Analytical Model of Epidemic Routing for Delay-Tolerant Networks Qingshan Wang School of Mathematics Hefei University of Technology Hefei, China

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1 Analytical Model of Epideic Routing for Delay-Tolerant etwors Qingshan Wang School of Matheatics Hefei University of Technology Hefei, China Zygunt J. Haas School of Electrical and Coputer Engineering Cornell University Ithaca, Y 4853, USA haas@ece.cornell.edu ABSTRACT Epideic routing is considered as one of the ost effective routing schees for delay-tolerant networs; indeed, epideic routing allows for shortest routing delays. Most of the analytical wor in the technical literature concentrates on the latency fro the pacet creation tie until its reception by the destination. However, gaining insight into the teporal nature of the epideic routing process can be of significant value; e.g., in designing new variations of the epideic routing protocol. In this paper, we derive an analytical iterative odel for pacet propagation in a obile networ, where the dynaic connectivity aong networ nodes is odeled by a rando graph. We investigate the probability distribution function of the nuber of infected nodes, the average nuber of infected nodes, and the probability of all nodes becoing infected, all as functions of tie. We also provide siulation results, which confir our analytical odel. As part of our study, we observed several interesting phenoena, e.g., the draatic ipact that even a odest increase of nodal speed has on the infection rate, especially at low speeds. Categories and Subject Descriptors C. [Coputer Counication etwor]: etwor Protocols Routing Protocols General Ters Algoriths, Perforance Keywords Delay tolerant networs, Epideic routing, Generating function, Infection odel, Rando graph. ITRODUCTIO Traditional counication networs eploy the notion of store-and-forward routing, where it is assued that there exists a path fro the source node to the destination node before counication coences. However, in soe counication environent, such as in sparse obile networs, lins aong nodes are established only for short periods of tie. Thus, due to the interittent nature of connectivity aong nodes, a coplete path between the source and the destination nodes is unliely to exist at any particular tie. Routing in such networs is still possible, but requires storing the pacets until a forwarding lin becoes available, leading to significant routing delays. Such a routing schee, often tered store-carry-forward [7], ay be Perission to ae digital or hard copies of all or part of this wor for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. HP-MOSyson, October 5,, Paphos, Cyprus. Copyright ACM //...$5.. useful only if the networ (and the underlying applications) can tolerate the resulting delays thus the nae Delay Tolerant etwors (DTs). DTs find application in habitat onitoring networs [], vehicular ad hoc networs [], underwater networs [3], etc. Epideic routing (ER) [4] has been proposed as an efficient routing schee for DTs. Analogous to the spread of an infectious agent [5], in ER a node that carries a pacet copies the pacet onto every other node with which it can establish counication. Due to obility, lins are sporadically created and copies of the pacet spread fro the pacet s source node to all the networ nodes. odes that carry the pacet are referred to as infected, while nodes that are yet to receive a copy of the pacet are called susceptible. In particular, the pacet will be received by the destination node when the destination node encounters an infected node. ER allows for quic distribution of copies of the pacet, so that it results in the shortest possible delivery delay, although not with the sallest possible energy consuption, as creation of each new copy of a pacet in the networ expends the energy needed for the pacet transission. Thus, in an attept to iniize the energy required for the routing operation, schees have been proposed that reduce the nuber of copies of a pacet. Typically, such studies involve odels based on Marov Chain ([6],[7]). For exaple, [8] calculates the pacet delivery delay in two relay protocols: the two-hop ultiple-copy protocol and the unrestricted ultiple-copy protocol, while obtaining the nuber of copies of a pacet at the tie of delivery to the destination node. In both protocols, the source node copies the pacet to all encountered nodes. However, while in the unrestricted ulti-copy protocol an infected node can transit the pacet to any susceptible node, in the two-hop ultiple-copy protocol an infected node can transit the pacet to the destination node only. SWIM (the Shared Wireless Infostation Model) [9] introduced the notion of an anti-pacet a sall pacet created by the sin which propagates within the networ to delete obsolete pacets fro the networ nodes. The paper studied the source-todestination delivery delay of epideic routing using Marovian odel. The reference [] proposed the (p, q)-epideic routing using the anti-pacet notion introduced by [9], and derived the distribution of the delivery delay. In the (p, q)-epideic routing, when a susceptible node encounters the source, it accepts a pacet copy with probability q ( ). When a susceptible node encounters an infected node (except the source), it accepts a pacet copy with probability p ( ). When the destination node eets an infected node (including the source) for the first tie, it accepts a pacet copy with the probability of. A rigorous, unified fraewor based on ordinary differential equations [] is used to study various forwarding and recovery schees which delete unnecessary pacets fro the networs to obtain the forulas for the pacet delivery delay, the nuber of copies sent, and the buffer occupancy.

