Modeling Time-variant User 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 32611-6120 †INRIA, Sophia-Antipolis, France ‡Dept. of Electrical Engineering, University of Southern California, Los Angeles, California 90089-2560 Email: wjhsu@ufl.edu, [email protected], [email protected], [email protected]fl.edu

Abstract— Realistic mobility models are important to under- schemes falling under the general framework of mobility- stand the performance of routing protocols in wireless ad hoc assisted routing have been proposed recently, as a measure to networks, especially when mobility-assisted routing schemes are improve the wireless network capacity[7] and increase the fea- employed, which is the case, for example, in delay-tolerant networks (DTNs). In mobility-assisted routing, messages are sibility of communication in more challenged environments[1]. stored in mobile nodes and carried across the network with nodal Mobility-assisted routing schemes, as opposed to path-based mobility. Hence, the delay involved in message delivery is tightly ad hoc routing protocols, utilize nodal mobility to dissemi- coupled with the properties of nodal mobility. nate messages in the network. In mobility-assisted routing, Currently, commonly used mobility models are simplistic transmissions from the senders to the receivers are not always random i.i.d. model that do not reflect realistic mobility charac- completed immediately through a connected, complete multi- teristics. In this paper we propose a novel time-variant community mobility model. In this model, we define communities that are hop path. Rather, when a sender moves to close proximity of visited often by the nodes to capture skewed location visiting some other nodes in the network, the packet is forwarded to preferences, and use time periods with different mobility pa- and stored in these intermediate nodes for potentially long rameters to create periodical re-appearance of nodes at the same time periods, waiting for the transmission opportunities to location. We have clearly observed these two properties based on other nodes in the network. Instead of being considered as a analysis of empirical WLAN traces. In addition to the proposal of a realistic mobility model, we derive analytical expressions detrimental factor that makes reliable communication difficult, to highlight the impact on the hitting time and meeting times if mobility provides communication opportunities in mobility- these mobility characteristics are incorporated. These quantities assisted routing. Hence, in these settings, mobility and nodal in turn determine the packet delivery delay in mobility-assisted encounter are crucial components to understand the network routing settings. Simulation studies show our expressions have performance. error always under 20%, and in 80% of studied cases under 10%. However, most research studies on mobility-assisted rout- ing assume simplistic mobility models, such as the random I.INTRODUCTION walk[3], [4], [5] (in general, i.i.d. models), or a priori knowl- edge of future mobility[2]. These assumptions provide sce- In recent years, there has been an exponential growth in narios amenable to mathematical analysis that provides good the popularity of portable computation and communication insights to system performance. However, these simple mobil- devices. Advances in wireless communication technologies ity models do not address the complexity of nodal mobility in and standards have made ubiquitous communication an emerg- real-life settings. In all these models, all mobile nodes behave ing reality. With the ever expanding deployment of these statistically identical to each other, and their behaviors do not wireless-capable devices, there is an increasing interest in a change with respect to time. As the underlying mobility model new communication paradigm and applications that are made is an important factor of the performance of mobility-assisted possible through the new opportunities. routing schemes, there is an increasing need for mobility Ad hoc networks are self-organized, infrastructure-less net- models that capture the realistic mobility characteristics and works consist of only wireless devices. In traditional ad hoc remain mathematically manageable. networks, it is generally assumed, albeit implicitly, that com- Our main contribution in this paper is the proposal of a munications between nodes occur through multi-hop, complete time-variant community mobility model. The model captures paths in space. However, this assumption is in question for several important mobility characteristics we observed from several reasons. First, multi-hop spatial routing increases the empirical wireless LAN (WLAN) traces. Specifically, we number of transmissions and channel contention, and hence utilize the WLAN traces from the archives at [22] and [23] reduces the capacity of scarce wireless bandwidth [6]. Second, to understand the prominent mobility characteristics of current such end-to-end paths may not always exist, given the wide wireless network users in university campuses and corporate variation of potential adverse settings (e.g., low node den- buildings. We have identified skewed location visiting prefer- sity, unpredictable mobility) in which wireless communication ences and periodical re-appearance at the same location as may take place. Due to the fore-mentioned reasons, routing two prominent trends existing in multiple traces[12]. These mobility characteristics are central in our daily activities but which the nodal movement preferences are not i.i.d. in space have not been addressed by existing mobility models. In the and not homogeneous across time. proposed model, we create communities to serve as popular Along a different line of research, to understand mobil- locations for the nodes, and implement time periods in which ity empirically, there has been WLAN measurement works the nodes move differently to induce periodical behavior. which reveal the important mobility characteristics of the To our best knowledge, this is the first mobility model that real-world wireless network users [9], [10], [11]. Large-scale captures non-homogeneous behavior in both space and time. deployments of WLANs in university and corporate campuses Moreover, the proposed time-variant community model can provide an excellent platform in which huge amount of user be mathematically treated to derive analytical expressions data can be collected and analyzed. Communities for WLAN for two important quantities of interest that determine the trace-related study are available at [23] and [22]. performance of mobility-assisted routing schemes: the hitting We combine the two streams of research in this paper by time and the meeting time. Starting from stationary nodal taking into account the mobility properties observed in WLAN distribution, the hitting time is the average time before a users and proposing a mathematical manageable mobility node moves towards the vicinity of a randomly chosen