Contention aware performance analysis of. mobility-assisted routing
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Contention aware performance analysis of. mobility-assisted routing Contention aware performance analysis of. mobility-assisted routing Document Transcript

  • 1 Contention-Aware Performance Analysis of Mobility-Assisted Routing Apoorva Jindal, Konstantinos Psounis University of Southern California E-mail: apoorvaj, kpsounis@usc.edu. Abstract— A large body of work has theoretically analyzed the very low traffic rates, irrespective of whether the network is sparseperformance of mobility-assisted routing schemes for intermit- or not. For higher traffic rates, contention has a significant impacttently connected mobile networks. But the vast majority of these on the performance, especially of flooding-based routing schemes.prior studies have ignored wireless contention. Recent papers To demonstrate the inaccuracies which arise when contentionhave shown through simulations that ignoring contention leadsto inaccurate and misleading results, even for sparse networks. is ignored, we use simulations to compare the delay of three In this paper, we analyze the performance of routing schemes different routing schemes in a sparse network, both with andunder contention. First, we introduce a mathematical framework without contention, in Figure 1. The plot shows that ignoringto model contention. This framework can be used to analyze contention not only grossly underestimates the delay, but alsoany routing scheme with any mobility and channel model. predicts incorrect trends and leads to incorrect conclusions. ForThen, we use this framework to compute the expected delays example, without contention, the so called spraying scheme hasfor different representative mobility-assisted routing schemes the worst delay, while with contention, it has the best delay.under random direction, random waypoint and community-based mobility models. Finally, we use these delay expressions Finally, note that a qualitatively different type of intermittentlyto optimize the design of routing schemes while demonstrating connected networks, that of non-sparse networks which are inter-that designing and optimizing routing schemes using analytical mittently connected due to severe mobility [25], will obviouslyexpressions which ignore contention can lead to suboptimal or suffer from contention too.even erroneous behavior. Epidemic Routing (No Contention) 600 Randomized Flooding (No Contention) Spraying Scheme (No Contention) Expected Delay (time slots) Epidemic Routing (Contention) 500 Randomized Flooding (Contention) I. I NTRODUCTION Spraying Scheme (Contention) 400 Intermittently connected mobile networks (also referred to as 300delay tolerant or disruption tolerant networks) are networks wheremost of the time, there does not exist a complete end-to-end path 200from the source to the destination. Even if such a path exists, 100it may be highly unstable because of topology changes due to 0 3.6% 4.8% 6.3% 8.6% 11.9% 17.3%mobility. Examples of such networks include sensor networks for Expected Maximum Cluster Sizewildlife tracking and habitat monitoring [1], military networks [2],deep-space inter-planetary networks [3], nomadic communities Fig. 1. Comparison of delay with and without contention for three different routing schemes in sparse networks. The simulations with contention use thenetworks [4], networks of mobile robots [5], vehicular ad hoc scheduling mechanism and interference model described in Section III. Thenetworks [6] etc. expected maximum cluster size (x-axis) is defined as the percentage of total Conventional routing schemes for mobile ad-hoc networks like nodes in the largest connected component (cluster) and is a metric to measure connectivity in sparse networks [17]. The routing schemes compared are:DSR, AODV, etc. [7] assume that a complete path exists between epidemic routing [8], randomized flooding [26] and spraying based routinga source and a destination, and they try to discover these paths [12].before any useful data is sent. Since, no end-to-end paths exist Incorporating wireless contention complicates the analysis sig-most of the times in intermittently connected mobile networks nificantly. This is because contention manifests itself in a number(ICMNs), these protocols will fail to deliver any data to all but of ways, including (i) finite bandwidth which limits the numberthe few connected nodes. To overcome this issue, researchers have of packets two nodes can exchange while they are within range,proposed to exploit node mobility to carry messages around the (ii) scheduling of transmissions between nearby nodes whichnetwork as part of the routing algorithm. These routing schemes is needed to avoid excessive interference, and (iii) interferenceare collectively referred to as mobility-assisted or encounter- from transmissions outside the scheduling area, which may bebased or store-carry-and-forward routing schemes. significant due to multipath fading [27]. So, we first propose a A number of mobility-assisted routing schemes for intermit- general framework to incorporate contention in the performancetently connected mobile networks have been proposed in the analysis of mobility-assisted routing schemes for ICMNs whileliterature [8–18]. Researchers have also tried to theoretically keeping the analysis tractable. This framework incorporates allcharacterize the performance of these routing schemes [17, 19– the three manifestations of contention, and can be used with24]. But, most of these analytical works ignore the effect of any mobility and channel model. In this framework, loss of acontention on the performance arguing that its effect is small in transmission opportunity due to contention is modeled by a losssparse, intermittently connected networks. However, recent papers probability. We propose a general analytical methodology to find[11, 17] have shown through simulations that this argument is not the exact expression for this loss probability in terms of thenecessarily true. The assumption of no contention is valid only for network parameters for any given routing scheme. (Note that a
  • 2 N Area of the 2D torusprevious paper [23] proposed modeling loss due to contention M Number of nodes in the networkwith a loss probability, but it did not discuss how to find this loss K The transmission rangeprobability analytically.) Θ The desirable SIR ratio sBW Bandwidth of links in units of packets per time slot We then use this framework to do a contention-aware perfor-mance analysis for the following representative mobility-assisted TABLE Irouting schemes for ICMNs: direct transmission [16] where the N OTATION USED THROUGHOUT THE PAPER.source waits till it meets the destination to deliver the packet, II. N ETWORK M ODELepidemic routing [8] where the network is flooded with the packetthat is routed, and different spraying-based schemes [12, 13, 22, We first introduce the network model we will be assuming23] where a small number of copies per packet are injected into throughout this paper. We assume that there are M nodes movingthe network, and then each copy is routed independently towards in a two dimensional torus of area N . The following two sectionsthe destination. For each of these schemes, we will find the loss present the physical layer, traffic and mobility models assumedprobability due to contention using the proposed framework, and in this paper.then find the expected end-to-end delay expressions. We firstderive the delay expressions for the two most commonly used A. Physical Layer and Traffic Modelmobility models, the random direction and the random waypoint 1) Radio Model: An analytical model for the radio has tomobility model. But real world mobility traces have shown define the following two properties: (i) when will two nodesthat random direction/random waypoint mobility models are not be within each other’s range, (ii) and when is a transmissionrealistic [28, 29]. So, we also analyze these routing schemes for between two nodes successful. (Note that we define two nodes tothe more realistic community-based mobility model proposed by be within range if the packets they send to each other is receivedSpyropoulos et al [20]. The analysis for the community-based successfully with a non-zero probability.) If one assumes a simplemobility model is similar to the derivations for the random di- distance-based attenuation model without any channel fading orrection/random waypoint mobility models. (Note that we include interference from other nodes, then two nodes can successfullythe analysis for the random direction/random waypoint mobility exchange packets without any loss only if the distance betweenmodels because it is simpler, easier to understand and naturally them is less than a deterministic value K (also referred to asextends to the derivations for the more complicated community- the transmission range), else they cannot exchange any packet atbased mobility model.) all. The value of K depends on the transmission power and the Note that other papers have studied the performance of these distance attenuation parameter. However, in presence of a fadingrouting schemes without contention in the network. For ex- channel and interference from other nodes, even though two nodesample, [11, 20] studied the performance of direct transmission, can potentially exchange packets if the distance between them is[19–21] studied epidemic routing, and [12, 17] studied different less than K , a transmission between them might not go through.spraying based schemes. [24, 30] are preliminary efforts of ours A transmission is successful only when the signal to interferenceto analyze the performance of routing schemes under contention. ratio (SIR) is greater than some desired threshold.Specifically, [24] studies the expected delay of epidemic routing We assume the following radio model: (i) Two nodes are withinunder the random walk mobility model and [30] studies random- each other’s range if the distance between them is less than K ,ized flooding and a spraying based scheme under the random and (ii) any transmission between the two is successful only if thewaypoint mobility model. Here, we generalize our prior work SIR is greater than a desired threshold Θ. Note that this model isand provide results for more efficient routing schemes under a not equivalent to a circular disk model because any transmissionmore realistic mobility model. between two nodes with a distance less than K is successful with Finally, we use these delay expressions to demonstrate that a certain probability that depends on the fading channel modeldesigning routing schemes using analytical expressions which and the amount of interference from other nodes.ignore contention can lead to inaccurate and misleading results. 2) Channel Model: The analysis works for any channel model.Specifically, we choose to study how to optimally design spraying 3) Traffic Model: Each node acts as a source sending packetsbased schemes, since it has been shown that they have superior to a randomly selected destination.performance [17]. We compare the design decisions that resultfrom analysis with and without contention, and highlight the B. Mobility Modelscenarios where ignoring contention leads to suboptimal or even We will first present the delay analysis for the random di-erroneous decisions. rection/random waypoint mobility models [31] which are the The outline of this paper is as follows: Section II presents most commonly used mobility models for analysis as well as forthe network model used in the analysis. Section III presents simulations. But, the real world mobility traces show that mobilitythe framework to incorporate contention in performance analysis, models which assume that all nodes are homogeneous and moveand then Sections IV and V find the expected delay expressions randomly all around the network, like the random direction andfor different mobility-assisted routing schemes for the random the random waypoint mobility models, are not realistic [28, 29].direction/random waypoint and community-based mobility mod- Nodes usually have some locations where they spend a largeels. Section VI studies the impact of some approximations made amount of time. Additionally, node movements are not identicallyduring the analysis on its accuracy by comparing the analytical distributed. Different nodes visit different locations more often,results to simulation results. Section VII then uses the expressions and some nodes are more mobile than others. Based on thisderived in the previous sections to demonstrate the inaccura- intuition, Spyropoulos et al [20] proposed a more realistic andcies introduced by ignoring contention in the design of routing analytically tractable community-based mobility model. Later,schemes. Finally, Section VIII concludes the paper. Hsu et al [32] showed that the statistics of real traces match
