . (10) An additional depend ency arises from the transitions between communities and the calcu lation of ter m P ( k 1 , k 2 , . . . , k n ) . If epoch E k m ( m ) is per formed in commun ity k m the next epoch E k m +1 ( m + 1) will be perfor med in commu nity k m +1 with proba bility p k m ,k m +1 (the tran sition prob ability in the Markov Chain governing th e commun ity transitions in the TVC model) . Let us assum e that N e denotes ag ain the total ep ochs needed (of any typ e) to hit the target. Further, let there be l j epochs of type j (i.e. p erformed in commun ity j ) in the above mix of N e total epo chs. When N e → ∞ , then l i → π i N e , that is, the total number of epo chs in commun ity i dep ends o nly on the stationary p robability of commu nity i , π i . Th us, P ( k 1 , k 2 , . . . , k n ) = π k 1 · π k 2 . . . π k n . (11) Consequently , Eq .(10) beco mes P ( N e > n ) = n Y i =1 π k i · P ( E k i = miss ) . (12) This implies th at, in th e limit 10 , th e to tal nu mber of epo chs needed to h it th e target can be appro ximated by a geom etric distribution, whe re the “average” epoch has a hit prob ability of S t X i π i · P ( E i = h it ) (13) As the ﬁna l step, we need to calculate the pr obability o f a giv en e poch in commun ity j to hit the target. Instead o f using this per epoch hit probability , we revert now to what we call the unit- step h it probab ilities, P h . The u nit-step prob ability is the pr obability of enc ountering th e target exactly within th e next time-unit (rather than within the dura tion o f a whole epoch) . This discrete appro ximation provid es an equiv alent formu lation to th e above co ntinuous case (see [29]), howe ver it is more convenient to man ipulate in the ca se o f time- period boun daries and mee ting times calculated later . (Note that this approxima tion is again only possible wh en the a verage epoch leng th is in the order o f the respective co mmunity size, ensuring mixing after one epoch .) Note that the hittin g event can only occur wh en the n ode is physically moving inside the co mmunity whe re th e target is located 11 . Whether th e target is lo cated inside co mmunity j is denoted using the indicator function I ( targ et ∈ C omm t j | w t ) . If the target is o utside the commun ity , then this pr obability of hit is zero. If the target lies within c ommunity j , the n 10 In practice, the requirement is that a large number of epochs is needed on av erage until the target is hit. In the sparse network s we’ re intere sted in, this is a reasonable assumption, and as we shall sho w in S ecti on VI the achie ved accura cy is indeed high. 11 W e neglect the small probabili ty that the targ et is chosen out of the community but close to it, and make the contributi ons from epochs in state j zero if the chosen target coordinate is not in community j . 10 when a nod e moves with average speed v j , on average it covers a ne w area of 2 K v j in unit time. Since a n ode following random direction mov ements visits the area it moves about with equal pro bability , and the target coor dinate is chosen a t random, it falls in this n e wly covered ar ea with probab ility 2 K v j /C t j 2 [29]. Hence th e contribution to the unit-time hitting p robability b y movements made in state j is P t mov e,j 2 K v t j /C t j 2 . T hus, in Eq.(13), π j is replaced by I ( tar g et ∈ C omm t j | w t ) P t mov e,j in the unit-step case, and P ( E i = hi t ) by 2 K v t j /C t j 2 . As a ﬁnal remark, the c ontribution of tr ansitional epoch s to th e un it-time hitting pr obability is not e quiv alent to other epochs (d ue to th e depende ncy of end-po ints on local co m- munities, which introdu ces bias after comm unities h av e been chosen). Ne vertheless, in a nor mal mobility scenario, we ex- pect a node to spend the majority of its time within one of the commun ities rath er than in transitiona l epochs. Speciﬁcally , we will assume th at com munity transition probab ilities