2 Probability In this paper, we attept to gain understanding of the teporal aspects of the process of pacet propagation in epideic routing. As an exaple, Fig. shows the probability of pacet infection as a function of tie. The results were obtained by siulation runs of a networ of size 4[]-by-4[], with 5 nodes that are oving with rando direction obility pattern ([]), where the tie has been slotted into 5 [sec] intervals. Rather intuitively, the probability first increases while the nuber of susceptible nodes is still large; but although the nuber of infected nodes continues to increase, after achieving its axiu, the probability of infection decreases due to the decreasing nuber of susceptible nodes. Thus, the graph in Fig. is a result of the two trends: increase in the nuber of infected nodes and decrease in the nuber of susceptible nodes. The location and the value of the axial probability, the average tie of infection, and the spread of the infection tie are all functions of the networ paraeters, such as the nodes obility pattern(s) and the nuber of networ nodes. The original contribution of this paper is in our use of the rando graph theory [] with the epideic odeling to study the teporal pacet spread process in DT. In particular, we develop an analytical iterative forulation that could be used to derive the teporal functions of the probability of infection, the distribution function of the nuber of infected nodes, the average nuber of infected nodes, and the probability of all nodes being infected. We conduct siulations to verify our theoretical odels. Finally, while studying our derivations, we are able to ae a nuber of interesting observations regarding the pacet spreading process. The reainder of this paper is organized as follows. In Section, we present our networ and obility odels. In Section 3, we derive the iterative analytical odel of epideic routing, and we study the teporal properties of a nuber of epideic spreading functions. In Section 4, we discuss the setting of the size of a tie-slot. results are presented in Section 5. Finally, Section 6 concludes the paper Figure : Probability of a node infection as a function of tie. ETWORK AD MOBILITY MODELS We assue that the networ consists of nodes, which are randoly placed within a square []-by- [] networ, which is assued to be closed (i.e., foring a toroid), so that there are no boundary effects to consider. odes are oving with soe obility pattern; for our analytical study, we do not specify a particular obility odel, and we only require that the obility pattern be rando and independent, unconstrained, and stationary. In other words, that the velocity (speed and direction) of a node is chosen randoly fro soe fixed distribution and independently of that of the other nodes; that a node can freely ove into any location within the networ (i.e., that there are no exclusion zones within the networ area); and that the obility paraeters (e.g., the spatial and teporal distribution) reain constant with tie. We further assue that all nodes ove with the sae obility pattern (although this restriction could be alleviated), and that the transission range of a node is fixed at [], where With this networ odel at hand, we postulate that the intereeting ties of any two particular nodes can be odeled as exponentially distributed rando variable [8] with rate β, where the inter-eeting tie is defined as the tie between two consecutive eeting instances of any two specific nodes. Because of the eory-less property of the exponential distribution, the probability of two nodes eeting within the next t is given by: () where we assue that t is sall enough, so that the probability of ore than one encounter between the two specific nodes is negligibly sall. There are two special nodes in the networ, the source node and the destination node, where the source node generates a essage that is intended for delivery to the destination node. Because the union of the transission areas of the nodes is significantly saller than the networ area (i.e.,, the average nuber of neighbors of a node is significantly less than one. Therefore, typically, when the essage is generated, there is no end-to-end path between the source