  • 3with a time varying version of this community-based mobility studied in [34]. Nodes which share the same community have dif-model further proving that this model captures real world mobility ferent statistics than nodes which belong to different communities.properties. So, we also present the delay analysis for different (Its easy to see that nodes which share the same community meetmobility-assisted routing schemes for the community-based mo- faster and stay in contact for a longer duration.) The two importantbility model. properties which we use during the course of the analysis are 1) Community-based Mobility Model: We first define the fam- as follows: (i) The expected meeting time for nodes belongingily of Community-based mobility models: The model consists of to different communities is equal to the inter-meeting time fortwo states, namely the ‘local’ state and the ‘roaming’ state. The these nodes. However, note that the expected meeting and inter-model alternates between these two states. Each node inside the meeting times for nodes belonging to the same community arenetwork moves as follows: (i) Each node i has a local community not equal. (ii) Even though the overall statistics of the meetingCi of size Ci = c2 N, c ∈ (0, 1]. A node’s movement consists of and inter-meeting times for a community-based mobility modellocal and roaming epochs. (ii) A local epoch is a random direction is not exponential, after conditioning on whether the two nodesmovement restricted inside area Ci . (iii) A roaming epoch is a under consideration share the same community or not, the tail ofrandom direction movement inside the entire network. (iv) If the the distribution of the meeting and inter-meeting times becomesprevious epoch of the node was a local one, the next epoch is a exponential.local one with probability pl , or a roaming epoch with probability1 − pl . (v) If the previous epoch of the node was a roaming one, III. C ONTENTION A NALYSISthe next epoch is a roaming one with probability pr , or a local This section introduces a framework to analyze any routingone with probability 1 − pr . scheme for ICMNs with contention in the network. We first The Community-based mobility model can be used to model identify the three manifestations of contention. Even though thea large number of scenarios by tuning its parameters. We choose proposed framework will work for any mobility model in whicha specific scenario closely resembling reality where there are the process governing the mobility of nodes is stationary and ther small communities. These communities are assumed to be movement of each node is independent of each other, for easesmall enough such that all nodes within a community are within of presentation, we first present it for a mobility model with aeach other’s range. We also assume that the nodes spend most uniform node location distribution in Section III-B (commonlyof their time within their respective communities. This scenario used mobility models like random direction and random waypointcorresponds to the real scenario of different nodes sharing fixed on a torus satisfy this assumption [20, 35]). We then show howcommunities like several office buildings on a campus or several to extend it to mobility models with a non-uniform node locationconference rooms in a hotel, which is more realistic than a distribution by presenting the framework for the community-basedscenario where all nodes choose their community uniformly at mobility model in Section III-C.random from the entire network. 2) Mobility Properties: We now define three properties of a A. Three Manifestations of Contentionmobility model. The statistics of these three properties will beused in the delay analysis of different mobility-assisted routing Finite Bandwidth: When two nodes meet, they might have moreschemes. than one packet to exchange. Say two nodes can exchange sBW (i) Meeting Time: Let nodes i and j move according to a packets during a unit of time. If they move out of the range ofmobility model ‘mm’ and start from their stationary distribution each other, they will have to wait until they meet again to transferat time 0. Let Xi (t) and Xj (t) denote the positions of nodes i more packets. The number of packets which can be exchanged inand j at time t. The meeting time (Mmm ) between the two nodes a unit of time is a function of the packet size and the bandwidth ofis defined as the time it takes them to first come within range of the links. We also assume that the sBW packets to be exchangedeach other, that is Mmm = mint {t : Xi (t) − Xj (t) ≤ K}. are randomly selected from amongst the packets the two nodes (ii) Inter-Meeting Time: Let nodes i and j start from within want to exchange1 .range of each other at time 0 and then move out of range of each Scheduling: We assume a CSMA-CA scheduling mechanism isother at time t1 , that is t1 = mint {t : Xi (t) − Xj (t) > K}. in place which avoids any simultaneous transmission within one + hop from the transmitter and the receiver. Nodes within rangeThe inter meeting time (Mmm ) of the two nodes is defined as thetime it takes them to first come within range of each other again, of each other and having at least one packet to exchange are + assumed to contend for the channel. For ease of analysis, wethat is Mmm = mint {t − t1 : t > t1 , Xi (t) − Xj (t) ≤ K}. (iii) Contact Time: Assume that nodes i and j come within also assume that time is slotted. At the start of the time slot, allrange of each other at time 0. The contact time τmm is defined node pairs contend for the channel and once a node pair capturesas the time they remain in contact with each other before moving the medium, it retains the medium for the entire time slot. (Seeout of the range of each other, that is τmm = mint {t − 1 : Section III-B for more details.) Xi (t) − Xj (t) > K}. Interference: Even though the scheduling mechanism is ensuring The statistics of the meeting time, inter-meeting time and that no simultaneous transmissions are taking place within onecontact time for the random direction/random waypoint mobility hop from the transmitter and the receiver, there is no restrictionmodels are studied in [20, 33]. The two important properties on simultaneous transmissions taking place outside the schedulingsatisfied by both these mobility models, which we use during 1 The instantaneous unfinished work in the queue in bits will be the samethe course of the analysis are as follows: (i) the expected inter- for any work conserving queue service discipline like FIFO, random queueingmeeting time is approximately equal to the expected meeting time, and LIFO. Hence, for constant size packets, the throughput and the expectedand (ii) the tail of the distribution of the meeting and the inter- queueing delay at each queue will also remain the same for all the three schemes. In practice, note that one might use priority queueing instead ofmeeting times is exponential. these traditional queue service disciplines to reduce the overall end-to-end These statistics for the community-based mobility model are delay, for example see the experimental results in [36].
  • 4area. These transmissions act as noise for each other and hence (i) Finite Bandwidth Ebw Event that finite link bandwidth allows exchangecan lead to packet corruption. of packet A In the absence of contention, two nodes would exchange all Event that i and j want to exchange s otherthe packets they want to exchange whenever they come within S Es packets given there are S distinct packets inrange of each other. Contention will result in a loss of such the systemtransmission opportunities. This loss can be caused by either of pR ex Probability that nodes i and j want to exchange a particular packet for routing scheme Rthe three manifestations of contention. In general, these three (ii) Schedulingmanifestations are not independent of each other. We now propose Esch Event that scheduling mechanism allows i and ja framework which uses conditioning to separate their effect and to exchange packetsanalyze each of them independently. Ea Event that there are a nodes within one hop from the transmitter and the receiver Ec Event that there are c nodes within two hops butB. The Framework not within one hop from the transmitter and the receiver Expected number of possible transmissions whose Lets look at a particular packet, label it packet A. Suppose two t(a, c) transmitter is within 2K distance from thenodes i and j are within range of each other at the start of a time transmitterslot and they want to exchange this packet. Let ptxS denote the ppkt Probability that two nodes have at least oneprobability that they will successfully exchange the packet during packet to exchangethat time slot. First, we look at how the three manifestations of (iii) Interference Einter Event that transmission of packet A is notcontention can cause the loss of this transmission opportunity. corrupted due to interference Finite Bandwidth: Let Ebw denote the event that the finite Event that packet A is successfully exchangedlink bandwidth allows nodes i and j to exchange packet A. The EM −a inspite of the interference from M − a nodesprobability of this event depends on the total number of packets outside the scheduling area E[x] Average number of interfering transmissionswhich nodes i and j want to exchange. Let there be a total of Sdistinct packets in the system at the given time (label this event TABLE IIES ). Let there be s, 0 ≤ s ≤ S − 1, other packets (other than N OTATION USED IN S ECTION III-Bpacket A) which nodes i and j want to exchange (label this event S the event that packet A is successfully exchanged inspite ofEs ). If s ≥ sBW , then the sBW packets exchanged are randomlyselected from amongst these s + 1 packets. Thus, P (Ebw ) = “P ” the interference caused by these M − a nodes as EM −a . Then,P sBW −1 PS−1 P (Es ) + s=sBW ss+1 P (Es ) . To sim- S S P (Einter | Ea ) = P (EM −a ). S P (ES ) BW s=0plify the analysis, we make our first approximation here by Packet A will be successfully exchanged by nodes i and j only if the following three events occur: (i) the scheduling mechanismreplacing the random variable S by its expected value in the allows these nodes to exchange packets, (ii) nodes i and j decideexpression for P (Ebw )2 . (Note that simulations results presented to exchange packet A from amongst the other packets they want toin Section VI verify that this approximation does not have a exchange, and (iii) this transmission does not get corrupted duedrastic effect on the accuracy of the analysis.) to interference from transmissions outside the scheduling area. Scheduling: Let Esch denote the event that the scheduling Thus,mechanism allows nodes i and j to exchange packets. The X ptxS = P (Ebw ) P (Ea , Ec )P (Esch | Ea , Ec )P (Einter | Ea )scheduling mechanism prohibits any other transmission within a,cone hop from the transmitter and the receiver. Hence, to find 0 1 sBW −1 E[S]−1 E[S]P (Esch ), we have to determine the number of transmitter-receiver X E[S] X sBW P (Es ) =@ P (Es ) + Apairs which have at least one packet to exchange and are contend- s+1 s=0 s=s BWing with the i-j pair. Let there be a nodes within one hop from X P (Ea )P (Ec | Ea )P (EM −a )the transmitter and the receiver (label it event Ea ) and let there × . (1)be c nodes within two hops but not within one hop from the a,c t(a, c)transmitter and the receiver (label it event Ec ). These c nodeshave to be accounted for because a node at the edge of the Note that even though the proposed framework models the im-scheduling area can be within the transmission range of these portant factors contributing to contention like bandwidth restric-nodes and will contend with the desired transmitter/receiver pair. tions, random access scheduling, fading effects and interferenceLet t(a, c) denote the expected number of possible transmissions from multiple nodes, it does not model everything. Specifically,contending with the i-j pair. By symmetry, all the contending it does not incorporate losses due to packet collisions and lossesnodes are equally likely to capture the channel. So, P (Esch | due to finite queue buffers. Even though it would be doableEa , Ec ) = 1/t(a, c). to incorporate these two effects in the framework 3 , this would Interference: Let Einter denote the event that the transmission complicate the analysis quite a bit. Given that the importantof packet A is not corrupted due to interference given that nodes aspects of contention in our setting have already been included ini and j exchanged this packet. Let there be M − a nodes outside the framework, we choose not to further complicate the model.the transmitter’s scheduling area (this is equivalent to event Ea ). 3 (i) Collisions: The collision probability of a link depends on the localIf two of these nodes are within the transmission range of each topology around that link. One could use results from [37, 38] to find theother, then they can exchange packets which will increase the collision probability for a given topology for practical CSMA-CA schemesinterference for the transmission between i and j . Lets label like 802.11, then find the probability of the topology occurring in the network and finally remove the condition on the topology by using the law of total 2 We incorporate the arrival process through E[S] in the analysis. E[S] probability. (ii) Finite buffers: The expected number of packets in the queuedepends on the arrival rate through Little’s Theorem. Thus, after deriving the of a node can be easily calculated as a function of E[S]. Then well knownexpected end-to-end delay for a routing scheme in terms of E[S], Little’s bounds like the Chernoff bound could be used to find the probability that theTheorem can be used to express the delay in terms of only the arrival rate. number of packets in the queue exceed the buffer size.