exhibit a strong po sitive corr elation , th at is, if a node resides in commun ity j , it has a higher probability of staying within this commun ity for the next epoch, rather than lea ving. In this case, the to tal contr ibution of tr ansitional epo chs is small, a nd can be safely ignor ed in or der to not co mplicate our analysis. The above is a reason able assumption for ma ny target s cenarios we can imagin e; simulatio n re sults sh ow further that the time a node r esides in tr ansitional state is ind eed less tha n 10 % in th e scenarios co nsidered, not sign iﬁcantly affecting the ac curacy of the above expression. Giv en the fore -mentione d assumptions abo ut unit-step hit- ting probab ilities, the coro llary below follows. Cor ollary 5.8: The pr obability fo r a t least one hitting event to occur during time period t un der scen ario w t is P t H ( w t ) = 1 − (1 − P t h ( w t )) T t . (14) Finally , u sing the la w of total p robability , we d eriv e th e condition al h itting tim e under a speciﬁc target-co mmunity relationship, H T ( w 1 , ..., w V ) . Theor em 5. 9: H T ( w 1 , ..., w V ) = V X t =1 H T ( w 1 , ..., w V | f ir st hit in per iod t ) · P ( w 1 , ..., w V , f i rst hit in per iod t ) , (15) wher e the pr o bability fo r the ﬁrst h itting event to h appen in time period t is P ( w 1 , ..., w V , f i rst hit in per iod t ) = Π t − 1 i =1 (1 − P i H ( w i )) · P t H ( w t ) P , (16) and the hitting time und er this speciﬁ c con dition is H T ( w 1 , ..., w V | f ir st hit in per iod t ) = V X i =1 T i · ( 1 P − 1) + t − 1 X i =1 T i + 1 P t h ( w t ) , (17) wher e P = 1 − Π V t =1 (1 − P t H ( w t )) is the hitting pr o bability for one full cycle of time pe riods. Pr oo f: Eq . (1 6) holds as each cycle of tim e periods follows the same repetitiv e structu re, an d for the ﬁrst hitting ev ent to occu r in time period t it must no t occur in time p eriod 1 , ..., ( t − 1) . The ﬁrst term in Eq. ( 17) co rrespon ds to th e expected duratio n of full time period cycles un til the h itting ev ent oc curs. Sinc e for each cycle the success pr obability o f hitting th e 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. (17) is the sum o f dur ation o f time perio ds before the tim e per iod t in which the hitting event oc curs in the last cycle. Finally , the th ird term is the fraction of the last time period bef ore the h itting event occurs. No te that the last part is an appro ximation wh ich holds if the time per iods we consider are m uch longer than unit-time . C. Derivation of th e Meeting T ime The procedures o f th e deri vation o f th e meeting tim e is similar to that of the hitting time d etailed in the last section . In short, we derive the unit-step (or un it-time) meeting pro b- ability , P m , and the meeting proba bility for e ach type of tim e period, P M , an d p ut them togeth er to g et the overall meeting time in a similar fashion as in T heorem 5.9. Similar to Lemm a 5.7, we ad d up the contributions to the meeting probab ility from a ll commun ity pairs from node a and b in the following Lemma . Lemma 5 .10: Let commu nity j of nod e a an d commu nity k of node b overlap with each other for an ar ea A ( a t j , b t k ) in time period t . Then, th e conditio nal u nit-time meeting pr o bability in time period t when node a and b are in its commun ity j and k , respectively , is P t m ( a t j , b t k ) = P t mov e,j ( a ) P t mov e,k ( b ) ˆ v 2 K v A ( a t j , b t k ) A ( a t j , b t k ) C t j 2 ( a ) A ( a t j , b t k ) C t k 2 ( b ) + P t mov e,j ( a ) P t stop,k ( b ) 2 K v A ( a t j , b t k ) A ( a t j , b t k ) C t j 2 ( a ) A ( a t j , b t k ) C t k 2 ( b ) + P t stop,j ( a ) P t mov e,k ( b ) 2 K v A ( a t j , b t k ) A ( a t j , b t k ) C t j 2 ( a ) A ( a t j , b t k ) C t k 2 ( b ) . (18) Pr oo f: Eq uation ( 18) co nsists o f two par ts: (I) Both of the nodes are moving within the ov erlapped area. This add s the ﬁrst te rm in Eq. (18) to the meeting pr obability . The two r