and the destination nodes. Routing is done using the Epideic Routing (ER) schee, where a node copies all its pacets to any node that it encounters. For siplicity, we consider propagation of a single pacet only and we assue that such copying occurs at very high transission data rate, so that any encounter between two nodes allows each node to copy the pacet fro its eory onto the other s node eory (see [9] for treatent of the case where this assuption does not hold). Such propagation continues uninterrupted until all the networ nodes receive the pacet. (In reality, a pacet delivered to the destination node could be erased fro the other nodes see [9] for an exaple of such a schee.) We are interested in understanding how the propagation of the pacet occurs within the networ as a function of tie. To accoplish this, we odel the pacet propagation as a spread of an infectious disease, which otivated the use of the ters: an infected node and a susceptible node. Thus we establish the following ain etrics: the probability function of the nuber of infected nodes as a function of tie, the average nuber of infected nodes at a particular tie, and the probability of all nodes being infected by a particular tie. To carry out our study, we use rando graph theory. The networ is odeled as a graph, where the nodes are odeled as vertices, and the encounter between any two nodes as an edge between the two corresponding vertices in the graph. Initially there exists one infected node (vertex), while all the reaining - nodes are susceptible. Finally, tie is slotted in slots of size. We define as the probability that, within the next slot of, a particular node eets exactly different nodes out of the other - networ nodes (i.e., this eans that in the corresponding graph the node s degree is ). Thus, because of the independent obility of the nodes, we have: As a coent, we note that this excludes the Rando Waypoint Model fro consideration. A neighbor is a node located with the distance fro the node in question.

3 ( ) ( () We apply the probability generating function to the above result, as described by the following lea. Lea : The probability generating function (PGF) of the degree of a vertex in the graph is: Where ( ( ( and = ( (3) is the average degree of the nodes in the graph. 3. Perforance Evaluation We calculate three etrics as a function of tie: the probability distribution of the nuber of infected vertices, the average nuber of infected vertices, and the probability of all vertices being infected. Our analysis is based on the derivation of the probability generating function of the nuber of infected nodes as a function of tie. We assue that tie is slotted and that every slot is of duration (i.e., first slot is (, ], second slot is (, ],, and the th slot is ((-)t, t]). We first introduce two basic definitions: Definition : ( ( is the generating function of the nuber of infected vertices at tie t, where t= t. Definition : ( is the average nuber of infected vertices at tie t, where t= t. Of course, ( (. Thus (, and. During the tie period of (,t], the single infected node (the source node) will infect new nodes based on the probability generating function of the degree of the infected vertex (.Therefore, the probability generating function of the nuber of infected vertices at tie t= t can be expressed as: ( ) ( ), Ge x G x x p x x p x p x (4) where. Fro Def. and Eq. (4), we obtain the current average nuber of infected vertices as: ' () s Ge z, (5) where =. We define the set of infected nodes at tie as (, and when not abiguous we drop the tie index and siply refer to the set as. (ote that represents the average size of ( ; i.e.,. ) aturally, we should now ove on to calculate ( for. However, we realize that there is a difficulty with the extension for large due to the following two scenarios. First, as for the set of infected nodes,, will typically be larger than. Second, within the sae tie-slot, for it is possible that within a node will be infected ore than once (i.e., by ore than one encounter, each with a different infected node). In both cases, not every encounter of an infected node leads to a new infection. In other words, for the probability generating