  • 5 Lemma 3.2: t(a, c) = 1 + pa ppkt a − 1 +acpc ppkt where ` `` ´ ´´ Next, we find expressions for the unknown values in Equation 2 RR A3 (x,y)(1). pa = x,y A1 f (x, y)dxdy is the probability that two nodes 1) Finite Bandwidth: To account for finite bandwidth, we have are within range of each other given that both of them are in E[S] RR πK 2 −A3 (x,y)to find P (Es ) (the probability that nodes i and j have s other the scheduling area, pc = x,y A2 f (x, y)dxdy is thepackets to exchange given there are E[S] distinct packets in the probability that two nodes are within range of each other givensystem). Let pR be the probability that nodes i and j want to ex that one of them is within the scheduling area and the other nodeexchange a particular packet for the routing scheme R. Now, since is outside the scheduling area but within two hops ´ from either E[S]there are E[S] − 1 packets other than packet A in the network, the transmitter or the receiver, ppkt = 1 − 1 − pR ` „ « ex is the E[S] E[S] − 1 ` R ´s ` ´E[S]−s−1 probability that two nodes have at least one packet to exchange,P (Es )= pex 1 − pR ex . The value s f (x, y)is the probability that a node within the scheduling areaof pR depends on the routing mechanism at hand because which ex is at a distance x and y from the transmitter and the receiverpackets should the two nodes exchange is dictated by the routing respectively, and A3 (x, y) is the average value of the area withinpolicy. Note that this is the only term affected by the routing the scheduling area and within one hop from a node at a distancemechanism in the analysis. We will derive its value for different x and y from the transmitter and the receiver (see Figure 2). Werouting mechanisms in Section IV. state the value of f (x, y) and A3 (x, y) in the proof. 2) Scheduling: To account for scheduling, we have to figure Proof: See Appendix.out P (Ea ) (the probability that there are a nodes within the 3) Interference: The interference caused by other nodes de-scheduling area), P (Ec | Ea ) (the probability that out of the pends on the number of simultaneous transmissions and theremaining M − a nodes, there are c nodes within two hops from distance between the transmitters of these simultaneous trans-the transmitter or the receiver but not in the scheduling area) and missions and the desired receiver. Given that there are M − at(a, c) (the expected number of possible transmissions competing nodes outside the scheduling area (event Ea ), let there be x interfering transmissions at a distance of r1 , r2 , . . . , rx from thewith the i-j pair). desired receiver. Then, using the law of total probability, we get X X P (EM −a ) = P (EM −a | x, r1 , r2 , . . . rx )× x r1 ,r2 ,...,rx Tx x Rx P (r1 , r2 , . . . , rx | x)P (x). (2) y u1 While it is possible to calculate P (x) to substitute in the ex- pression of P (EM −a ), the resulting expression will be very complicated. Motivated by this, we replace x by its expected A3 (x,y) value. (Simulations results presented in Section VI verify thatFig. 2. Tx and Rx denote the transmitter and the receiver. Node u1 is a this approximation does not have a drastic effect on the accuracynode within the scheduling area and at a distance x from the transmitter and of the analysis.) First, we compute E[x]. There are M2 ` −a´at a distance y from the receiver. The shaded area represents A3 (x, y). possible pairs of nodes, and for a particular pair of nodes to interfere with the Each of the other M − 2 nodes (other than i and j ) are equally transmission between i and j , they should be within range of eachlikely to be anywhere in the two dimensional space because other, have at least one packet to exchange, and the schedulingthe mobility model has a uniform stationary distribution. So, we mechanism should allow them to exchange packets. Hence, theuse geometric arguments to figure out how many transmissions 2 expected number of interfering transmissions equals πK ppktcontend with the transmission between i and j . “P ”` M −a N 1 ´ Lemma 3.1: P (Ea ) = M −2 (p1 )a−2 (1 − p1 )M −a where p1 = a,c t(a,c) P (Ea )P (Ec | Ea ) . ` ´ 2 a−2A1 is the probability that a particular node lies within the Now we compute f (r) which denotes the probability density Nscheduling area, which ” an average value equal to A1 = has function of the distance between any two nodes. (Since each node“ √ is moving independently of each other, f (r) is the same for all 2π + 4 9 2 − 2cos−1 1 K 2 . ` ´ 3 the nodes.) The following equation states the expression for f (r) Proof: The node is equally likely to be anywhere in the for a torus of area N :two dimensional space. Consequently, p1 = Pr[a particular node ( 2πr √ N N r≤ 2is within one hop of either the transmitter or the receiver] = f (r) = 4r “ π “ √ ”” N √ N √ . (3)Pr[a particular node is within one hop of the transmitter] + Pr[a N 2 − 2cos −1 2r 2 < r < √N2particular node is within one hop of the receiver] - Pr[a particular P (EM −a | x, r1 , r2 , . . . rx ) is the complement of the outagenode is within one hop of both the transmitter and the receiver]. probability and depends on the channel model. The channel modelReplacing the distance between the transmitter and the receiver “ √ ” 2π+ 4 2 −2cos−1 ( 1 ) K 2 only affects this term in the entire analysis. The outage proba- 9 3by its expected value yields p1 = N . Recall bilities have been calculated for several realistic channel modelsthat nodes i and j are already within the scheduling area. So, „ « including the Rayleigh-Rayleigh fading channel [39] (both the M −2 desired signal and the interfering signal are Rayleigh distributed),P (Ea ) = (p1 )a−2 (1 − p1 )M −a . a−2 the Rician-Rayleigh fading channel [40] (the desired signal has Corollary 3.1: P (Ec | Ea ) = Mc (p2 )c (1 − p2 )M −a−c ` −a´ Rician and the interfering signal has Rayleigh distribution), andwhere p2 = A2N 1 is the probability that a particular −A the superimposed Rayleigh fading and log normal shadowingnode lies within two hops from either the transmitter or channel [41]. The results from these papers can be directly used“ receiver but not ` ´” the scheduling area, and A2 =the √ within here to make the framework work for any of these channel 2 35 8π + 9 − 8cos −1 1 6 K 2 is the average value of the area models. The following lemma uses the result from [39] to derivewithin two hops from either the transmitter or the receiver. P (EM −a ) for the Rayleigh-Rayleigh fading channel model.