atios, A ( a t j ,b t k ) C t j 2 ( a ) and A ( a t j ,b t k ) C t k 2 ( b ) , cap ture the prob abilities that the nod es are in th e overlappe d area o f the co mmunities. The contribution to the unit-time mee ting proba bility is the produ ct of probab ilities of both nodes moving within the overlapped a rea an d th e term 2 K v A ( a t j ,b t k ) , which reﬂects the covered area in unit time . W e use the fact tha t when both nodes move acco rding to the rando m direction model, o ne c an calculate the e ffecti ve (extra) area covered by assuming that one node is static, and the other is moving with the (h igher) r elative speed between the two. T his difference is cap ture with the m ultiplicative factor ˆ v [29]. (II) One node is movin g in the overlapped are a, and the other one pauses within the area. T his adds the remainin g two terms in Eq. (18) to the unit-time m eeting pro bability . These 11 terms follow similar rationale as the previous one , with the difference that now only on e node is movin g. The second term correspo nds to the case whe n n ode a moves (and b is static), and the th ird ter m cor respond s to the case when n ode b m oves (and a is static). The deriv ation of the unit-tim e meetin g pr obability b etween nodes a and b f or tim e period t includ es all p ossible scenarios of comm unity overlap. If nod e a has S t ( a ) commu nities and node b has S t ( b ) commu nities, there can b e at most S t ( a ) S t ( b ) commu nity-overlapping scenarios in time per iod t . For similar r eason detailed in th e proof of Lemma 5.7, we neglect the con tribution of tr ansitional ep ochs to the unit-time meeting prob ability . Note th at (1 8) is the g eneral fo rm of Equatio n (13) and (1 4) in [15]. If we assume perfect overlap an d a single comm unity from bo th nodes, we ar riv e at (14) . I f we assume no over- lap, we result in (13). Also note in the genera l expressions presented in this pa per , the whole simulatio n area is also considered as a c ommunity . Therefor e we d o not have to include a separate term to capture the roam ing epo chs. Cor ollary 5.11: Th e pr ob ability for at lea st one meetin g event to o ccur d uring time p eriod t is P t M =1 − X ∀ ( j,k ) { P ov ( a t j , b t k ) · (1 − P t m ( a t j , b t k )) T t } , (19) wher e P ov ( a t j , b t k ) is the pr o bability that the commu nity j of node a overlaps with commu nity k of node b . This qua ntity is simply 1 if th e co mmunities h ave ﬁxed assignments and A ( a t j , b t k ) 6 = 0 . If th e communities are chosen rando mly , th is pr o bability can be derived by Lemma 4.5 in [15]. Due to space constraint, the Lemma is not repr oduce d here . Finally , similarly to Theorem 5.9, the e xpected meeting time can be calculated using the results in the L emmas in this section. Theor em 5. 12: The expected mee ting time is M T = V X t =1 M T ( meet in per iod t ) P ( meet in pe riod t ) . (20) Wher e th e qu antities in the above equa tion are calcu lated by P ( meet in per iod t ) = Π t − 1 i =1 (1 − P i M ) · P t M Q , (21) M T ( meet in peri od t ) = V X i =1 T i · ( 1 Q − 1) + t − 1 X i =1 T i + 1 P t m , (22) wher e Q = 1 − Π V i =1 (1 − P i M ) is the meeting pr obability for one full cycle of time period s. Pr oo f: The proof is para llel to th at o f Theorem 5.9 and is o mitted due to space limitations (see [40] f or details). As a ﬁnal no te, we can easily modify the ab ove the ory to acco unt fo r potential “off ” p eriods (e.g. b y intro ducing a p er step or per epoch “off ” pro bability , and a respective multiplicative factor). Due to space limitations, we do not include h ere these modiﬁcatio ns. V I . V A L I DA T I O N O F T H E T H E O RY W I T H S I M U L A T I O N S In this section , we comp are the theoretical deriv ations of the previous section against the correspo nding simulation r esults, for various param eter settin gs. Th rough extensive simulations with multiple scena rios and par ameter settings, we establish