ge- model. In [12] we identified several prominent properties that ographical location, and the meeting time is the average time are common in multiple WLAN traces collected from vari- before two nodes move to the vicinity of each other. These ous environments, including on-off behavior, skewed location quantities capture the time between available communication visiting preferences, and periodical re-appearing behavior of opportunities under the mobility model, and can be used as nodes. Hence, we believe a good mobility model for wireless building blocks to analyze the performance of more complex network users should preserve these characteristics. In this packet forwarding schemes [4], [5]. We further show that our paper we extend the concept of community model proposed in theoretical derivation is accurate through simulation cases with our previous paper [16] further to include time-variant, non- a wide range of parameter sets. In all cases, the error between i.i.d. behavior of mobile nodes. simulation results and theoretical values is less than 15% for There are several previous attempts to build models the hitting time and 20% for the meeting time, and for 80% for WLAN users with the properties observed in WLAN of studied cases the error is below 10%. traces[13], [14], [15]. These models match the preference and The remaining of the paper is organized as follows: In the pause duration with the users in observed traces. They fall section II we discuss related work. For clarity, a simplified into the category of the WLAN association model, in which version of our time-variant community mobility model is currently associated access points are used as the indicator for introduced in section III, and the expressions for the hitting mobile node locations. Hence the applicability of the models time and the meeting time are derived in section IV-A and is specific to WLANs. However, in the case of mobility- IV-B, respectively, and validated with simulation in section assisted routing, nodes communicate not only when they are IV-C. In section V, we show a good matching between the associated with fixed network infrastructure (such as WLAN), mobility characteristics of our model and the real WLAN but also when they meet when moving between places. Hence traces and further discuss about the possible extensions of we abstract the mobility characteristics from WLAN traces, the time-variant community model. We conclude the paper in and propose a continuous mobility model (i.e., node locations section VI. are given by (x, y) coordinates) which has wider applicability. Note that it would be more relevant to compare our model to II.RELATED WORK such continuous human mobility traces. However, due to the unavailability of such traces, our best choice is WLAN-based Mobility-assisted routing has been proposed for various ones. Moreover, modeling the exact trajectory may not be as purposes in the literature. In [7], a two-hop routing scheme crucial to our goal as providing a continuum in the mobility has been shown to improve the network capacity. In [1], process. it is proposed to overcome the intermittent communication There are several other efforts to collect mobile node opportunity in challenged network settings (generally known encounter traces with hand-held devices [20], [21]. In these as delay-tolerant networks (DTNs)), with low node density works the performances of routing protocols are directly ob- or unpredictable mobility. So far, most studies on mobility tained by finding the inter-encounter time distribution, instead assisted-routing in the literature assume either a complete of being derived from a mobility model. This approach is knowledge of future mobility and encounters[2] or an i.i.d. different but complementary to our approach. However, note mobility pattern[7] with each individual node following simple that it is possible to derive encounter patterns from mobility mobility models, such as a random walk[3], [4], [5], mainly models, but not verse visa, so we choose to focus on the more for the sake of theoretical tractability. In [17] the authors fundamental task (i.e., formulate a realistic mobility model). assume a constant meeting rate between the mobile nodes to We are interested in comparing the inter-meeting times of our derive the inter-meeting times. These previous works focus model with these traces in the future. on deriving the performance of mobility-assisted routing with The concept of community is also mentioned in [19]. mathematical analysis. In this work we extend the scope of However, the authors assume the attraction of a community the analysis by proposing a time-variant community model, in to a mobile node is derived from the number of friends of this o eaeietclt n nte ie,non- (i.e., another and one do campus), hence to on and identical communities building behave different visited not pick most can (e.g., nodes node different a geographical visited of most the area of describe notion to the created that is the Note the community areas. other defining in than often by nodes more community so the achieve for We location(s) communities model. popular mobility define continuous to need we to wish integer we of features proposed major the time-gap two in the a retain are after These days. few probability of higher as multiples with at display AP time we same time online location 1(b) online same of Fig. the its 95% In at than of APs. more 65% 5 and as than AP, more one spends with average on display node we 1(a) part Fig. locations, inseparable preferences In an visited spend lives. is frequently to schedule our of weekly tend of or handful typical daily us a recurrent of of at a observations and time Most the the beings: of on human most of based activities is daily belief This model. location same two traces[12], the that the preferences believe from visiting observed we and location we Nonetheless, on, points features access networks. always of important these coverage not the the in in are of gaps (APs) some devices movement are the there continuous typically particular, register We WLAN In not [23]). from devices. do or made which [22] are observations at traces, these available several that by traces acknowledge collected (e.g., traces LAN groups wireless characteristics research the with mobility realistically, consult mobility important understand we To lives. the daily from captures observed that model Traces WLAN in Observed Characteristics Mobility A. hc sa motn raudrdvlpet ecos to choose work. We future development. as under this area structure, leave network important dependency social an inter-node the is for with which independently understanding models solid moves a Mobility require node others. each the make of and to preference decisions, as- location-based we follow movement paper nodes our the In that community. sume the in residing currently node Average fraction of online time I.D III. i.1 w motn oiiyfaue bevdfo LNtraces. WLAN from observed features mobility important Two 1. Fig.