  • 6 a+k − a − 1 + pa a + acpc . ``` ´ ` ´ ´ ` ´ ´ Lemma 3.3: For the Rayleigh-Rayleigh fading channel model, „ «−E[x] (c) t(a, c, k) = 1 + ppkt 2 2 2 4 R Θ( 2K ) 3 Proof: The probability of loss due to finite bandwidth isP (EM −a ) = r 1 + 4 f (r)dr. r derived using the same arguments made in Section III-B.1. To Proof: Kandukuri et al [39] evaluated the outage prob- derive the probability of loss due to scheduling, we have to Rayleigh-Rayleigh fading channel to be 1 −ability“ for the ” find the number of nodes within the scheduling area, while to R −1Qx ΘPi R i=1 1 + P R , where P0 is the received power from the derive the probability of loss due to interference, we have to 0desired signal and PiR is the received power from the ith find the number of simultaneous transmissions within the networkinterferer. Assuming all the nodes are transmitting at the same and the distance between the transmitters of these simultaneouspower level and α = 4 in the “distance ”attenuation model, transmissions and the desired receiver. The proof of this lemma Θr04 −1 is based on the following observation: All nodes within theP (EM −a | x, r1 , r2 , . . . rx ) = x Q i=1 1 + r 4 , where r0 is the community are within the scheduling area while the remaining idistance between nodes i and j . Replacing r0 and x with their nodes will be uniformly distributed over the entire network.expected values and removing the condition on ri ’s by using the (a) The probability that a particular node is in the local state orlaw of total probability yields the result. the roaming state (πl and πr ) was derived in [20]. Since each of Now, we have all the components to put together to find ptxS these nodes in moving independently of each other, the numberin Equation (1). In Sections IV and V, we present case studies to of nodes in their local states is binomially distributed. Since thedemonstrate how the framework is used for performance analysis transmitter and the receiver are in their local states (as stated ` M −2´ k−2 M −kof routing schemes. earlier), P (Ek ) = k−2 πl πrr . r (b) The remaining M − k nodes are uniformly distributed overC. Extension: The Framework for the Community-based Mobility the entire network. The probability that a of these nodes are withinModel the scheduling area (P (Ea | Ek )) and the probability that c of Equation (1) is independent of the mobility model, and hence these nodes are within the two hops from the transmitter or thestill holds. But, to extend the framework to a mobility model with receiver but not within the scheduling area ( P (Ec | Ea , Ek )) isa non-uniform node location distribution, the values of P (Ea ), derived using the same arguments as used in Lemma 3.1 to be equal to P ` a | Ek ) = Ma (p1 )a−2 (1 − p1 )M −a and P (Ec | ` −k´P (Ec | Ea ), t(a, c) and P (EM −a ) will have to be re-derived. In (E M −k−a (p2 ) (1 − p2 )M −a−c . p1 and p2 were defined c ´general, these expressions will be evaluated after conditioning on Ea , Ek ) = cthe current transmitter and receiver location, and then, the law of and derived in Lemma 3.1 and Corollary 3.1 respectively. Finally,total probability would be used to remove the condition. note that there are M − k − a nodes outside the scheduling area, For the community-based mobility model described in Sec- which if within range of each other, are allowed to simultaneouslytion II-B, we will condition over whether the current transmitter transmit. Since, these M − k − a nodes are uniformly distributedand receiver belong to the same community or to different over the entire network, the probability of loss due to interferencecommunities and whether they meet within a community or is equal to P (EM −a−k ) whose value was derived in Section III-outside. For nodes belonging to the same community who meet B.3.within their common community, let pR denote the probability txS1 (c) The k nodes within the community are within each other’sthat these two nodes are able to successfully exchange a particular range. The a nodes within the scheduling area but not within thepacket inspite of contention. The probability of nodes belonging community, will be within the transmission range of the k nodesto different communities meeting within a community is negligi- within the community. However, they will be within each other’sble as the communities are very small. (Similarly, the probability transmission range with probabiity pa (defined and derived inof nodes belonging to the same community meeting within a Lemma 3.2). The c nodes within two hops from the transmittercommunity not their own is also negligible.) If two nodes do not or the receiver but not within the scheduling area will not liemeet within any community, let pR denote the probability that txS2 within range from the k nodes within the community, howeverthese two nodes are able to successfully exchange a particular they will be within range of the a nodes within the schedulingpacket inspite of contention (irrespective of whether the two area with probability pc (defined and derived in Lemma 3.2).nodes belong to the same community or different communities). All the node pairs within each other’s range will contend with RThe following lemma derives the value of ptxS1 . For ease of the desired transmission only if they have at least one packet topresentation, the following lemma assumes that the number of exchange (probability of this event is equal to ppkt and its valuenodes sharing a community is equal to M , and the position of was derived in Lemma 3.2). Putting ´ everything together yields r ```a+k´ `a´ − 2 − 1 + pa a + acpc . ` ´ ´each community is chosen uniformly at random from the entire t(a, c, k) = 1 + ppkt 2 2network. “P The value of pR is also derived in a similar fashion. txS2 sBW −1 E[S] PE[S]−1 BW Lemma 3.4: pR txS1 = s=0 P (Es ) + s=sBW ss+1 ” „P M P IV. D ELAY A NALYSIS FOR P OPULAR M OBILITY M ODELS E[S] 1P (Es ) × r k=2 a,c P r(Ek ) t(a,c,k) P (Ea | Ek )P (Ec | Ea , Ek ) In this section, we find the expected end-to-end delay ofP (EM −a−k )), where: four different mobility-assisted routing schemes for intermittently(a) Ek is the event that there are k nodes in the community. connected mobile networks, when nodes move according to the ` M −2´ k−2 M −k 1−pr random direction or random waypoint mobility models. For each P (Ek ) = k−2 πl πrr r where πl = 2−pl −pr is the routing scheme R, we first define the routing algorithm, then probability that a particular node is in the local state and 1−pl derive the value of pR and finally derive the value of the expected ex πr = 2−pl −pr is the probability that a particular node is in 4 end-to-end delay . Parameters that depend on the mobility model, the roaming state. are denoted by using ‘mm’ as a super- or sub-script.(b) P (Ea | Ek ), P (Ec | Ea , Ek ) and P (EM −a−k ) are derived in a manner similar to the derivation of the corresponding 4 Note that pR is the only parameter in the framework which depends on ex probabilities in Section III-B.2. the routing scheme.
  • 7A. Direct Transmission nodes not having a copy, there is a transmission opportunity to Direct transmission is one of the simplest possible routing increase the number of copies by one. Since ICMNs are sparseschemes. Node A forwards a message to another node B it networks, we look at the tail of the distribution of the meetingencounters, only if B is the message’s destination. We now time which is exponential for both the random direction andanalyze its performance with contention. the random waypoint mobility models. The time it takes for Lemma 4.1: pdt = M (M −1) . ex 2 one of the m nodes to meet one of the other M − m nodes Proof: In direct transmission, each packet undergoes only is equal to the minimum of m(M − m) exponentials, whichone transmission, from the source to the destination. A packet is again an exponential random variable with mean m(Mmm ] . E[M −m) 1has node i as its source with probability M . The probability that Now when they meet, the probability that the two nodes are able epidemicj is the destination given i is the source is M1 (the destination to successfully exchange the packet is psuccess . If they fail to −1is chosen uniformly at random from amongst the other M − 1 exchange the packet, they will have to wait one inter-meeting +nodes). Thus, the probability that i and j want to exchange a time to meet again. But, since E[Mmm ] = E[Mmm ] for both the 2particular packet is equal to M (M −1) (i is the source and j is the random direction and the random waypoint mobility model, anddestination or vice versa). both meeting and inter-meeting times have memoryless tails, the mm Theorem 4.1: Let E[Ddt ] denote the expected delay of direct expected time it takes for one of the m nodes to meet one oftransmission. Then, E[Ddt ] = E[Mmm ] , where E[Mmm ] is the mm the other M − m nodes again is still equal to m(Mmm ] . Hence, E[M pdt success −m)expected meeting time of the mobility model ‘mm’, pdt success = E[Depidemic (m)] = pepidemic m(Mmm ] +2pepidemic (1−pepidemic ) mm success E[M −m) success success ´E[τmm ] E[Mmm ] E[Mmm ] epidemic1− 1 − pdt ` txS is the probability that when two nodes come m(M −m) + ... = epidemic . The value of psuccess can m(M −m)psuccesswithin range of each other, they successfully exchange the packet be derived in a manner similar to the derivation of pdt success inbefore going out of each other’s range (within the contact time Theorem 4.1.τmm ). Now, we find the value of pepidemic and then find the expected ex Proof: The expected time it takes for the source to mm end-to-end delay for epidemic routing (denoted by E[Depidemic ]).meet the destination for the first time is E[Mmm ] (the ex- Lemma 4.3: pexepidemic = PM −1 2m(M −m) PM −1 1 m=1 M (M −1) i=m M −1pected meeting time). With probability 1 − pdt , the two nodes txS 1 m(M −m)are unable to exchange the packet in one time slot due to Pi 1 . j=1 j(M −j)contention. They are within range of each other for E[τmm ] Proof: Let there be m copies of a particular packet in thenumber of time slots. (We are making an approximation here network. Then the probability that node i has a copy is equal toby replacing τmm by its expected value.) Thus pdt success = m and the probability that node j does not have a copy given ´E[τmm ] M 1 − pdt` txS is the probability that the source and the that node i has one is equal to (M −m) . Thus, the probability that M −1destination fail to exchange the packet while they are within nodes i and j want to exchange the packet given that there are mrange of each other. Then they will have to wait for one inter- copies of the packet in the network is equal to 2m(M −m) . Now, M (M −1)meeting time to come within range of each other again. If they we find the probability that there are m copies of the packet infail yet again, they will have to wait another inter-meeting time the network. The copies of a packet keep on increasing till the mm dt` come within range. Thus, E[Ddt ] = E[Mmm ] + psuccessto + 2 + packet is delivered to the destination. The probability that the (1 − pdt dt ´ success )E[Mmm ] + 2(1 − psuccess ) + E[Mmm ] + . . . = destination is the kth node to receive a copy of the packet is (1−pdt success )E[Mmm ]E[Mmm ] + pdt . Since + E[Mmm ] = E[Mmm ] for equal to M1 for 2 ≤ k ≤ M . A packet will have m copies −1 successboth random direction and random waypoint mobility models, in the network only if the destination wasn’t amongst the firstE[Ddt ] evaluates to E[Mmm ] . mm pdt m − 1 nodes to receive a copy. The amount of time a packet success mm has m copies in the network is equal to E[Depidemic (m)]. Hence,B. Epidemic Routing the probability