the accura cy of the theoretical framework. Due to space limitations, we can only sho w s ome e xamples of the simulation results we h av e. Mo re co mplete resu lts can be found at [40]. W e summarize the parameter s fo r the te sted scena rios in T able II. W e use two d ifferent setup of the TVC model fo r the simulation cases. Th e pa rameters listed in T able I I are for the simp le m odels ( M odel 1 − 4 ), where we hav e two time periods with two co mmunities in e ach time per iod (one of the commun ities is the whole simulation ﬁeld) . W e also simulate for mo re g eneric setup of th e TVC model ( M odel 5 − 7 , ref er to [40] for its para meters), where we have three co mmunities (one o f th em is the simulatio n ﬁeld) in each time p eriod. For the g eneric mod els, we h a ve exp erimented with two ways of commun ity pla cement: in a tiered fashion (as drawn in TP2 in Fig. 2), or in a ran dom fashion. Our discrete-time simulato r is written in C++. Mo re details ab out the simulato r , as well a s the so urce code, can be foun d at [40]. A. The A v erag e Node De gr ee For the average nod e degree, we create simu lation scenario s with 50 n odes in th e simulation ar ea, and ca lculate the average node degree of eac h nod e by taking the tim e average acr oss snapshots taken ev ery secon d d uring the simulation , and th en av erage across all nodes. All the simulation ru ns last for 60000 seconds in this subsection. As we show in Corollary 5.4, when the commu nities are random ly chosen, the av erage n ode d egree turn s ou t to be the av erage numb er of n odes falling in the comm unication range of a given no de, as if all n odes ar e uniformly distrib uted . Hence the average node degree do es not depend o n th e exact choices of co mmunity setup (i.e. single, mu ltiple, or multi- tier commun ities) or other par ameters. In Fig. 7 (a), we see the simulation cu rves follow th e pr ediction o f the theory we ll. T o m ake the scenario a bit more realistic, w e simulate some more scenar ios when the commu nities are ﬁxed. Among the 50 no des, we m ake 2 5 o f them pick th e commun ity centered at (30 0 , 300 ) and the other 25 pick the commun ity centered at (7 00 , 70 0) . W e simulate scenario s for all seven sets of par ameters, an d show some example results in Fig. 7 (b). In the simulatio ns, when the com munication ranges are small as com pared to the edge of th e comm unities, the relativ e error s ar e low , ind icating a g ood match between the theory and the simulation. Howe ver , as the commu nication range in creases, the area covered by the com munication disk becomes co mparable to the size of the com munity an d E q. ( 4) is no long er accurate sin ce the commun ication disk extend s out of the overlapped area in m ost cases. T hat is the rea son for the discrepan cies between the th eory and simulatio n. Besides Model-3 , we ob serve at most 20 % of error wh en th e commun ication disk is less th an 20% th e size of the inner- most c ommunity , indicating that our th eory is valid when the commun ication ra nge is relativ ely sma ll. 12 T ABLE II P A R A M E T E R S F O R T H E S C E N A R I O S I N T H E S I M U L AT I O N W e use the sa me mov ement spee d for all node: v max = 15 and v min = 5 in all sce narios. In all cas es we use two time periods and they are named as time period 1 and 2 for consistency . W e only list the parameters for the simple models ( M odel 1 − 4 ) here. Please refer to [40] for the details of the generic models ( M odel 5 − 7 ). Model name Description N C 1 l C 2 l D max,l D max,r L l L r π 1 l π 1 r π 2 l π 2 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 2880 Model 2 Highly attractive c ommunities 1000 200 50 100 200 52 520 0 . 667 0 . 333 0 . 889 0 . 111 3000 2000 Model 3 Not attracti ve communities 1000 100 100 50 200 80 800 0 . 5 0 . 5 0 . 667 0 . 333 2000 1000 Model 4 Large-size communities 1000 200 250 50 100 20 0 800 0 . 7 0 . 3 0 . 889 0 . 