associated with the AP 1.E+00 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 nodrt capture to order In mobility a propose to is paper the of objective main The a kwdlcto visiting location Skewed (a) 12 14 16 18 11111121 111 101 91 81 71 61 51 41 31 21 11 1 SRPINOF ESCRIPTION AP sorted by total amount of time associated with i with associated time of amount total by sorted AP USC preferences. MIT rpryosre rmWA rcs hr a where traces, WLAN from observed property Dart-04 o ahnd n aigand iisisown its visits node a making and node each for Dart-03 hudrmi norpooe mobility proposed our in remain should , UCSD rpry hr oer-per tthe at re-appears node a where property, ievratcmuiymblt model mobility community time-variant kwdlcto iiigpreferences visiting location skewed T IME t and -

VARIANT Prob. (Node re-appears at the same b eidclr-pernea the at re-appearance Periodical (b) AP after the time gap) 0.1 0.2 0.3 0.4 0.5 0.6 eidclr-perneat re-appearance periodical 0 7 6 5 4 3 2 1 0 eidclre-appearance periodical kwdlcto visiting location skewed aelocation. same M MIT OBILITY Time(days) gap USC i.i.d. Dart-03 Dart-04(March) M behavior skewed ODEL , . iuainae.A h einn fec pc,tenode whole the the epoch, in each between move distributed of to uniformly beginning community. speed free a the is its picks At node within the area. confined epoch, simulation is roaming node a the In of mobility the movement: a derivations. proposing following on the on focus in heavily we model depend while model scenario, targeted our the the for that, Note chosen area. parameters simulation the detailed within random at chosen are a iuainae.W eoetecmuiysz nteNMP the in size the community than the a as denote smaller of We the period much area. this of are simulation during abstraction communities node an The the as time. for location serves at visited community frequently period The movement node. normal each the of In boundary 2. t starting starts Fig. model the mobility in the from that illustrated assume we as discussion, following alternatively, occur periods uigteCP oi eapasa t omnt ihhigh with community its community at period its re-appears visit with it to probability so node CMP, a the We for during 1(b): probability Fig. high in assign shown can behavior re-appearance periodical the (CMP) lengths fixed periods: of of types V. two section fore- into in to two show extended the we be as of mobility, can simplicity, trends complex model its more major The model of the features. spite capture mobility In to period, mentioned presentation. sufficient time clear each is in of it community sake single the a for and periods time of model mobility community variant Model Mobility Community Time-variant changes the B. capture to order time. in across is behavior behavior modeled Time-variant This of explicitly [18]: probability. be in higher to suggested has with line same work the after along noon home during to and lunch go for time, people restaurants e.g., hour, working lives, during daily only offices our not omnipresent in the time behavior captures time-dependent in naturally structure also but such periodicity, even creates that with Note communities their probability. to higher go nodes which during periods capture location to same order In space). in movement model. concentration mobility and community periods time-variant movement the normal in Alternating periods 2. Fig. 0 = C nec iepro,and a w ifrn oe of modes different two has node a period, time each In oipeettm tutr,i u oe iei divided is time model our in structure, time implement To the of version simplified a describe we section this In c C -by- ihnec iepro,w sinacmuiyto community a assign we period, time each Within . n -by- ffie lengths fixed of C c oa epoch local C qae easm httecmuiylocations community the that assume We square. Normal movementNormal n eetbihsrcue ntm,b enn time defining by time, in structures establish we , qaeadtecmuiysz nteCPas CMP the in size community the and square periods T n Tn and , cycle and ( Tc T T n c omlmvmn eid (NMP) periods movement Normal h ae scetdt capture to created is later The . ocnrto oeetperiods movement concentration omn epoch roaming + movement periods eidclr-pernea the at re-appearance periodical Concentration T c ) hc ossso w types two of consists which hs w ye ftime of types two These . nalclepoch, local a In . imet [ v min v , max N generic -by- time- ] and N 1-pl 1-pr pl πlTl Roaming Local Ppause,l = , (3) epoch epoch πl(Ll/vl + Tl) + πr(Lr/vr + Tr) pr πrTr Fig. 3. Illustration of the 2-state Markov chain model of local/roaming epoch. Ppause,r = , (4) πl(Ll/vl + Tl) + πr(Lr/vr + Tr) TABLE I where πr = pr/(pr + pl) and πl = pl/(pr + pl). The above PARAMETERSOFTHETIME-VARIANTCOMMUNITYMOBILITYMODEL stationary probabilities can be calculated for NMP and CMP independently. N Edge length of simulation area Proof: The derivation follows easily from calculating the C Edge length of community stationary distribution of the local and the roaming epochs vmin, vmax, v Minimum, maximum, and average of movement speed Tmax, T Maximum and average pause time after each epoch using basic Markov chain theory[8], and from taking into L Average epoch length account the average duration of each epoch. Probabilities of choosing a roaming (local) p (p ) r l epoch after a local (roaming) epoch Stationary distribution of percentage IV. DERIVATION OF THEORETIC EXPRESSIONS π (π ) l r of local (or roaming) epoch among all epochs Stationary distribution of fraction of time In this section we derive the theoretic expressions of the P (P ) move pause the node is in moving (or pause) status hitting and the meeting times under the time-variant com- K Transmission range of nodes munity model and validate it with simulation. The expected HTcase Expected hitting time under the given ”case” MTcase Expected meeting time under the given ”case” hitting time is the average time for a node, starting from the stationary distribution, to move into the transmission range of a fixed, randomly chosen target coordinate (i.e., ”hits” the target) in the simulation area. The expected meeting time is a movement angle uniformly between [0, 2π], and performs the expected time for two mobile nodes, both starting from the a constant speed, random direction movement in the corre- stationary distribution, to move into the transmission range of sponding area. The movement length of each epoch is drawn each other. Note that this definition is different from inter- from an exponential distribution with average epoch length of meeting time used in [17], [20], which accounts