that there are m copies of a packet in the network mm E[Depidemic (m)] equals M −1 M1 Pi E[Dmm P Epidemic routing [8] extends the concept of flooding to i=m −1 (j)] . Applying the law of total j=1 epidemicICMN’s. It is one of the first schemes proposed to enable message probability over the random variable m and substituting the valuedelivery in such networks. Each node maintains a list of all of E[Depidemic (m)] from Lemma 4.2 gives pepidemic . mm exmessages it carries, whose delivery is pending. Whenever it Theorem 4.2: P E[Mmm ] E[Depidemic ] = M −1 M1 mm Piencounters another node, the two nodes exchange all messages i=1 m=1 epidemic . −1 m(M −m)psuccessthat they don’t have in common. This way, all messages are Proof: The probability that the destination is the ith nodeeventually spread to all nodes. The packet is delivered when the to receive a copy of the packet is equal to M1 for 2 ≤ i ≤ M .first node carrying a copy of the packet meets the destination. The −1 The amountP time it takes for the ith copy to be delivered ofpacket will keep on getting copied from one node to the other is equal to i mm m=1 E[Depidemic (m)]. Applying the law of totalnode till its Time-To-Live (TTL) expires. For ease of analysis, we probability over the random variable i and substituting the valueassume that as soon as the packet is delivered to the destination, mm mm of E[Depidemic (m)] from Lemma 4.2 yields E[Depidemic ].no further copies of the packet are spread. To find the expected end-to-end delay for epidemic routing, we mmfirst find E[Depidemic (m)] which is the expected time it takes for C. Spraying a small fixed number of copiesthe number of nodes having a copy of the packet to increase fromm to m + 1. Another approach to route packets in sparse networks is that of mm Lemma 4.2: E[Depidemic (m)] = E[Mmm ] controlled replication or spraying [12, 13, 22, 23]. A small, fixed epidemic , where “ ”E[τmm ] m(M −m)psuccess number of copies are distributed to a number of distinct relays.pepidemic = 1 − 1 − pepidemic success txS . Then, each relay routes its copy independently towards the des- Proof: When there are m copies of a packet in the network, tination. By having multiple relays routing a copy independentlyif one of the m nodes having a copy meets one of the other M −m and in parallel towards the destination, these protocols create
  • 8 „ mm « PL−1 2m(M −m) PL f sw E[Df sw (m)]enough diversity to explore the sparse network more efficiently m=1 M (M −1) i=m pdest (i) i P mm , k=1 E[Df sw (k)]while keeping the resource usage per message low.  1 1≤i<L Different spraying schemes may differ in how they distribute where pf sw (i) = dest M −1 M −L is the probability thatthe copies and/or how they route each copy. We study two M −1 i=L thdifferent spraying based routing schemes here. These two differ the destination is the (i + 1) node to receive a copy of thein the way they distribute their copies. packet. 1) Source Spray and Wait: Source spray and wait is one of Proof: The proof runs along the same lines as the proof ofthe simplest spraying schemes proposed in the literature [12]. For Lemma 4.3. Theorem 4.4: E[Df sw ] = L pdest (i) i mm f sw mm P Pthis scheme, the source node forwards all the copies (lets label i=1 m=1 E[Df sw (m)].the number of copies being sprayed as L) to the first L distinct Proof: The proof runs along similar lines as the proof ofnodes it encounters. (In other words, no other node except the Theorem 4.2.source node can forward a copy of the packet.) And, once these V. D ELAY A NALYSIS OF ROUTING S CHEMES WITH THEcopies get distributed, each copy performs direct transmission. C OMMUNITY- BASED M OBILITY M ODEL First, we find the value E[Dssw (m)], then we find pssw and mm ex In this section, we derive the expected delay values for fourfinally, we derive the expected end-to-end delay for source spray mm different mobility-assisted routing schemes with the community-and wait (denoted by E[Dssw ]). ( E[Mmm ] based mobility model. We first analyze direct transmission and mm (M −1)pssw 1≤m<L Lemma 4.4: E[Dssw (m)] = success E[Mmm ] epidemic routing as these two form the basic building block for Lpssw m=L all routing schemes. Then, we analyze two different spraying successwhere pssw success = 1 − (1 − ptxS )E[τmm ] . ssw based schemes: fast spray and wait and fast spray and focus Proof: See Appendix. which differ in the way they route each individual copy towards 2Lpssw (L) mm “ ” E[Dssw (L)] Lemma 4.5: pssw ex = dest M (M −1) PL mm E[Dssw (k)] + the destination after the spray phase. Note that the value of pR ex“ mm ”k=1 for each routing scheme remains the same as derived in Section 2 PL−1 PL ssw E[Dssw (m)] M −1 m=1 i=m pdest (i) i E[D mm (k)] , where P 8 “Q k=1 ” ssw IV. The derivation of the expected delay for the community- i−1 M −j−1 i < 1≤i<L based mobility model uses arguments similar to the ones usedpssw (i) = “j=1 M −1 M −1 is the dest : Qi−1 M −j−1 ” i=L in the derivation of the expected delay for the random direction j=1 M −1 / random waypoint mobility model. The proofs which are veryprobability that the destination is the (i + 1)th node to receive a similar are not discussed to keep the exposition interesting. Tocopy of the packet. simplify the presentation in this section, we assume that the Proof: The proof runs along the same lines as the proof of number of nodes sharing a community is equal across all rLemma 4.3. communities, that is the number of nodes sharing a community is Theorem 4.3: E[Dssw ] = L pdest (i) m=1 E[Dssw (m)]. mm P ssw Pi mm i=1 equal to M . Finally, we define the notation related to the statistics r Proof: The proof runs along similar lines as the proof of of the mobility properties for the community-based mobilityTheorem 4.2. + model. Let E[Mcomm,same ] (E[Mcomm,dif f ]), E[Mcomm,same ] 2) Fast Spray and Wait: In Fast Spray and Wait, every relay + (E[Mcomm,dif f ]) and E[τcomm,same ] (E[τcomm,dif f ]) denotenode can forward a copy of the packet to a non-destination node the expected meeting time, inter-meeting and contact time forwhich it encounters in the spray phase. (Recall that in source nodes which belong to the same community (belong to differentspray and wait, only the source node can forward copies to communities) respectively. Please refer to [34] for their exactnon-destination nodes.) There is a centralized mechanism which values.ensures that after L copies of the packet have been spread, nomore copies get transmitted to non-destination nodes. Note that A. Direct Transmissionthis is not a practical way to distribute copies, however we include comm Let E[Ddt ] denote the expected delay of direct transmissionit in the analysis because it spreads copies whenever there is any dt for the community-based mobility model. Further, let psuccess1opportunity to do so and hence has the minimum spraying time be the probability that when two nodes belonging to the samewhen there is no contention in the network. Once these copies community come within each other’s range, they successfullyget distributed, each copy performs direct transmission. We now exchange the packet before going out of each other’s range andderive the expected delay of fast spray and wait with contention let pdt success2 be the probability that when two nodes belongingin the network. to different communities come within each other’s range, they First, we find the value E[Df sw (m)], then we find pf sw and mm ex successfully exchange the packet before going out of each other’sfinally, we derive the expected end-to-end delay for fast spray and mm range.wait (denoted by E[Df sw ]). All the derivations are very similar comm (r−1)m E[Mcomm,dif f ] Theorem 5.1: E[Ddt ] = r(m−1) pdt +to the corresponding derivations for epidemic routing. The only „ + success2 « (1−pdt success1 )E[Mcomm,same ]difference is that when m = L nodes have a copy of the packet, a m−r E[Mcomm,same ] + , r(m−1) pdt success1transmission opportunity will arise only when one of these m = Lnodes meet the destination. 8 where pdt success1 = 1 − (1 − pdt )E[τcomm,dif f ] txS1 and < E[Mmm ] 1≤m<L pdt success2 = 1 − (1 − ptxS2 )E[τcomm,same ] . dt f sw mm m(M −m)psuccess Lemma 4.6: E[Df sw (m)] = E[Mmm ] Proof: The probability that the destination belongs to a : f sw m=L different community than the source is equal to (r−1)m . The Lp ”E[τmm ]success r(m−1) derivation of the expected delay after conditioning on whether the “ f swwhere pf sw success = 1 − 1 − ptxS . source and the destination belong to the same community or not Proof: The proof runs along the same lines as the proof of mm is similar to the derivation of E[Ddt ] in Theorem 4.1. Finally,Lemma 4.4. using the law of total probability to remove the conditioning „ f sw mm « 2Lpdest (L) E[Df sw (L)] Lemma 4.7: pf sw ex = M (M −1) PL E[D mm (k)] + comm yields E[Ddt ]. k=1 f sw
  • 9 E[Tm,mp ]B. Epidemic Routing of time the system remains in state (m, mp ) is equal to self . 1−pm,mp In a manner similar to the derivation of pself p , the probability m,m This section derives the expected delay of epidemic rout-ing for the community-based mobility model. Since each node that the system moves to (m,« p + 1) is derived to be (1 − „ m mp E[Tm,mp ]spends most of its time within its community (which implies pepidemic ) 1 − success1 + . The transmission is successful E[Mcomm,same ]E[Mcomm,dif f ] >> E[Mcomm,same ]), we make an approximation with probability pepidemic , in which case the system moves to the success1to simplify the exposition by assuming that with high probability, mp state (m + 1, mp − M −m ). Since each node not having a copya node starting from its stationary location distribution will first r “ ” mpmeet a node within its own community than a node belonging to of the packet has met on an average M −m nodes which have ra different community. This implies that once a node gets a copy a copy of the packet, when a new node receives the packet, thisof a packet, with high probability, all members of its community number has to be subtracted from mp .will get the copy before any node outside its community. A simple Now, we find the probability that the system will visit theoutcome of this is that the first M − 1 nodes to get a copy of the r state (m, mp ) (denoted by pm,mp ). The system can move mppacket belong to the source’s community. to state (m, mp ) from states (m − 1, mp + M −m−1 )(with r We first study how much time it takes for all nodes within probability pepidemic ) and (m, mp − 1) (with probability success1the source’s community to get a copy of the packet. This „ « (mp −1)E[Tm,m −1 ]derivation is different from all the derivations in Section IV (1 − pepidemic ) 1 − success1 + p ). Thus, E[Mcomm,same ] + 8 epidemicbecause E[Mcomm,same ] = E[Mcomm,same ]. Thus, we need to > psuccess1 pm−1,mp + mp M −m−1 >keep track of which pair of nodes have met in the past but were > > > „ r « > + 1 − (mp −1)E[Tm,mp −1 ] if m > 1 >unable to successfully exchange the packet. We model the system > + > E[Mcomm,same ] >using the following state space: (m, mp ) where 1 ≤ m ≤ M > > r (1 − pepidemic )pm,mp −1 > >is the number `of nodes which have a copy of the packet and > < success10 ≤ mp ≤ m M − m is the number of node pairs such that ´ pm,mp = „ « r > > (mp −1)E[Tm,mp −1 ]only one node of the pair has a copy of the packet, they have > > 1− + > > E[Mcomm,same ] if m = 1, mp > 0met at least once after the node (which has the copy) received its > (1 − pepidemic )pm,mp −1 > > > success1copy, and they were unable to successfully exchange this packet > > > >in their past meetings. Let E[Din (m)] denote the expected time > 1 if m = 1, mp = 0. :it takes for the number of nodes having a copy of the packet to Solving this set of linear equations yields pm,mp .increase from m to m + 1 given m < M (which implies that r comm Now, we find E[Depidemic (m)] which is the expected time itall nodes within the source’s community have not yet received a takes for the number of nodes having a copy of the packet tocopy of the packet). increase from m to m + 1. Pm( M −m) r E[Tm,mp ] Lemma 5.2: Lemma 5.1: E[Din (m)] = mp =0 pm,mp self , ( ` ` M ´´ E[Din rem m, ` M´ if rem m, =0 1−pm,mp r r commwhere E[Tm,mp ] is the expected time elapsed till one of the nodes E[Depidemic (m)] = E[Mcomm,dif f ] ` if rem m, M ´ =0 epidemic r m(M −m)psuccess2not having a copy meets a node having a copy of the packet “ ”E[τcomm,dif f ]given that the system is in state (m, mp ), pself p is the probability m,m where pepidemic = 1− 1 − ptxS2 success2 epidemic and rem (x, y)that the system remains in the state (m, mp ) after these nodes is the remainder left after dividing x by y .