function of the degree of a vertex, (, overestiates to the nuber of new infections in a tie-slot. We discuss this proble next in ore details. More specifically, when the nuber of infected vertices is larger than, we consider an encounter event where two already infected nodes eet with each other within a tie-slot. We refer to this type of an event as a repeated infection or an RI event. The probability of an RI event is provided by the following theore: Theore : Suppose that the nuber of infected vertices equals. The probability of two infected nodes eeting within a tieslot is at ost. Proof: When the size of the set of infected vertices equals, and since the probability of a vertex eeting any other randoly chosen vertex is equal for all vertices, the probability of an infected vertex eeting another infected vertex equals. To proceed now with our calculation of the probability generating function of the nuber of infected nodes as a function of tie, we ae a siple assuption that, for large networs and for sall values of, the lielihood of an RI event is negligible. In particular, when the size of the set of infected vertices is very sall relative to the total nuber of vertices; i.e., based on Theore, the chance of an RI event can be neglected. Thus, at the beginning of the second slot, ( t, t ], the nuber of infected vertices is with the probability of p (,,, ),. Therefore, based on the property of probability generating functions ([3]), the distribution of the nuber of infected vertices at the end of the second slot is ( G ( x)) x with the probability of p (,,, ),. Therefore we obtain the probability generating function of the nuber of infected vertices at tie t = t as: ( ),( ( )),, (6) Ge x p G x x p x and the average nuber of infected vertices is: s Ge () ( z ) ' (7) In general, for t = t (=,, ) the probability generating function of the nuber of infected vertices can be written as: where p, ( ), ( ( )),, (8) Ge x p G x x p x u! ( ) u u u pu, p p p u u u u,!!! u u u u u u ( ) u u, where u, u,, u are integers in [,u]. Based on Eq. (8), the probability distribution of the nuber of infected vertices at tie t = t ( =,, ) is given by p, ( ), and the average nuber of the infected vertices is given by the follow theore. Theore : The average nuber of the infected vertices at tie t = t ( =,, ) is s ( ) z (,, ) when Proof: We prove the theore by induction..

4 Initial step: When, Eq. (5) states that, which satisfies Theore. Inductive step: ow suppose that Theore is valid for soe value of t= (-)t. In other words, for t= (-)t, the average nuber of infected nodes is ( ' ' Then, s Ge () ( p, ( G ( x)) x ) x = p ( ( G ( x)) ( G ( x)) x ( G ( x)) x ) ', x =, p ( z ) = ( z ) p, = ( z ), where the last equality follows fro the inductive step of the proof. Using Eq. () for large and sall p, we obtain that p p ( p) ( ) e i i where ( ) p p. Thus, the rando graph has a Poisson distribution of vertex s degrees and z p. Consequently, the ean size of the set of the infected vertices is given by: s z p. (9) ( ) ( ) Finally, the probability of all nodes being infected at tie ( can be obtained fro Eq. (8) by calculating p,. We now consider our assuption that the probability of an RI event is negligible. Clearly, as increases, the validity of our assuption diinishes and, based on Theore, the error introduced in Eq. (8) becoes ore significant. We choose soe probability P th and we find the largest tie t= ax t at which the probability of an RI event is still below P th. Using Theore and Eq. (9), we obtain: ( ( ( ) () Therefore, we state that Eq.(8) is accurate for within the probability error of P th. For copleteness of discussion, we also present an alternative approach, which is valid for t t, ax. This approach will allow us to fully copare our analytical results with siulation in following sections. In this approach, we assue that the tie-slot is sall enough so that the probability of ore than one encounter between any two nodes within a tie-slot is negligible. We ephasize that this includes encounters between infected and susceptible nodes. Let q be the nuber of infected vertices at tie t t, ax, and p( q ) be the probability that the nuber of infected vertices is ( We define ( and ( as the probabilities of an infection and of no infection within a tie-slot, respectively, when the nuber of infected vertices is. We calculate ( ((, because any of the - susceptible vertices don t eet any of the infected vertices. Based on the assuption that there is at ost one infection within the tie-slot, we conclude that ( (. Using the above definitions and the Marovian odel in Fig., we write down an iterative equation for p( q ) ( ), as: ( ( ( ( ( () and ( is as defined above. The initial condition for Eq. () could be the value of p( q ) p, where ( ) ( ) p ( ) p( p) ( p) ax, ax p, (,,, ) is calculated based on Eq. (8) at tie ax. We note that the iteration of Eq. () could also be started fro in which case we use the fact that ( and ( (. The average nuber of infected nodes by tie t = t can then be written as follows: s p( q ). () Finally, the probability of all nodes being infected at tie t t, ax can be obtained fro Eq. () by calculating p( q ). ( ( ( - ( ( ( ( Figure : The Marov Chain of the nuber of infected nodes 4. SIZIG THE TIME-SLOT The deterination of the size of the tie-slot, t, is iportant to ensure the validity of our analytical odel; e.g., if the slot-size is too large, the probability of ore than one event with the tieslot cannot be neglected, and our odel would introduce an error by underestiating the nuber of encounters. In a networ with nodes, the infection rate is axial when there are approxiately / infected vertices and / susceptible nodes. In this situation, the probability that a particular susceptible node encounters any (but exactly one) infected node within t ( ) equals ( ) p ( p ). The probability that a particular susceptible node does not encounter any infected node within t ( ) equals ( ) p. ow, when there are / susceptible vertices, the probability that any (but exactly one) susceptible node encounters any (but exactly one) infected node within t p ( ), is given by: (3) Siilarly, the probability that no susceptible node eets any other infected node within t, p ( ), is given by:

5 Cuulative Distribution Function ( ) ( ) ( ) p p t( seconds) Figure 3: The distribution of β. (4) ow, cobining Eqs. (3) and (4) gives the probability that, within t, there is at ost a single infection event; i.e., that either there is no encounter between an infected node and a susceptible node, or that exactly one susceptible node eets exactly one infected node. We further assue that the probability of the copleentary event (i.e., ore than one encounter) equals arbitrarily sall value, where <<, and we write: [ p( p) ][( p) ] [( p) ] =. (5) Siplifying the left-hand-side of Eq. (5) yields: ( ) ( ) ( ) ( ) ( ) [ p( p) ][( p) ] [( p) ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) p( p) ( p) ( ) ( ) [ ( p p ) ] [ p( ) ] [ p( ) ] 4 [ p ( ) ]. Thus, Eq. (5) can be rewritten as: 4 p ( ). (6) Based on Eqs. (6) and (), to ensure that at ost a single infection event occurs within t, t should be: t ln[ / ( / ) ]/. (7) Depending on the choice of, Eq. (7) can be used to deterine the size of the tie-slot t. 5. SIMULATIO RESULTS 5. Model We use siulation to verify our analytical odel. In our siulation, nodes are randoly placed in a square area of 4 4 [ ]. odes are oving according to the rando direction (RD) obility ([4]). The speed and travel tie of a node are chosen uniforly fro the interval [, [/sec]] and [, [sec]], respectively. The networ area wraps around ([5]), so a node that hits the area boundary reenters the networ at the opposite boundary. Transission range of a node is assued to be r = 5[]. One node is selected as the source of the epideic routing. Each configuration is siulated for top different and independent runs. For coparison purposes, we assue in our analytical odel that and P th equal. - and.5, respectively. Unless otherwise stated, =5 and top =. We choose the following four perforance etrics: (a) The probability distribution of the nuber of infected nodes as a function of tie. (b) The average nuber of infected nodes as a function of tie, s(,, ) (c) The cuulative distribution function (CDF) of the tie by which all the nodes becoe infected. (d) The average tie by which all the nodes becoe infected, referred