for the time L. If the node hits the boundary of the simulation area in a duration between a meeting event and the subsequent one. roaming epoch, or hits the community boundary in a local Both quantities have important implications for the message epoch, it is re-inserted from the other end of the area (i.e., the delivery time in mobility-assisted routing scenarios. The hit- boundaries are ”torus” boundaries). At the end of the epoch, ting time is important when we have some fixed points in the node picks a pause time at the end point uniformly from the network for the mobile nodes to collect information (e.g., [0,Tmax], and picks the movement mode for the next epoch moving sensors find an event of interest occurring at random according to a two-state Markov model, as shown in Fig. 3. location) or deliver information (e.g., drop of its readings to For example, if the node has just finished a local epoch, it fixed sink). The meeting time is especially important if store- chooses the next epoch to be roaming with probability pr or and-carry message forwarding across multiple mobile nodes local with probability 1 − pr. These probabilities can be set is an application of interest. It is the average time before differently in NMP and CMP, hence we denote them as prn two given nodes encounter with each other, and hence it is and prc, respectively. Note that in the remainder of the paper closely related to the packet delivery delay in sparse networks we may append subscript l or r to indicate the parameters for where nodal encounters are the main opportunities for message local or roaming epochs and subscript n or c to indicate the delivery. parameters for NMP or CMP. We summarize the notations we used to describe the mobility model in Table I. Also note that A. Derivation of the Hitting Time these parameters can be set differently for different nodes to The sketch of the derivation of the hitting time is as follows: induce heterogeneous nodal behavior. However, for simplicity, We first derive the probability of hitting the target in a unit- we leave the details as future work. time slice, Ph, under the mobility model. Since the node Lemma 3.1: At any given time instant, the node is in one is performing random direction movements and the target is of the following four states: (a) moving in a local epoch, (b) also chosen at random, each time unit can be considered as moving in a roaming epoch, (c) pause after a local epoch, (d) an independent Bernoulli trial with success probability Ph. pause after a roaming epoch. The stationary distribution of We can then calculate the expected hitting time using the probability in each state is: geometric distribution. The details are given below. With the existence of communities, whether the chosen πl(Ll/vl) target is in one of the communities changes the expected Pmove,l = , (1) πl(Ll/vl + Tl) + πr(Lr/vr + Tr) hitting time value significantly. The hitting times for four possible cases are hence calculated separately, and then take πr(Lr/vr) weighted average. In the following, the first and the second Pmove,r = , (2) πl(Ll/vl + Tl) + πr(Lr/vr + Tr) subscripts in HTi,j, where i, j ∈ {in, out}, correspond to whether the target is in the community during the NMP and the Theorem 4.4: The expected hitting time for the {out, in} CMP, respectively. Pi,j is the probability of the corresponding case can be calculated as: HTi,j. HTout,in = P rob(hit in NMP )HTout,in,hit in NMP Lemma 4.1: + P rob(hit in CMP )HTout,in,hit in CMP HT = P HT + P HT overall in,in in,in in,out in,out PH,n (5) = (Tn + Tc)(1/P − 1) +Pout,inHTout,in + Pout,outHTout,out, P Tn 2 2 2 2 2 2 PH,n 1 − (1 + TnPh,n)(1 − Ph,n) where Pin,in = (Cn/N )(Cc /N ), Pin,out = (Cn/N )(1 − + (9) P P (1 − (1 − P )Tn ) C2/N 2), P = (1 − C2/N 2)(C2/N 2), and P = h,n h,n c out,in n c out,out (1 − P )P T + T (1 − C2/N 2)(1 − C2/N 2). + H,n H,c ( n c − T ) n c P P c Proof: The proof follows from basic probability and sim- (1 − P )P 1 − (1 + T P )(1 − P )Tc ple geometric arguments. Note that communities are chosen + H,n H,c c h,c h,c , P P (1 − (1 − P )Tc ) independently at random for both the NMP and the CMP. h,c h,c In order to calculate the expected hitting time for each where P = PH,n + PH,c − PH,nPH,c is the probability for at case, the unit-time hitting probability should be calculated least one hitting event to occur in one full cycle of NMP and separately. For the sake of simplicity, we only show the details CMP. of deriving HTout,in in the following Lemma 4.2 and Theorem Proof: The expected hitting time for the {out, in} case 4.4. The hitting time for the other three cases can be derived can be calculated as the weighted average of two separate sub- in a similar manner. cases: The first hitting event occurs during NMP or CMP. We Lemma 4.2: The unit-time hitting probabilities for the case can view the occurrence of hitting events in two types of time {out, in} in NMP and CMP are: periods as independent coin toss trials, which give head with probability P and P , respectively. We want to calculate P = P (2Kv /N 2) and (6) H,n H,c h,n move,r,n r the number of flips needed until we get the first head, when 2 2 Ph,c = Pmove,r,c(2Kvr/N ) + Pmove,l,c(2Kvl,c/Nc ). (7) we flip these two coins alternatively, starting from the NMP. Proof: The hitting event can only occur when the node The success probability for each full cycle (containing one is physically moving, as a non-moving node cannot encounter NMP and the subsequent CMP) is P = PH,n + PH,c − the fixed target which is not in its transmission range. Since PH,nPH,c. The probabilities for the first hitting event to occur the target location is chosen out of the community for the in NMP and CMP are PH,n/P and (1 − PH,n)PH,c/P , NMP, it can only be hit during roaming epochs in the NMP. respectively, as in each cycle of time periods NMP precedes We neglect the small probability that the target is chosen out CMP, and the time period structure is repetitive in itself. of the community but close to it, so the node can actually For each sub-case, the time until the first hitting event hit the target during a local epoch. When a node moves with can be further divided into two parts: The complete time average speed vr, on average it covers a new area of 2Kvr periods before the last time period in which the hitting event in unit time. Since random direction movement covers the occurs, and the fraction of the last time period until the whole simulation area with equal probability [16], and the hitting event occurs. If the hitting event occurs during the target coordinate is chosen at