(which met after E[Tm,mp ]) are unable to successfully exchange Proof: As previously discussed, the first M − 1 nodes rthe packet, and pm,mp is the probability that the system visits to receive a copy of the packet are the nodes belonging tostate (m, mp ). the source’s community. Then, a node belonging to another Proof: Let the system be in state (m, mp ). We first derive community (lets label it community Y ) will receive a copy fromthe expected time duration after which the system moves to one of the nodes belonging to the source’s community. Afteranother state. A transmission opportunity will arise only when that, the next M − 1 nodes to get a copy of the packet are the rone of the m nodes carrying a copy of the packet meet one ones which belong to community Y . Even though there are otherof the M ` m not´ having a copy of the packet. There are a − nodes which have a copy of the packet (belonging to the source’s rtotal of m M − m such node pairs of which mp have already community), with high probability, the nodes in community Y rmet before. Since, both the meeting and inter-meeting times will receive a copy of the packet from a node belonging tohave exponential´ tails, the expected time elapsed till one of its own community. Thus, the expected time for the copies tothese m M − m node pairs come within range is E[Tm,mp ] = spread within community Y is equal to the expected time for the ` r„ m( M −m)−mp mp «−1 copies to spread within the source’s community. Similarly, the r E[Mcomm,same ] + + . If the two nodes which expected time for the copies to spread within any community E[Mcomm,same ]met are not able to successfully exchange the packet, then the after a node belonging to that community obtains a copy, issystem will remain in the same state if these two nodes were equal to the expected time for the copies to spread within theone of the mp node pairs which have already met at least once source’s community (irrespective of how many nodes outside thein the past, otherwise the system will move to (m, mp + 1). community have copies of the packet). Finally, for the scenarioThus, the probability that the system remains in the same state when for all communities, either all or no nodes in a community mp E[Tm,mp ]is pself p = (1 − pepidemic ) E[M + m,m success1 , where pepidemic = success1 have a copy of the packet, the expected time for the copies to ] “ ”E[τcomm,samecomm,same ] increase can be found in a manner similar to the derivation of1 − 1 − pepidemic txS1 . If the system remains in the mm E[Depidemic (m)] in Lemma 4.2.same state, then it will take yet another time duration equal to Finally, we derive the expected delay of epidemic routing forE[Tm,mp ] for a transmission possibility. Again, with pself p the m,m comm the community based mobility model (denoted by E[Depidemic ]) commsystem will remain in the same state. Thus, the expected amount in terms of E[Depidemic (m)] using the same argument used to
  • 10 1800 2000 400 400 Simulation (M=50) Simulation (M=50) Simulation (M=50) Simulation (M=50) Expected Delay (time slots) Expected Delay (time slots) Expected Delay (time slots) Expected Delay (time slots) 1600 Theoretical (M=50) Theoretical (M=50) Theoretical (M=50) Theoretical (M=50) 350 1400 Simulation (M=200) Simulation (M=200) Simulation (M=200) Simulation (M=200) 1500 300 Theoretical (M=200) Theoretical (M=200) Theoretical (M=200) Theoretical (M=200) 1200 300 1000 1000 250 200 800 200 600 500 100 400 150 200 0 100 0 5 6 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 K (distance units) K (distance units) K (distance units) K (distance units) (a) Direct Transmission. (b) Epidemic Routing. (c) Source Spray and Wait with (d) Fast Spray and Wait with L = L = 5. 5.Fig. 3. Simulation and analytical results for the expected delay for the random waypoint mobility model. Network parameters: N = 120 × 120 square units,Θ = 5, sBW = 1 packet/time slot. The expected maximum cluster size varies from 5%(K = 5, M = 50) to 23%(K = 10, M = 200). (Note that thenumber of instances for which each Monte Carlo simulation is run is chosen so as to ensure that the 90% confidence interval is within 5% of the simulationvalue.) mmderive E[Depidemic ] in Theorem 4.2. gains over spray and wait for heterogeneous networks (networks Theorem 5.2: P where each node is not the same). Community-based mobility M −1 iE[Depidemic ] = i=1 M1 comm comm P −1 m=1 E[Depidemic (m)]. model introduces an inherent heterogeneity in the network as nodes differ depending on which community they belong to. So, we study a spray and focus scheme for the community-basedC. Spraying a small fixed number of copies mobility model, and later we compare it to the corresponding 1) Fast Spray and Wait: This section derives the expected spray and wait scheme.delay of fast spray and wait routing scheme for the community- Fast spray and focus performs fast spraying in the spray phase.based mobility model. As before, first we derive the value of To be able to do utility-based forwarding in the focus phase, [15] commE[Df sw (m)]. For m < L (in the spray phase), the value of maintained last encounter timers to build the utility function. For commE[Df sw (m)] is derived in a manner similar to the derivation of community-based mobility models, [18] proposed the use of a commE[Depidemic (m)] as flooding is used to spread the L copies in the simpler function as a utility function for their ‘Label’ scheme: If commspray phase. Now, we derive the value of E[Df sw (L)] which is a relay meets a node which belongs to the same community asthe expected time to find the destination in the wait phase. ! the destination, the relay hands over its copy to the new node. comm M −ˆ l Pˆ( M −ˆ) l r l We use this simple utility function to route copies of the packet Lemma 5.3: E[Df sw (L)] = M −L r mp =0 pˆ p E[T mp ] l,m M −l ˆ in the focus phase. r “ M −ˆ ” l E[Mcomm,dif f ] This section derives the expected delay of fast spray and focus , where ˆ = rem L, r , E[Ts ] ` M´+ 1 − M −L r l f sw Lpsuccess2 for the community-based mobility model. pf sf can be derived in exis the expected time till the destination receives a copy a manner similar to the derivation of pf sw . To avoid repetition, exof the packet given there are s nodes belonging to the we skip the derivation of pf sf here. exdestination’s community which were unable to successfully comm As before, first we derive E[Df sf (m)]. Since flooding isexchange the packet with the destination in the past, and used to spread the copies in the spray phase, E[Df sf (m)] “ ”E[τcomm,dif f ] comm f sw f swpsuccess2 = 1 − 1 − ptxS2 . for m < L can be derived in a manner similar to the deriva- comm Proof: After the spray phase (after L copies have been tion of E[Depidemic (m)]. The next lemma derives the value of commspread), there is a community which has only ˆ = rem L, r E[Df sf (L)] which is the expected time it takes for the packet ` M´ lnodes carrying a copy of the packet. The probability that the to get delivered to the destination in the focus phase.destination is one of the remaining M − ˆ nodes belonging to this r l 0 comm Lemma 5.4: E[Df sf (L)] = 1 M −ˆ “ ” l M −l Pl M − l ˆ ˆ M −l „ « ˆ ˆ E[Mcomm,dif f ]community is equal to M −L . First we derive the expected delay in M −L @ mpr=0 pl,m E[T mp ]A + 1 − M −L r r ˆ p r f sf M −l ˆ L M psuccess2the wait phase when the destination belongs to this community. r r !! M −1 f sf + (1−psuccesss1 )E[Mcomm,same ]Then, we derive the expected delay when the destination does + rM E[Mcomm,same ] + f sf , where psuccess1not belong to this community. Finally we use the law of total r “ ”E[τcomm,dif f ]probability to combine everything together and get the result. ˆ = rem L, M , pf sf f sw ` ´ l r success1 = 1 − 1 − ptxS1 andPlease see the Appendix for proof details. f sf “ f sw ”E[τcomm,same ] Finally, we derive the expected delay of fast spray and wait for psuccess2 = 1 − 1 − ptxS2 .the community based mobility model (denoted by E[Df sw ]) in comm Proof: See Appendix. commterms of E[Df sw (m)] using the same argument used to derive Now we derive the expected delay of fast spray and focus for comm mmE[Df sw ]. the community based mobility model (denoted by E[Df sf ]) in comm Theorem 5.3: E[Df sw ] = i=1 pf sw (i) m=1 E[Df sw (m)]. terms of E[Df sw (m)]. comm PL Pi comm dest Theorem 5.4: E[Df sf ] = L pf sf (i) i comm comm P P 2) Fast Spray and Focus: Spray and Focus schemes [15] differ i=1 dest m=1 E[Df sf (m)],  1 i<Lfrom spray and wait schemes in how each relay routes the copy where pf sf (i) = dest M −1 M −L .towards the destination. Instead of doing direct transmission, each M −1 i=L Proof: The proof runs along similar lines as the proof ofrelay does a utility-based forwarding towards the destination, that Theorem 4.2.is, whenever a relay carrying a copy of the packet meets anothernode (label it node B ) which has a higher utility, the relay givesits copy to node B . Node B now does a utility based forwarding VI. ACCURACY OF A NALYSIStowards the destination and the relay drops the packet from its In the previous sections, we made a number of approxima-queue. [15] showed that spray and focus has huge performance tions to keep the analysis tractable. Here, we assess to which