111 2000 1000 Communication Range (K) Average Node Degree 0 0.5 1 1.5 2 0 20 40 60 8 0 10 0 The ory M odel 1 Model5 (a) Randomly placed community . Communication Range (K) Average Node Degree 0 1 2 3 4 5 6 7 8 0 10 20 30 4 0 50 Model5-sim Model5-t heor y Model7-sim Model7-t heor y Model3-sim Model3-t heor y (b) Fixed communities. Fig. 7. Examples of s imulati on results (the avera ge node degree) . B. The Hitting T ime and the Meetin g T ime W e per form simulations fo r the hitting and the meeting times for 50 , 000 ind ependen t iterations for e ach scen ario, and compare the a verage results with the theo retical values deri ved from the cor respondin g eq uations (i.e. (6) and (2 0)). T o ﬁnd out the h itting o r the meeting time, we move the nodes in the simulator ind eﬁnitely u ntil they hit the target or mee t with each other, resp ecti vely . Again we show some example results in Fig. 8. For all the scenario s (inclu ding the on es not listed her e), the r elati ve errors are within acc eptable range . Th e ab solute values fo r the e rror are w ithin 1 5% fo r th e hitting time and within 20% fo r the m eeting time. For more than 70% of the tested scenarios, the error is below 10% (refer to [40] fo r oth er ﬁgures). These results display the accuracy of our theory under a wide rang e of parameter settings. The err ors between the theoretical and simulatio n results are main ly due to some of the approximatio ns we made in the various deriv ations. For example, th e appro ximation of the hitting and meeting processes with discrete, unit-time Bernou lli tr ials is valid on ly for the epochs that are long enoug h (in the order of community size) and if there are a lot o f e pochs. Fu rthermor e, there exist some bord er effects – when a nod e is close to the bor der o f a comm unity , it could a lso “see” some oth er nodes outside of th e co mmunity if its tran smission range is large enoug h. Howe ver , we have ch osen to igno re such occu rrences to keep our a nalysis simp ler . Nevertheless, as sh own in the ﬁgures, the error s are always within acceptable rang es, justifyin g o ur simplifying assumptions. V I I . U S I N G T H E O RY F O R P E R F O R M A N C E P R E D I C T I O N Although th e various the oretical qu antities derived fo r th e TVC model in Section V are in teresting in th eir own merit, they are particularly useful in predicting protocol performanc e, which in tur n can gu ide the decision s o f system opera tion. W e illustrate this poin t with two examples in th is sectio n. H i tt i ng Time (s) Communication range (K) 0 1000 0 2000 0 3000 0 4000 0 5000 0 10 2 0 30 40 5 0 60 70 Model 1-sim Model 1-theo ry Model 2-sim Model 2-theo ry Model 6(mult i_ti er)-si m Model 6(mult i_ti er)-th eory (a) Hitting Time. M ee t ing Time (s) Communication range (K) 0 3000 6000 9000 12000 15000 10 2 0 30 40 50 60 70 Model 5(ti ered_com m)-sim Model 5(ti ered_com m)-the ory Model 7(mul ti_c omm)-si m Model 7(mul ti_c omm)-t heo ry Model 3-sim Model 3-the ory (b) Meeting Time. Fig. 8. Examples of s imulati on results. A. Estimation of the Numbe r of Nodes Needed for Geographic Routing It has been shown in g eograph ic ro uting that the average node degree d etermines the su ccess rate of m essages d eli v- ered [28]. Thus, using the results of Section V -A we can estimate the nu mber of nod es (as a function of th e average node degree ) needed to achieve a target perform ance for geogra phic routing , f or a giv en scenario . W e con sider the same setu p as in Section VI-A, where half of the n odes are assigned to a commu nity cen tered at (300 , 3 00) and the other half are assigne d to another com- munity cen tered at (700 , 700) . W e ar e in terested in rou ting messages across o ne of the c ommunities, fro m coo rdinate (250 , 2 50) to coordinate (35 0 , 35 0 ) with simple g eograph ic routing (i.e., greedy forwardin g only , without face routin g [21]). Using simulations we o btain th e success rate of geo- graphic routing un der various