to here as T. First, to estiate the encounter rate, β, we run our siulation odel with only two nodes, and we record the tie at which the two nodes firstly eet. The cuulative distribution function of β is illustrated in Fig. 3, where the results were obtained by siulation runs. 5. Evaluation Methodology In general, to copare the analytical and the siulation results, we use the (Chi-Square) statistical test ([6]). In particular, for the perforance etric of distribution of the nuber of infected nodes as a function of tie, the range of integers fro to is divided into bins, where a bin is assigned a range of integers that correspond to the nuber of infected nodes at a particular tie instance. The i th bin counts the nuber of siulation runs, O i, in which the nuber of infected nodes fits within the range assigned to the bin. Furtherore, we define E i as the expected nuber of runs (calculated based on our analytical odel), whose nuber of infected nodes belongs to the i th bin. In particular, we choose the - integers ni ( i ) to partition the integers range fro to, as to define the bins. We deterine n i to be the sallest integer, so that the nuber of siulation runs whose nuber of infected nodes is expected to fall within the [, n i ] range is at least top i /, where top is the nuber of siulation runs. For the perforance etric of the average nuber of infected nodes as a function of tie, the bins represent tie intervals, where the i th bin, O i, counts the nuber of nodes which are infected within the interval that corresponds to the i th bin. Siilarly, E i is the expected nuber of nodes (calculated based on our analytical odel), which are infected within the i th tie interval. In particular, we choose the - integers t, t,, t to partition the tie interval [, ), as to define the bins, and we deterine t i to be such that E /. i For the perforance etric of the cuulative distribution function of the tie by which all nodes are infected, bins represent tie intervals, where the i th bin, O i, counts the nuber of siulation runs where all the networ nodes becoe infected at a tie that is within the tie interval assigned to the i th bin. Siilarly, E i is the expected nuber of topologies (calculated

6 The average nuber of infected nodes The average nuber of infected nodes The average nuber of infected nodes Cuulative distribution function The average nuber of infected nodes based on our analytical odel), with all nodes becoing infected at a tie that corresponds to the interval assigned to the i th bin. In particular, we choose the - integers t, t,, t to partition the tie interval [, ), as to define the bins and we deterine t i to be such that E /. i 4 top. ( ) (9).67. In the last subsection, we study the perforance of the etric (a) by setting = 4. The critical value is,. therefore, ( ) (3).35 ([6]). Table : The average tie until all nodes becoe infected vs. P th Analysis Analysis Analysis (P th=.5) (P th =.5) (P th =.) T (a) (P th =.5) (P th =.5) (P th =.) (P th =.5) (P th =.5) (P th =.) (b) Figure 4: Perforance coparison for different values of P th The statistical test is defined by the following forula: i ( Oi Ei), (8) E and the hypothesis that the siulation results agrees with the analytical is rejected if: i ( ), (9) where is the significance level, and ( ) is the critical value of the test. Otherwise, the hypothesis is accepted. In our evaluation, we assue that.. Furtherore, in the next three subsection, we evaluate the perforance of the etrics (b), (c), and (d) by setting =. Thus, based on the Chi-Square distribution table ([6]), we obtained the critical value 5.3 The ipact of P th In order to study the influence of P th on the perforance, P th is varied fro.5 to.. The results are shown in Fig. 4 and Table. Fig. 4(a) shows the perforance coparison of the average nuber of infected nodes as a function of tie. The statistics for P th =.5,.5,. are.4,., and.5, respectively. These statistics are all saller than the critical value. Fig. 4(b) shows the perforance coparison of CDF of the tie by which all nodes becoe infected. The statistics for P th =.5,.5,. are.8, 4.39, and 4.58, respectively. The first two statistics are saller than the critical value. Thus