  • 11 4 x 10 1.6 1200 2500 1200 Simulation (r=6, M=30) Simulation (r=6, M=30) Simulation (r=6, M=30) Simulation (r=6, M=30) 1.4 Theoretical (r=6, M=30) Theoretical (r=6, M=30) Theoretical (r=6, M=30) Theoretical (r=6, M=30) Simulation (r=4, M=40) 1000 Simulation (r=4, M=40) Simulation (r=4, M=40) 1000 Simulation (r=4, M=40) 2000 Expected Delay Theoretical (r=4, M=40) Expected Delay Expected Delay Theoretical (r=4, M=40) Theoretical (r=4, M=40) Theoretical (r=4, M=40) 1.2 Expected Delay 800 800 1 1500 0.8 600 600 0.6 1000 400 400 0.4 10 12 14 16 18 20 10 12 14 16 18 20 10 12 14 16 18 20 10 12 14 16 18 20 K K K K (a) Direct Transmission. (b) Epidemic Routing. (c) Fast spray and wait with L = (d) Fast spray and focus with L = 10. 10.Fig. 4. Simulation and analytical results for the expected delay for the community-based mobility model. Network parameters: N = 500 × 500 squareunits, Θ = 5, pl = 0.8, pr = 0.2, sBW = 1 packet/time slot. The expected maximum cluster size varies from 15%(K = 10, r = 6, M = 30) to24%(K = 20, r = 4, M = 40). (Note that the number of instances for which each Monte Carlo simulation is run is chosen so as to ensure that the 90%confidence interval is within 5% of the simulation value.)extent these approximations create inaccuracies. We focus on approximation of assuming the entire meeting and inter-meetingthe following approximations: (i) replacing S by E[S] in the time distribution to be exponential creates a noticeable error.expression of P (Ebw ) in Section III-B, (ii) replacing the random (Replacing the values derived based on this approximation byvariable representing the number of interfering transmissions (x) actual values derived from simulations makes the simulationby its expected value in Section III-B.3. (iii) replacing the contact and analytical curves indistinguishable.) The effect of this ap-time by its expected value in the expression of pR success in the proximation worsens as the node density increases (either Kdelay analysis of all routing schemes, (iv) assuming the entire or M increases). For the community-based mobility model, themeeting and inter-meeting time distribution to be exponential in assumption of exponential distribution for the inter-meeting timethe delay analysis of flooding-based routing schemes, and (v) for nodes belonging to different communities results in underes-assuming that a node starting from its stationary distribution will timating the expected delay. This effect of this approximation ismeet a node belonging to its own community before a node significant for smaller values of K . Also, the following additionalfrom some other community with high probability in the delay approximation plays a noticeable role: assuming that starting fromanalysis of routing schemes for the community-based mobility its stationary distribution, a node will meet a node belonging to itsmodel. We use simulations to verify that these approximations own community before a node from some other community: thisdo not have a significant impact on the accuracy of the analysis. approximation results in overestimating the expected delay andWe use a custom simulator written in C++ for simulations. worsens as the number of nodes in other communities increases (rThe simulator avoids excessive interference by implementing a increases). The first approximation dominates for lower values ofscheduling scheme which prohibits any simultaneous transmission K and r, and the second approximation becomes more dominantwithin one hop from both the transmitter and the receiver. It as K and r increases.incorporates interference by adding the received signal from othersimultaneous transmissions (outside the scheduling area) andcomparing the signal to interference ratio to the desired threshold. VII. A PPLICATION : D ESIGN OF S PRAYING - BASED ROUTINGThe simulator allows the user to choose from different physical S CHEMESlayer, mobility and traffic models. We choose the Rayleigh-Rayleigh fading model for the channel and Poisson arrivals in The design of spraying-based routing schemes poses the fol-our simulations. lowing three fundamental questions: (i) How many copies to spray? (ii) How to spray these copies in the spraying phase? We study the robustness of all the approximations by varying (iii) How to route each individual copy towards the destinationthe level of connectivity in the network (which in turn is achieved after the spraying phase? [12, 15, 17] answered these questionsby altering the transmission range K , the number of nodes in the assuming there is no contention in the network. In this section,network M and the number of communities in the network r for we use the expressions derived in the previous sections to study ifthe community-based mobility model). As a connectivity metric incorporating contention introduces significant differences in thewe use the expected maximum cluster size, which is defined as the answers to these questions.percentile of nodes that belong to the largest connected cluster,and denote its value in the figures’ captions. Figures 3(a)-3(d)and 4(a)-4(d) compare the expected end-to-end delay for different 25 Without Contention 250 Expected Delay (time slots) With Contentionrouting schemes obtained through analysis and simulations for 20different values of K and M for the random waypoint mobility 200 15model and for different values of K , M and r for the community Lbased mobility model. (Note that K is expressed in the same √ 10 150distance units as N .) We have compared the analytical and 5simulation results for a large number of scenarios, but due to 0 100 50 100 150 200limitations of space, we present some representative results for Target Expected Delay (time slots) 0 5 10 L 15 20 25each routing scheme. Since both the simulation and the analytical (a) (b)curves are close to each other in all the scenarios, we concludethat the analysis is fairly accurate. Fig. 5. (a) Minimum value of L which achieves the target expected delay for source spray and wait. (b) L against expected delay (with contention). Now we comment on which approximations create small yet Network parameters: N = 100 × 100, K = 8, M = 150, Θ = 5, E[S] =noticeable errors. For the random waypoint mobility model, the 70, T stop = 0, v = 1, sBW = 1.
  • 12A. How Many Copies to Spray elapsed since the packet was generated. Somewhat surprisingly, This section studies the error introduced by ignoring contention depending on the density of the network, source spray and waitwhen one has to find the minimum value of L (the number of can spray copies faster than fast spray and wait. This occurscopies sprayed) in order for a spraying-based scheme to achieve a because fast spray and wait generates more contention aroundspecific expected delay. (Note that we want the minimum value of the source as it tries to transmit at every possible transmissionL which achieves the target delay as bigger values of L consume opportunity. Such a behavior is expected for dense networks, butmore resources.) We choose the source spray and wait scheme these results show that increased contention can deteriorate fastwith the random waypoint mobility model as the case study in spray and wait’s performance even in sparse networks. In general, rwpthis section. We numerically solve the expression for E[Dssw ]in unless the network is very sparse, strategies which spray copiesTheorem 4.3 to find the minimum value of L which achieves slower yield better performance than more aggressive schemesa target delay and plot it in Figure 5(a) both with and without thanks to reducing contention. In ongoing work, we are tryingcontention for a sparse network. (For the expected delay of source to find the optimal spraying algorithm and design practical andspray and wait without contention, we use the expression derived implementable heuristics which achieve performance very closein [12].) This figure shows that an analysis without contention to the optimal. [42] is a first step in this direction. It derives thewould be accurate for smaller values of L (smaller values of optimal spraying scheme and a simple heuristic which performsL generate lower contention in the network), however it would very close to the optimal, but it assumes that there is no contentionpredict that one can use a large number of copies to achieve in the network. Currently, we are merging this work with thea target expected delay which actually will not be achievable contention framework proposed in this paper to find the optimalin practice due to contention. For example, the analysis without spraying scheme with contention in the network.contention indicates that a delay of 50 time units is achievablewith L = 23 while the contention-aware analysis indicates that C. How to Route Individual Copiesit is not achievable. Figure 5(b) shows that L = 23 results inan expected delay of more than 118 time units, which is also Without contention, performing utility-based forwarding onachievable by L = 5. Thus choosing a value of L based on each individual copy outperforms spray and wait schemes be-predictions from a contention-ignorant analysis led to a value cause it identifies appropriate forwarding opportunities that couldof delay which is not only much higher than expected but also deliver the message faster [15]. But, utility-based forwardingwould have been achieved by nearly four times fewer copies. requires more transmissions and hence, increases the contention in the network. So we study how much performance gains are achieved by spray and focus over spray and wait (for theB. How to Spray Multiple Copies community-based mobility model) both with and without con- Intuitively, spraying copies as fast as possible is the best way tention in the network by plotting the minimum value of theto spread copies if all the relay nodes are equal/homogeneous. average number of transmissions it takes to achieve a given target(One might want to bank copies for future encounters with expected delay for both the schemes in Figure 7. We first find the‘super nodes’ when relay nodes are heterogeneous, see our prior minimum value of L which achieves the given target expectedwork [42].). To answer whether spraying the copies as fast as delay for both the schemes and then find the average number ofpossible is optimal under a homogeneous relays scenario, we transmissions which is equal to L ipR (i). (The minimum P i=1 destcompare the two different spraying schemes introduced in Section value of L is computed using the analytical expressions derivedIV-C, source spray and wait and fast spray and wait for the in Section V-C. The value of pR (i) for both the schemes was destrandom waypoint mobility model. Since fast spray and wait derived in Theorems 5.3 and 5.4.) We observe that fast sprayspreads copies whenever there is any opportunity to do so, it has and focus outperforms fast spray and wait even with contentionthe minimum spraying time when there is no contention in the in the network, with gains being larger with contention. Sincenetwork [17]. On the other hand, since source spray and wait does E[Mcomm,dif f ] >> E[Mcomm,same ], forwarding a copy to anynot use relays to forward copies, it is one of the slower spraying node in the destination’s community in the focus phase signif-mechanisms when there is no contention in the network. icantly reduces the delay for the same L without significantly 35 increasing the contention as it requires only one extra message Expected number of copies spread Source spray and wait(M=100) 30 Fast spray and wait(M=100) per copy. Hence, fast spray and focus shows more performance 25 Source spray and wait(M=250) Fast spray and wait(M=250) gains over fast spray and wait after incorporating contention. 20 15 VIII. C ONCLUSIONS 10 In this paper, we first propose an analytical framework to 5 model contention to analyze the performance of any given 0 0 200 400 600 mobility-assisted routing scheme for any given mobility and Time (time slots) channel model. Then we find the expected delay for represen-Fig. 6. Comparison of fast spray and wait and source spray and wait: tative mobility-assisted routing schemes for intermittently con-Expected number of copies spread vs time elapsed since the packet was nected mobile networks (direct transmission, epidemic routinggenerated. Network parameters: N = 100 × 100 square units, K = 5, Θ = and different spraying based schemes) with contention in the5, sBW = 1 packet/time slot, L = 20. Expected maximum cluster size(metric to measure connectivity) for these network parameters is equal to network for the random direction, random waypoint and the more4.6% for M = 100 and 5.2% for M = 250. realistic community-based mobility model. Finally, we use these Now we study how fast the two schemes spread copies of delay expressions to demonstrate that designing routing schemesa packet when there is contention in the network. Figure 6 using analytical expressions which ignore contention can lead toplots the number of copies spread as a function of the time suboptimal or even erroneous decisions.