communication r anges wh en 200 nodes m ove according to the mobility parameter s of Mod el- 1 (T able II). Results are shown in Fig. 9 (each point is the percentag e of su ccess ou t of 2000 trials). If we assume the mobility mod el is different, say Mo del-3 , we would like to know how m any nodes we need to achieve similar perfor- mance. Using Eq. (5) we ﬁnd that 7 6 0 no des are needed to create a similar average no de degree fo r Mo del-3 . T o validate this, we also simulate g eograph ic routing for a scenario where 760 nod es f ollow Mode l-3 . Comparing the resulting message delivery ratio for this scenario to th e o riginal scenario ( 20 0 nodes with Mo del-1 ) in Fig. 9, we see that similar suc cess rates are achieved under the same transmission range, which conﬁrms th e accuracy o f ou r an alysis. B. Pr ed icting Message Delive ry Dela y with Epidemic R outing Epidemic routing is a simple and po pular p rotocol that has b een pro posed for networks where no dal c onnectivity is intermittent (i.e., in Delay T olerant Networks) [34]. It has been shown that message pro pagation un der epidemic routin g can be mod eled with sufﬁ cient accura cy using a simple ﬂuid -based 13 0 0.2 0.4 0.6 0.8 1 10 15 20 25 30 35 4 0 Model 1_200no des Model 3_760no des Nodal communication range S u c c e s s r a t e Fig. 9. Geograp hic routing success rate under differe nt m obilit y parame- ter sets and node numbers. Simulation time N o d e s r e c e i v e d t h e m e s s a g e 0 10 20 30 40 50 60 0 50 0 1000 1500 2000 Theo ry (SI mo del ) Simul ati o n Fig. 10. Packe t propagation with epi- demic routing with two node groups with diffe rent communities. model [35]. (Note that its perfor mance has also been analyzed using Mar kov Chain [8] and Rand om W alk [30] models.) Th is ﬂuid model has been borrowed fr om the Mathematical Biology commun ity , and is u sually refer red to as the SI (Susceptib le- Infected) e pidemic mod el. T he gist of the SI mod el is th at the rate b y whic h the numb er of “inf ected” nodes increa ses (“infected” no des here are no des who have received a copy of the m essage) can b e appro ximated by the prod uct of three quantities: the numb er of already infected n odes, the number of susceptible (not y et infected) n odes, and the pair-wise contact rate, β ( assuming n odes m eet in depende ntly – this contact rate is equ i valent to the unit-step meeting p robabilities calculated in (18)). Thus, one could p lug-in these meeting probab ilities in to the SI mo del eq uations an d c alculate the delay f or epid emic routing . Y et, in the TVC model (and often in rea l life) there a re mu ltiple group s of nodes with d ifferent commun ities, and thus different pair-wise co ntact rates that depend o n the com munity setup. For example, n odes with the same o r overlapping commun ities ten d to meet much mor e often than n odes in far away com munities. For this reason , we extend the basic SI m odel to a more general scenario. W e consider the following setup in the ca se stu dy: W e use Model-3 (T able I I) for th e mobility parameters. A total o f M = 5 0 nodes are divided into two gr oups o f 2 5 nodes e ach. One g roup h as its com munity cen tered at (300 , 300) and the other at (700 , 700) . On e packet starts fro m a rando mly p icked source no de and spre ads to all other no des in the network. The propag ation of the message can be d escribed by the f ollowing equations:        dI 1 ( t ) dt = β ov I 1 ( t ) S 1 ( t ) + β no ov I 2 ( t ) S 1 ( t ) dI 2 ( t ) dt = β ov I 2 ( t ) S 2 ( t ) + β no ov I 1 ( t ) S 2 ( t ) S 1 ( t ) + I 1 ( t ) = M / 2 S 2 ( t ) + I 2 ( t ) = M / 2 . (23) where S x ( t ) an d I x ( t ) d enote the numb er o f susceptib le and infected nodes at time t in g roup x , respectively . Parameters β ov and β no ov represent the pair-wise unit-time meeting probab ility when the com munities are overlapped (i.e., for nodes in the same gr oup) and n ot overlapped (i.e., nodes in different