random, it falls in this newly NMP, the expected duration of whole periods before that is covered area with probability 2Kv /N 2, and hence the unit- r (Tn + Tc)(1/P − 1), since it takes 1/P full cycles for one 2 time hitting probability is Ph,n = Pmove,r,n(2Kvr/N ), i.e., success event to occur if the success probability is P . The when the node performs a roaming movement and the target time until the first hitting event occurs in the last NMP is is in the newly covered area in the time-unit. The target location is chosen in the community for the CMP, XT n i−1 so it can be hit during both roaming and local epochs. The Ph,n(1 − Ph,n) i T hitting probability is the sum of these two cases. For roaming 1 − (1 − Ph,n) n epochs the argument is the same as in the last paragraph, i=1 (10) 1 − (1 + T P )(1 − P )Tn 1 and for local epochs, the node covers randomly chosen new = n h,n h,n ≈ . P (1 − (1 − P )Tn ) P area of 2Kvl,c in the community, hence it hits the target with h,n h,n h,n 2 probability 2Kvl,c/N . c The last approximation holds if Tn is large. If the first hitting event occurs in the CMP, the expected Note that the movement made in each time unit does not duration of whole periods before that is (T + T )(1/P − increase or decrease the probability of hitting the target in n c 1) + Tn, and the remaining fraction of the last CMP can subsequent time units, therefore each time unit can be consid- be calculated in a similar fashion to (10). Putting all these ered as an independent Bernoulli trial with success probability components together, we arrive at (9). given in Lemma 4.2. The corollary below immediately follows. Finally, the overall expected hitting time is derived by Corollary 4.3: The probability for at least one hitting event solving all four cases in (5) following the procedure outlined to occur in the whole NMP and CMP, respectively, are: in Lemma 4.2 and Theorem 4.4. Note that the last equation Tn Tc PH,n = 1 − (1 − Ph,n) ,PH,c = 1 − (1 − Ph,c) . (8) for the expected hitting time, (9), applies to all four cases. The N We now move on to derive the unit-time meeting probability. The meeting probability for nodes with overlapped communi- ties and non-overlapped communities are derived separately. N Lemma 4.6: The unit-time meeting probability for nodes C+2K with non-overlapped communities is C 2Kv × 2P (P + P ) Fig. 4. Illustration of the expansion of the ”footage” of community. P = r move,r pause,r pause,l m,no ov N 2 2Kv × 2P P C2 l move,l pause,r (13) + 2 × 2 only difference is in the unit-time hitting probability, Ph,n and C N 2 2 P . The general expression for P is 2Kvvˆ((Pmove,r + Pmove,l) − P ) h,c h,n + move,l , N 2 2 Ph,n = I(target in comm. in NMP )Pmove,l,n2Kvl,n/Cn and the unit-time meeting probability for nodes with over- 2 lapped communities is + Pmove,r,n2Kvr/N , (11) 2 2KvlvPˆ 2Kv × 2P P where I(·) is the indicator function. The expression for P P = move,l + l move,l pause,l h,c m,ov C2 C2 can be derived similarly. 2Kvr × 2Pmove,r(Ppause,r + Ppause,l) + 2 B. Derivation of the Meeting Time N (14) 2Kv × 2P P C2 The derivation of the meeting time is similar to the hitting + l move,l pause,r × C2 N 2 time detailed in the last section. In the following derivation, 2 2 2Kvvˆ((Pmove,r + Pmove,l) − P ) again we first arrive at the meeting probability between two + move,l . N 2 nodes in a unit-time slice and the meeting probability of each Proof: If the communities do not overlap, the nodes time period. Then, in a very similar fashion to Theorem 4.4, can only meet when at least one of them is out of the we derive the meeting time by separating the sub-cases of community (i.e., roaming). The first and the second terms in meeting in the NMP or the CMP, and adding up the time (13) correspond to the scenario when one node is moving components of whole periods and fraction of the last period and the other is not. In the first term, the moving node is in in which the meeting event occurs. roaming state and the non-moving node can be in either local The meeting time calculation heavily depends on the relative or roaming state. The moving node covers 2Kvr new area each location of the communities of the involved nodes. Since time unit. Since it performs a roaming movement, it meets nodes move within their corresponding communities more 2 with the other node with probability 2Kvr/N as it does not often than roaming out of the communities, it is obvious that have a priori knowledge about where the paused node is. In two nodes with overlapping communities should meet each the second term the moving node performs a local movement other much faster. We first derive the probability of nodes and the paused node in roaming epoch happens to pause having overlapped communities, and then derive the meeting within the community of the moving node, which happens with probabilities for both cases, overlapped or non-overlapped probability C2/N 2. Since the moving node moves locally, it communities. 2 meets with the other node with probability 2Kvl/C . Lemma 4.5: For a specific type of time period, the commu- The third term corresponds to the scenario when both nodes nities of two nodes overlap with probability are moving. We make use of the fact that when both nodes (C + 2K)2 move according to the random direction model, the effective P = . (12) x N 2 extra area covered can be captured by the multiplicative factor Proof: As shown in Fig. 4, when a mobile node moves of relative speed, vˆ, which is 1.27 [16]. Note that the two within its community, the area covered by the node (i.e., the nodes cannot meet if they both perform local movement, hence area that could fall in the communication range of the node) we have to multiply the meeting probability by the factor of actually extends out of the community by the transmission 2 2 ((Pmove,r + Pmove,l) − Pmove,l) (i.e., at least one of them is range of the node. The ”footage” of the community is hence moving in roaming epoch). larger than C2. We approximate this area by (C + 2K)2, If the communities overlap, the nodes meet with higher ignoring the small differences at the corners. probability when they both perform local movements. Here If the other node has its community chosen within this area, we make the simplifying assumption that the two communities the meeting probability between the nodes would be much are perfectly overlapped. As we will show in section IV-C, larger. Since each node selects its community at random within the theory is reasonably close to the simulation despite such the simulation area, the probability that part of the footage of simplification. The first two terms in (14) correspond to the the community of node 1 is chosen as part of the community scenario when both nodes are in local epochs. Under such (C+2K)2 of node 2 is simply N 2 . scenario, the new area covered by a moving node contains the 25000 2 meeting time other node with probability 2Kvl/C . The first term captures (Theory) 20000 meeting time the scenario when both nodes move locally and the second (Simulation) term captures the scenario when only one node moves, with 15000 hitting time (Theory) similar reasonings as above. The remaining terms correspond 10000 hitting time (Simulation) to the scenario when at least one of the nodes is in roaming 5000 epoch. They are exactly the same as in the sub-cases with time / Meeting Hitting 0 non-overlapped communities. 