we accept the hypothesis that the analysis odel fits the siulation data well for P th =.5 and.5. Finally, as shown in Table, the analytical value of T is only.7% and % less than the corresponding siulation results for P th =.5 and.5, respectively. Based on Eq. (), we obtained the value of ax of 66 and that correspond to P th =.5,.5, respectively. The results in Fig. 4 and Table show that the analytical results gradually diverge fro the siulation curve as P th increases, the reason being that the probability of an RI event increases with P th. In particular, the results deonstrate that the analytical odel overestiates the infection rate, thus underestiating the tie by which all networ nodes becoe infected. 5.4 The ipact of the nuber of nodes We investigate the effect of the variation of the nuber of nodes in our analytical odel and confir the results against siulation using the Chi-Square test. Figs. 5 (a), (b), and (c) depict the perforance when the nuber of nodes in the networ is 5,, and, and the corresponding Chi-Square test statistics are.3,., and.76, respectively. The CDF of the tie by which all networ nodes are infected is shown in Fig. 6, and the corresponding statistics are 9., 5.35, and.87, respectively. These statistics, which are all saller than the critical value, confir that the siulation results atch the results predicated by the analytical odel. Siilarly, the analytical T (a) =5 (b) = (c) = Figure 5: The average nuber of infected nodes for various nuber of networ nodes

7 Cuulative distribution function Cuulative distribution function Cuulative distribution function The average nuber of infected nodes The average nuber of infected nodes The average nuber of infected nodes Cuulative distribution function Cuulative distribution function Cuulative distribution function (a) =5 (b) = (c) = Figure 6: The CDF of the tie by which all networ nodes are infected for various nuber of networ nodes (a)v=5[/s] (b) v=[/s] (c) v=5[/s] Figure 7: The average nuber of infected nodes for various nodal speeds values depicted in Table are.5% above,.% below, and.6% below the siulation results of the corresponding cases of 5,, and nodes, respectively. Table :The average tie of all nodes becoing infected vs. T =5 = = Analysis The ipact of the speed of node We evaluate the perforance of our analytical odel by varying the nodal speed which is uniforly chosen fro [, v], where v equals 5[/s], [/s], and 5[/s], and we copare these results with the siulation results in Figs. 7 and 8. For the three choices of nodal velocity, the statistics for Fig. 7 are results atch well those predicated by the analytical odel. As can be observed fro the graphs in Fig. 7, even a oderate increase in the speed of the nodes leads to a considerable increase in the average nuber of infected nodes. For exaple, at 5[sec] and v=5[/s], there are on the average about 5 infected nodes, at v=[/s] this nuber increases to close to, and at v=5[/s] the nuber of infected nodes grows to over 35. The reason for this behavior is that a faster oving infected node encounters ore nodes and, therefore, is able to infect ore nodes. The sae behavior is deonstrated in Fig. 8 by the CDF of the tie by which all nodes becoe infected. It is worth noticing that especially at low speeds, even a sall increase in the speed ay have a significant ipact on the overall infection rate. Lastly, it can be seen fro Table 3 that the values of T of the analytical odel for the above three values of the speed are:.9%,.6%, and.8% less than that of corresponding siulation results, respectively (a) v=5[/s] (b) v=[/s] (c) v=5[/s] Figure 8: The CDF of the tie by which all networ nodes becoe infected for various nodal speeds.5,.7, and.4, respectively, and for Fig. 8 are 8., 8.35, and 4.39, respectively. These statistics indicate that the siulation Table 3: The average tie until all nodes becoe infected vs. v T v=5[/s] V=[/s] V=5[/s]