g roups), respectively . W e u se Eq. (18) to obtain these qu antities. This mod el is an extension from the standar d SI m odel [3 5] and similar extensions c an be made for mo re than two group s [ 32]. Th e ﬁrst equatio n governs the chang e of in fected nodes in the ﬁrst group. Notice that the infection to susceptible no des in the group ( S 1 ( t ) ) can come fro m the infected node s in the same gro up ( I 1 ( t ) ) or the oth er grou p ( I 2 ( t ) ). W e can solve the system of equatio ns in (23) to get the ev olution of th e total infe cted n odes in th e network. As can be seen in Fig. 10, the theory curve closely follows the trend in the simulation curve. This indicate s ﬁrst that scenar ios generated by o ur m obility mod el are still amen able to ﬂuid model based mathem atical analysis (SI), despite th e increased complexity intro duced by the co ncept of commu nities. It also shows that results pro duced thus can be used b y a system designer to pr edict how fast m essages p ropaga te in a given network environment. This might, fo r example, determ ine if extra nodes are ne eded in a wireless co ntent distribution network to sp eed u p m essage d issemination. As a ﬁnal no te, in a ddition to epid emic routin g, the theor eti- cal results for the hitting and meeting times could be applied to predict the delay of various other DTN routing pro tocols (see e.g. [29], [30], [35]), f or a large range of mobility scenar ios that ca n be captured by the TVC mo del. V I I I . C O N C L U S I O N A N D F U T U R E W O R K W e have proposed a time-va riant commun ity mobility mod el for wireless mobile networks. Our model preserves comm on mobility characteristics, namely skewed loca tion visiting pref- er ences an d period ical re-appearance observed in empir ical mobility traces. W e have tuned the TVC m odel to match with the mobility ch aracteristics of various traces (WLAN traces, a vehicle mobility trace, and an enco unter trace of moving human beings), displayin g its ﬂexibility an d generality . A mobility trace generator of o ur model is a v ailable at [40]. In addition to p roviding realistic mo bility p atterns, the TVC model can be mathematically analyz ed to d eriv e several quantities of interest: th e average node d e gr ee , the hitting time and the meeting time . Throu gh extensive simulations, we have veriﬁed the accu racy of our theory . In the f uture we would like to further an alyze the p erfor- mance of various routing pr otocols (e.g., [3 0], [31]) under the time-variant com munity mobility mo del. W e also would like to con struct a systematic way to autom atically ge nerate th e conﬁgur ation ﬁ les, such that th e commu nities and time period s of node s ar e set to capture the in ter-node enco unter prope rties we observe in various trace s ( for example, the Small W orld encoun ter p atterns o bserved in WLAN traces [1 3]). R E F E R E N C E S [1] F . Bai, N. Sadagopan, A. Helmy , ”The IMPOR T ANT Frame work for Analyzi ng the Impact of Mobili ty on Performance of Routing for Ad Hoc Netw orks”, AdHoc Networks Journal - E lse vie r , V ol . 1, Issue 4, pp. 383-403, November 2003. [2] M. Balazinska and P . Castro, ”Charact erizin g Mobili ty and Network Usage in a Corporate W irele ss L ocal-Are a Network, ” In Proceeding s of MOBISYS 2003, May 2003. [3] C. Bettstet ter , G. Resta, P . Santi, ”The Node Distrib utio n of the random waypoi nt mobility model for wireless ad hoc netw orks, ” in IEEE Trans. on Mobile Computing, vol. 2, issue 3, pp. 257-269, Jul. 2003. [4] J. 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Ragha vend ra, ”Performanc e Anal- ysis of Mobility -assisted Routing, ” In Proceedings of AC M MOBIHOC, May 2006. [30] T . Spyropoulo s, K. Psounis, and C. S. Ragha v endra, ”Ef ﬁcient Routing in Intermitte ntly Connected Mobil e Netwo rks: The Single-cop y Case, ” to appear in A CM/IEEE Transact ions on Networking, Feb . 