0 10 20 30 40 50 60 70 Communication range (K) Following the unit-time meeting probability for the sub- Fig. 5. Comparing the simulation results and theoretical values for the hitting cases of overlapped or non-overlapped communities in Lemma and the meeting times (model 1). 4.6 and the probability for community overlap derived in 0 0.25 Lemma 4.5, we have: 0 20 40 60 80 -0.02 0.2 Model 1 -0.04 Model 2 Corollary 4.7: The probability for at least one meeting 0.15 Model 3 -0.06 Model 4 0.1 event to occur during NMP and CMP, respectively, are -0.08 0.05 -0.1 Model 1 Model 2 0

Relative Error (HT)Relative -0.12 Model 3 Relative Error (MT) 0 20 40 60 80 Tn Tn -0.14 -0.05 PM,n = 1−Px,n(1−Pm,ov,n) −(1−Px,n)(1−Pm,no ov,n) , Model 4 -0.16 -0.1 (15) Communication range (K) Communication range (K) (a) Hitting time. Tc Tc (b) Meeting time. PM,c = 1−Px,c(1−Pm,ov,c) −(1−Px,c)(1−Pm,no ov,c) . (16) Fig. 6. Relative error between theory and simulation values. In an almost parallel fashion to Theorem 4.4, the expected meeting time can be calculated using the results in the Lemmas or the meeting time, we move the nodes in the simulator in this section. indefinitely until they hit the target or meet with each other, Theorem 4.8: The expected meeting time is: respectively. As shown in Fig. 5 for model 1, the simulation results and theoretical values are very close to each other. The MT = P rob(meet in NMP )MTmeet in NMP relative error between theory and simulation results is between + P rob(meet in CMP )MTmeet in CMP 4.62% to 10.38% for the hitting time and between 0.59% to PM,n = (Tn + Tc)(1/Q − 1) 4.27% for the meeting time. Note that the simulator is not a Q repetition of theoretical calculations and hence the matching P 1 − (1 + T P )(1 − P )Tn M,n n m,n m,n (17) results validate each other. More details about the simulator + T Q Pm,n(1 − (1 − Pm,n) n ) and the codes can be found at [24]. (1 − PM,n)PM,c Tn + Tc + ( − Tc) We show the relative errors between the theoretical val- Q Q ues and the simulation results for other models in Fig. 6. Tc (1 − PM,n)PM,c 1 − (1 + TcPm,c)(1 − Pm,c) The relative error is calculated as Error = (T heory − + T , Q Pm,c(1 − (1 − Pm,c) c ) Simulation)/Simulation. Hence a positive error indicates where Q = PM,n + PM,c − PM,nPM,c is the probability for the theoretical value is larger than the simulation result, while at least one meeting event to occur in one full cycle of NMP a negative error indicates the converse. From Fig. 6 we see that and CMP. for all four models, the relative errors are within acceptable Proof: The proof is parallel to the proof of Theorem 4.4 range. The absolute values for the error are within 15% for and we omit it due to space constraints. the hitting time and within 20% for the meeting time. For 80% of the tested parameter sets, the error is below 10%. C. Validation of Theory with Simulations These results display the accuracy of our theory under a wide In this section we compare the theoretical values of the parameter settings, especially when the communities are small hitting time and the meeting time to the corresponding simula- compared to the simulation area. tion results, under various parameter settings. We summarize the parameters for tested models in Table II. Among them, V. MATCHING REALAND SYNTHETIC TRACES model 1 is the case when the model behaves similar to the We have shown in the previous section that the hitting time MIT WLAN trace (details in the next section). Model 2 is and the meeting time can be derived for the proposed time- a scenario that communities have strong attraction. Model 3 variant community mobility model. In this section we in turn is a scenario that communities are not very attractive and the show that in addition to theoretical tractability, the model is nodes have equal probability to perform local and roaming also generic enough to be fine-tuned and display matching movements in the NMP. Model 4 features large communities. mobility characteristics with the WLAN traces, in terms of We perform simulations for the hitting and the meeting both skewed location visiting preferences and periodical re- times for 50, 000 independent iterations and compare the appearance at the same location. average results with theoretical values derived from (5) and To tally location preferences and re-appearance probability (17). Our discrete-time simulator is written in C, and nodes for the continuous time-variant community mobility model, we move as per descriptions in section III. To find out the hitting first create a synthetic WLAN trace from the mobility model. TABLE II PARAMETERSFORTHESCENARIOSINTHESIMULATION

Model name Description N Cn Cc vmax, vmin Tmax,n Tmax,c Lr Ll pl,n pr,n pl,c pr,c Tn Tc Model 1 Match with the MIT trace 1000 100 100 15, 5 100 50 520 80 0.5 0.2 0.8 0.2 5760 2880 Model 2 Highly attractive communities 1000 200 50 15, 5 100 200 520 52 0.6 0.3 0.8 0.1 3000 2000 Model 3 Not attractive communities 1000 100 100 15, 5 50 200 800 80 0.5 0.5 0.6 0.3 2000 1000 Model 4 Large-size communities 1000 200 250 15, 5 50 100 800 200 0.7 0.3 0.8 0.1 2000 1000

AP sorted by total amount of time associated with it 0.3 1 11 21 31 41 51 61 71 81 91 1.E+00 0.25 Community 1.E-01 0.2 Model-simplified 1.E-02 0.15 MIT Model-simplified 1.E-03 0.1 2nd-tier

1.E-04 Community MIT 0.05 associated with the AP associated with the 1.E-05 Model-complex Model-complex 0 Average fraction of fractiontimeof Averageonline AP after the the given time gap) afterAP 1.E-06 0 2 4 6 8 Prob. (Node re-appears (Node Prob.at the samere-appears Time gap (days) Fig. 7. Matching location visiting preferences of Fig. 9. Illustration of multi-tier community (3- Fig. 8. Matching periodical re-appearance of the the synthetic traces to the WLAN trace. tiers). synthetic traces to the WLAN trace.