8 The probability distribution The probability distribution Analysis Probability distribution of the nuber of infected nodes as a function of tie The probability distribution of the nuber of infected nodes as a function of tie provides an iportant insight into the process of the pacet spread. Fig. 9 shows the probability distribution of the nuber of infected nodes at the ties of 6[s], 34[s], 4[s], and 5[s], when top =. (Fig. 9(b) shows an expended y-axis of Fig. 9(a).) For the probability distribution of infected nodes at 6 [s], the statistic in Fig. 9, as copared with the analytical results, is.45, again confiring a very good atch between analysis and siulation. As a point of reference, the axial gap between our analytical odel and siulation at the above four tie instances is less than.5%. Fro the results presented in this section, we conclude that our analytical odel faithfully represents the in fectionbasedspreading process of pacets in a DT t=6 second() t=34 second() t=4 second() t=5 second() t=6 second() t=34 second() t=4 second() t=5 second() The nuber of infected nodes (a) t=6 second() t=34 second() t=4 second() t=5 second() t=6 second() t=34 second() t=4 second() t=5 second() The nuber of infected nodes (b) Figure 9: The probability distribution of the nuber of infected nodes; (b) is a zooed-in version of (a) 6. COCLUSIOS In this paper, we investigated the process of a pacet spread in obile wireless networs. The pacet propagation fro a single node is odeled as spread of an infectious disease, and the networ connectivity is odeled by using rando graph theory. In particular, we proposed an iterative odel for calculating the probability of a nuber of infected nodes as a function of tie. Our basic odel is based on probability generating functions and is valid for relatively sall nuber of infected nodes. When the nuber of infected nodes increases, we proposed another iterative odel based on the Marovian property of the syste. Using our odels, we derived three etrics: the probability distribution of the nuber of infected nodes, the average nuber of infected nodes, and the probability of all nodes being infected, all the three etrics being functions of tie. We deonstrated that our siulation results are very close to the results predicted of our analytical odel, confiring the correctness of our odel. 7. REFERECES [] A. Mainwaring, D. Culler, J. Polastre, R. Szewczy, and J. Anderson, Wireless sensor networs for habitat onitoring, in Proc. ACM International Worshop on Wireless Sensor etwors and Applications (WSA), Sept.. []. An, J. Riihijarvi, and P. Mahonen, Studying the delay perforance of opportunistic counication in VAETs with realistic obility odels, in Proc. IEEE VTC Spring 9, Apr. 9. [3] E. Magistretti, J. Kong, U. Lee, M. Gerla, P. Bellavista, and A. Corradi, A obile delay-tolerant approach to long-ter energy-efficient underwater sensor networing, in Proc. IEEE WCC, Mar. 7. [4] A. Vahdat and D. Becer, Epideic routing for partiallyconnected ad hoc networs, Technical Report, Due University CS-6, April. [5] F. Brauer and C. Castillo-Chávez, Matheatical odels in population biology and epideiology, Springer-Verlag ew Yor, Inc.,. [6] S.-K. Yoon, Z. J. Haas, and J. H. Ki, Efficient tradeoff of restricted epideic routing in obile ad-hoc networs, in Proc. IEEE MILCOM, Oct. 7. [7] H. Zhu, S. Chang, M. Li, K. ai, and S. Shen, Exploiting teporal dependency for opportunistic forwarding in urban vehicular networs, in Proc. IEEE IFOCOM, Apr.. [8] R. Groenevelt, P. ain, and G. Koole, The essage delay in obile ad hoc networs, Perforance Evaluation, 6(- 4):-8, Oct. 5. [9] Z. J. Haas and T. Sall, A new networing odel for biological applications of ad hoc sensor networs, IEEE/ACM Trans. on etworing, 4():7-4, Feb. 6. [] T. Matsuda and T. Taine, (p, q)-epideic routing for sparsely populated obile ad hoc networs, IEEE Journal on Selected Areas in Counications, 6(5): , Jun. 8. [] E. Zhang, G. eglia, J. Kurose, and D. Towsley, Perforance odeling of epideic routing, Coputer etwors, 5():867-89, Jul. 7. [] B. Bollobás, Rando graphs, nd ed., Cabridge University Press, ew Yor, 999. [3] H. S Wilf, Generating functionology, nd ed., Acadeic Press, London, 994. [4] P. ain, D. Towsley, B. Liu, and Z. Liu, Properties of rando direction odels, in Proc. IEEE IFOCOM, March 5. [5] Z. J. Haas, The routing algorith for the reconfigurable wireless networs, in Proc. ICUPC, Oct [6] P.E. Greenwood and M.S.iulin, A guide to chi-squared testing, Wiley, ew Yor, 996.