2008. [31] T . Spyropoulo s, K. Psounis, and C. S. Ragha v endra, ”Ef ﬁcient Routing in Intermitte ntly Connected Mobile Networks: The Multi -copy Case, ” to appear in A CM/IEEE Transact ions on Networking, Feb . 2008. [32] S.T anachai wiwat and A. Helmy , ”Encounter -based W orms: Analysis and Defense”, IEEE Conference on Sensor and Ad Hoc Communication s and Networ ks (SECON) 2006 Poster/Demo Session, September 2006. [33] C. T uduce and T . Gross, ”A Mobility Model Based on WLAN Traces and its V ali datio n, ” In Proceedings of IEEE INFOCOM, Mar . 2005. [34] A. V ah dat and D. Becke r , ”Epidemic Routing for Parti ally Connected Ad Hoc Networks, ” T echn ical Report CS-20000 6, Duke Univ ersity , April 2000. [35] X. Zhang, G. Ne glia, J. Kur ose, and D. T o wsle y , ”Performance Modeling of Epidemic Routing, ” in Proceedings of IFIP Networking 2006. [36] Cab Spotting, a project that tracks taxi mobility in the San Francisco Bay Area. Trac e ava ilabl e at http://c abspott ing.org/ap i . [37] CRA W D AD: A Community Resource for Archivi ng Wi reless Data At Dartmouth. http://c rawda d.cs.dartmouth.edu/inde x.php . [38] MobiLib: Comm unity-wi de Library of Mobility and Wi reless Netw orks Measurement s. http://ni le.usc.edu/ MobiLib . [39] The Network Simulator - NS-2. http://www .isi.edu/nsna m/ns/ [40] Simulat ion codes used in this work and its detaile d description are av ailable at http:/ /nile.c ise.uﬂ.edu/˜weijenhs/TVC model W ei-jen Hsu was born in T aipei, T aiwan , in March 1977. He rece i ved the B.S. de gree in Electrical Engineeri ng and the M.S. degree in Communica- tion Engineeri ng, respecti v ely , from National T aiwan Uni ve rsity , in June 1999 and June 2001. He recei ved the Engineer degree in Electric al Engineering from Uni ve rsity of Southern Californ ia, in August 2006. He is currently a Ph.D. student in the CISE De- partment , Univ ersit y of Florida. His main research intere st inv olv es the utiliz ation of realisti c measure- ment data in variou s tasks in computer netw orks, includi ng user modeling and beha vior-a ware protocol design. Thrasyv oulos Spyr opoulos w as born in Athens, Greece, in July 1976. He recei ved the Dip loma in Electric al and Computer Engineering from the Nationa l T echni cal Univ ersit y of Athen s, Greece, in February 2000. In May 2006, he recei ved the Ph.D degre e in Electrical Enginee ring from the Unive rsity of Southern Califor nia. In 2006-07, he was a post- doctora l research er at INRIA, Sophia-Antipol is. He is currently a senior researcher with the Swiss Fed- eral Institute of T echno logy (ETH), Z urich. Konstanti nos Psounis Konstanti nos Psounis is an assistant professor of EE and CS at the Uni versi ty of Southern California. He recei ved his ﬁrst de- gree from National T echnica l U ni ve rsity of Athens, Greece, in 1997, and the M.S. and Ph.D. degre es from Sta nford Uni versity in 1999 and 2002 re- specti vely . Konstant inos models and analyz es the performanc e of a varie ty of networ ks, and designs methods and algorithms to solve problems related to s uch systems. He is the author of more than 40 research papers, has recei ved facul ty awards from NSF and the Zumberge foundation, and has been a Stanford graduate fello w throughout his graduat e studies. Ahmed Helmy Dr . Ahm ed Helmy recei ved his Ph.D. in Computer Science (1999), M.S. in Elec- trical Engineering (EE) (1995) from the Univ ersity of Southern California (USC). He is Associate Pro- fessor and directo r of the wireless netw orking lab at the CISE Dept, Uni versi ty of Florida. From 1999 to 2006, he was faculty with EE-USC. He was a ke y research er in the network simulator (NS-2) and the protocol indepen dent multicast (PIM-SM) projects at USC/ISI. In 2002, he recei ve d the NSF CAREER A war d. His intere sts include network protoc ol design and analysis for mobile ad hoc and sensor networks, and m obilit y modeling.

Modeling Spatial and Temporal Dependencies of User Mobility in Wireless Mobile Networks

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