We do so by introducing the notion of virtual access points in that the CMP captures the working hours (i.e., nodes go to the simulation area and defining when the mobile nodes are office with very high probability) and the NMP models off- associated with these virtual APs. Note that the behavior of work hours. The synthetic trace mimics the peaks of the re- mobile nodes do not change from the description in section III appearance probability. However, since we have only two types by the introduction of these virtual APs. They are introduced of time periods in the model, the peaks repeat itself with the solely for the purpose of tallying mobility characteristics. We same value, while in the WLAN trace we also observe weekly divide the whole simulation area into 100 regular, equal-sized periodicity: the probability for nodes to re-appear at the same grids. Each grid is covered by one virtual AP. Since devices AP is higher if the time gap is seven days. are usually turned off when users start moving them in the real Our simplified model, as described in section III, makes WLANs, we make a similar assumption that the mobile nodes the theoretical derivation of the hitting time and the meeting are considered associated with the virtual AP in the current time cleaner to handle. But, on the other hand, it may be grid they reside in only when they are not moving. We then insufficient to fully capture complex user behavior, as shown tally the mobility characteristics from the synthetic trace by by comparing curves of model-simplified and MIT in Fig. the same way as we treat real WLAN traces. 7 and 8. To resolve the mismatch, we introduce more fine- We use the MIT WLAN trace (presented in [9]) as an grained model both in space and time. In the space domain, example to show that our model, with parameters carefully we introduce multi-tier communities, as illustrated in Fig. adjusted, could generate a synthetic trace that mimics the be- 9 with a three-tier example. The node visits each tier of havior of real WLAN traces. We also achieved good matching the community with decreasing probability. This extension is with the USC[22] or the Dartmouth[11] traces, but do not suggested by the intuition that we move locally close to homes show it here due to space constraints. In Fig. 7, we show and offices more than go far from these places. This way, we the skewed location visiting preferences property. In both the allow a smooth decaying tail rather than the horizontal tail MIT trace and the synthetic trace from our simple model (the in the simple model. In the time domain, we add more time curve labeled as model-simplified, using parameters of Model periods with different mobility parameters. These additional 1 in Table II), the nodes stay within their favorite locations time periods can be used to capture either more detailed (i.e., the communities) for high probability. However, for the mobility behavior for different time periods within a day or WLAN trace the curve shows a fast decaying tail, while for differences of mobility in weekdays and weekends. Finally, our model the tail of the curve levels off. This is because the we use a more complex model with six-tier communities node roams within the whole simulation area and pauses at and three distinct time periods to model weekday work hour, any location with equal probability in roaming epochs. We weekend day time, and night time separately. The synthetic will show a method to extend our simple model to resolve trace derived from this model matches well with the MIT trace this issue below. in terms of both location visiting preferences and periodical In Fig. 8, we show the periodical re-appearance property. re-appearance. The curves labeled as Model-complex in Fig. 7 In the MIT trace, the nodes re-appear at the same AP with and Fig. 8 show the mobility characteristics for this complex higher probability if the time gap between the considered model. time instants is an integer multiple of days. As shown in the Fine-tuning our model to match with the mobile node figure (model-simplified), we can set the time periods, such behavior in the WLAN trace is more of a trial and error process. We first adjust the attraction from each tier of the we will have a mobility model to describe an environment community (i.e., the likelihood of having an epoch in the tier) including users with diverse mobility characteristics. We be- and the pause times of the node in each tier to shape a decaying lieve such a model is a very important building block for tail of location visiting preference. This is typically achieved understanding protocol performances in real-life settings. by a power-law decaying attraction and pause time at each ACKNOWLEDGEMENT community tier. The time ratio nodes spend on moving and pausing is then adjusted to change the peaks in the curve The authors would like to thank the anonymous reviewers of periodical re-appearance. The key point is, once these for providing valuable comments. This work is supported by properties are collected from the targeted environment, our NSF awards CNS-0520017 and career award 0134650. model provides a flexible platform which can be tuned to REFERENCES match with the desired scenario. [1] Delay tolerant networking research group. http://www.dtnrg.org. Note that the theory derivations of the hitting time and the [2] S. Jain, K. Fall, and R. Patra, ”Routing in a delay tolerant network,” In meeting time with these extensions are technically feasible Proceedings of ACM SIGCOMM, Aug. 2004. by simply adding more sub-cases. For the hitting time, we [3] R. C. Shah, S. Roy, S. Jain, and W. Brunette, ”Data mules: Modeling and analysis of a three-tier architecture for sparse sensor networks,” Elsevier split the sub-cases that the target falls in each tier of the Ad Hoc Networks Journal, 2003. community in (5). For the meeting time, we need to add [4] T. Spyropoulos, K. Psounis, and C. S. Raghavendra, ”Single-copy routing more sub-cases that the communities of two nodes overlap in intermittently connected mobile networks,” In Proceedings of IEEE SECON, Oct. 2004. with various degrees (i.e., up to a certain tier), in addition to [5] T. Spyropoulos, K. Psounis, and C. S. Raghavendra, ”Spray and wait: (13) and (14). Due to lack of space, we do not present this Efficient routing in intermittently connected mobile networks,” In Pro- analysis in detail here. ceedings of ACM SIGCOMM workshop on Delay Tolerant Networking (WDTN), Aug. 2005. Currently we assume that each node randomly picks its own [6] P. Gupta and P. Kumar, ”Capacity of wireless networks,” Transactions on community from the simulation area. However, in realistic Information Theory, 46(2), 2000. settings there should be points of common interest in the envi- [7] M. Grossglauser and D. Tse, ”Mobility increases the capacity of ad hoc wireless networks,” IEEE/ACM Transactions on Networking, 10(4), 2002. ronment, at least during certain time periods (e.g., restaurants [8] S. Ross, ”Introduction to Probability Models,” 8th edition, Academic on campus during lunch time, hotels in an amusement park Press, published Dec. 2002. during nights). To capture such common attraction points, we [9] M. Balazinska and P. Castro, ”Characterizing Mobility and Network Usage in a Corporate Wireless Local-Area Network,” In Proceedings of could use an alternative way to set up communities: We can MOBISYS 2003, May 2003. fix the locations of multiple communities serving as known [10] M. McNett and G. Voelker, ”Access and mobility of wireless PDA point-of-interest in the targeted scenario, and let the nodes users,” ACM SIGMOBILE Mobile Computing and Communications Review, v.7 n.4, Oct. 2003. choose one of them at random with given probabilities during [11] T. Henderson, D. Kotz and I. Abyzov, ”The Changing Usage of a Mature concentration periods. This set up captures that nodes will Campus-wide Wireless Network,” In Proceedings of ACM MOBICOM, meet at these attraction points with higher probability during Sep. 2004. [12] W. Hsu and A. Helmy, ”On Important Aspects of Modeling User CMP. The only change needed in the theory is the probability Associations in Wireless LAN Traces,” In Proceedings of the Second of community overlap in the derivation of the meeting time. International Workshop On Wireless Network Measurement, Apr. 2006. Instead of using (12), the probability of community overlap, [13] C. Tuduce and T. Gross, ”A Mobility Model Based on WLAN Traces and its Validation,” In Proceedings of IEEE INFOCOM, Mar. 2005. Px, is simply the probability of choosing the same attraction [14] R. Jain, D. Lelescu, M. Balakrishnan, ”Model T: An Empirical Model for point to visit during CMP, which can be easily derived. User Registration Patterns in a Campus Wireless LAN,” In Proceedings of ACM MOBICOM, Aug. 2005. VI.CONCLUSIONANDFUTUREWORK [15] D. Lelescu, U. C. Kozat, R. Jain, and M. Balakrishnan, ”Model T++: An Empirical Joint Space-Time Registration Model,” In Proceedings of We have proposed a time-variant community mobility ACM MOBIHOC, May 2006. model for wireless mobile networks. Our model preserves [16] T. Spyropoulos and K. Psounis, ”Performance Analysis of Mobility- assisted Routing,” In Proceedings of ACM MOBIHOC, May 2006. two characteristics of mobility observed in current WLANs, [17] R. Groenevelt, P. Nain, and G. Koole, ”The Message Delay in Mobile namely skewed location visiting preferences and periodical Ad Hoc Networks,” In Proceedings of PERFORMANCE, Oct. 2005. re-appearance at the same location. In addition to providing [18] M. Kim, D. Kotz, and S. Kim, ”Extracting a mobility model from real user traces,” In Proceedings of IEEE INFOCOM, Apr. 2006. realistic mobility patterns, the proposed model can be math- [19] M. Musolesi and C. Mascolo, ”A Community Based Mobility Model for ematically analyzed to derive two quantities of interest, the Ad Hoc Network Research,” In Proceedings of the Second International hitting time and the meeting time. These two quantities are Workshop on Multi-hop Ad Hoc Networks (REALMAN), May 2006. [20] A. Chaintreau, P. Hui, J. Crowcroft, C. Diot, R. Gass, and J. Scott, closely related to the routing performance in delay-tolerant ”Impact of Human Mobility on the Design of Opportunistic Forwarding networks or mobility-assisted packet forwarding. Through Algorithms,” In Proceedings of IEEE INFOCOM, Apr. 2006. extensive simulation studies, we have verified the accuracy [21] J.Su, A. Chin, A. Popivanova, A. Goel, and E. de Lara, ”User Mobility for Opportunistic Ad-Hoc Networking,” In Proceedings of the 6th IEEE of our theory - the error is always below 20% and under 10% Workshop on Mobile Computing Systems and Applications, Dec. 2004. for most cases. [22] MobiLib: Community-wide Library of Mobility and Wireless Networks In the future we would like to extend our theory to the Measurements. http://nile.usc.edu/MobiLib. [23] CRAWDAD: A Community Resource for Archiving Wireless Data At scenario in which nodes display more heterogeneous behavior. Dartmouth. http://crawdad.cs.dartmouth.edu/index.php. Nodes may have different set of parameters and the time period [24] Simulation codes used in this work and its detailed description are structure can be different for the nodes. With these extensions, available at http://nile.cise.ufl.edu/˜weijenhs/TVC model