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  • 14 „„ « « a When there are L copies of a packet in the network, there1 + pa ppkt − 1 . To figure out the value of pa , lets 2 are L nodes which can deliver a copy to the destination butchoose a pair of nodes amongst these a nodes and label the nodes even if the source meets some other node which does not have au1 and u2 . Let f (x, y) denote the pdf that a node u1 is a distance copy, it cannot attempt to transmit a copy to the other node. Thex from from the transmitter and at a distance y from the receiver. mm expression for E[Dssw (L)] is derived in a manner similar to theThen, using simple combinatorics, we derive f (x, y) to be equal to8 0 4K 2 +2Kd+d2 1 derivation of Lemma 4.2 to be Lpssw ] . E[Mmm> 9 success 2cos−1 @ Proof: (Lemma 5.3) After the spray phase (after L copies> 4K (K+d) A if K < x = K + d < K + 2K ,> 3> 3> have been´spread), there is a community which has only ˆ =< 2K −d A1 3 ( ) d+ K <y <K . l 3 rem L, M nodes carrying a copy of the packet. The probability `>> if 0q x ≤ K, 0 ≤ θ < 2π, < r 1 that the destination is one of the remaining M − ˆ nodes belonging> l> A1 4K 2 4Kxcos(θ) y= x2 + −>: r 9 3 M −ˆlNow, conditioned over the fact that node u1 is at a distance x to this community is equal to M −L . First we will derive the rfrom the transmitter and y from the receiver, we determine the expected delay in the wait phase when the destination belongs toprobability that node u2 is within range from u1 . By assumption, this community. The probability that the system state is (ˆ mp ) l,u2 is within one hop from the transmitter and the receiver. If (where mp denotes the number of node pairs in the communityu2 is also within a distance K from u1 , that is, u2 lies in the which want to exchange this packet, and had an opportunity inarea marked by the intersection of the following three circles: (i) the past to exchange this packet but were unable to do so duecentered at the transmitter with radius equal to K , (ii) centered to contention) is equal to pˆ p . (The value of pˆ p was derived l,m l,mat the receiver with radius equal to K , and (iii) centered at u1 in Lemma 5.1.) Given the system state in which the spray phasewith radius equal to K , then u2 is within range of u1 . Thus, the ended is (ˆ mp ), the number of nodes which had an opportunity l,probability that u1 and u2 are within range of each other given to deliver the packet to the destination but were unable to do mpthat u1 is at a distance x and a distance y from the transmitter and so is equal to M −ˆ. (As discussed in the proof of Lemma 5.1, l (x,y) rthe receiver respectively, is equal to A3A1 where A3 (x, y) = each node not having a copy of the packet has met on an averageA√ y) + A5 (x, y) − A6 (x, y), A4 (x, y) ´= 2K 2 cos−1 2K − x mp ` ´ 4 (x, M −ˆ nodes which have a copy of the packet.) To derive the delay l 2 − x2 , A (x, y) = 2K 2 cos−1x ` y y p r 4K 2 − y 2 and2 5 ` x ´ 2K − 2y ´4K associated with the wait phase, we define a new system state: (s) K 2 sin−1 2K + sin−1 2K + sin−1 1 ` ` ` ´´A6 (x, y) r“ = 3 where s is the number of nodes in the destination’s community ”`+1 (x + y)2 − 4K 2 x − y + 2K y − x + 2K ´` ´ which had an opportunity to deliver the packet to the destination 4 9 3 3 √ √ but were unable to do so due to contention. Let Ts denote the x 4K 2 −x2 y 4K 2 −y 2 √ 2 2K 2 additional time it will take to deliver the packet to the destination− 4 − 4 − 9 . The value of A1 wasderived in Lemma 3.1. Removing the condition on the location given the current system state is (s). Then, given that nodes in the commof u1 using the law of total probability yields the value of pa . destination’s community have a copy of the packet, E[Df sw (L)] !The value of ppkt can be derived from simple combinatorics to Pˆ( M −ˆ) l r l ´E[S] is equal to mp =0 pˆ p E[T l,m mp ] .be 1 − 1 − pR ` ex . M −l r ˆ To complete the previous proof, we now describe Now, we quantify the contention due to the c nodes within two how to derive the value of E[Ts ]. One of the nodeshops from the either the transmitter or the receiver but not in carrying the packet meets the destination after an expectedthe scheduling area. Contention arises when one of the a nodes „ « −1 ˆ l−s sis within range of one of the c nodes. There are ac such pairs. time duration of E[Mcomm,same ] + + . E[Mcomm,same ]Lets choose one such pair and label the corresponding nodes u1 With probability pf sw success1 , this node is able to deliverand u3 , where u1 lies in the scheduling area while u3 is within the packet to ” the destination (where pf sw = success1two hops from the either the transmitter or the receiver but not “ E[τcomm,same ]in the scheduling area. Define pc = Pr[u1 and u3 are within 1 − 1 − pf sw txS1 ). With probability ps = „ «„ «−1range of each other]. Then, the expected number of transmissions ˆ l−s s + E[Mcomm,same ] E[Mcomm,same ] + + s ,contending are acpc ppkt . pc is derived in a manner similar to the E[Mcomm,same ]derivation of pa using the following two observations: (i) u3 can the node which meets the destination is one of the s nodes whichlie anywhere within two hops from either the transmitter or the have missed an opportunity to deliver the packet to the destination “ ”receiver, and (ii) Conditioned over the fact that node u1 is at a in the past. Hence, with probability ps 1 − pf sw success1 the packetdistance x from the transmitter and y from the receiver, u3 will does not get delivered to the destination and the system remainsbe within a distance K from u1 only if it lies in the circle of in state s and will take an additional E[Ts ] time to deliver theradius K centered at u3 but not in the scheduling area. packet to the destination. On the other hand, with probability “ ” (1 − ps ) 1 − pf sw success1 , the packet does not get delivered to Proof: (Lemma 4.4) The proof runs along the same lines the destination and the system moves to state s + 1 (as one moreas the proof of Lemma 4.2. When there are 1 ≤ m < L copies of node belonging to the destination’s community has missed ana packet in the network, there are m nodes which can deliver a opportunity to deliver the packet to the destination) and will takecopy to the destination only, and there is one source node which an additional E[Ts+1 ] time to deliver the packet to the destination.can deliver a copy to any of the M − m − 1 other nodes which „ « −1 l−s ˆ sdo not have a copy of the packet. Hence, there are a total of Thus, E[Ts ] = E[Mcomm,same ] + + +m + M − m − 1 = M − 1 node pairs, which when meet, have an “ ” “ E[Mcomm,same ] ” f sw f swopportunity to increase the number of copies from m to m + 1. ps 1 − psuccess1 E[Ts ] + (1 − ps ) 1 − psuccess1 E[Ts+1 ].The expected time it takes for one of these M − 1 node pairs This set of linear equations can be solved to find E[Ts ]. “ M ˆ”to meet is E[M−1 ] . Using the same argument as in the proof of M mm −l Now, with probability 1− M −L , none of the nodes belonging rLemma 4.2, E[Dssw (m)] can be derived to be (M −1)pmm ] . mm E[M ssw to the destination’s community have a copy of the packet and the success
  • 15expected time it takes for the L nodes to deliver the packet todestination can be derived in manner similar to the derivation of E[Mcomm,dif f ]Lemma 4.2 to be equal to f sw . Lpsuccess1 Finally combining everything together by using the law of totalprobability to remove the condition on whether a node belongingto the destination’s community had a copy of the packet after thespray phase or not, yields the result. Proof: (Lemma 5.4) After the spray phase (after L copieshave been´spread), there is a community which has only ˆ = lrem L, M nodes carrying a copy of the packet. The probability ` rthat the destination is one of the remaining M − ˆ nodes belonging r l M −ˆ lto this community is equal to M −L . The expected delivery delay rto the destination for this scenario is derived in a manner similar commto the derivation of E[Df sw (L)] in Lemma 5.3. Now we derive the delivery delay for the scenario when thenodes in the destination’s community do not have a copy of thepacket. The expected time it takes for the L nodes carrying acopy to deliver a copy to one of the M in the destination’s r E[Mcomm,dif f ]community is equal to f sf . (This is derived in a L M psuccess2 rmanner similar to the derivation of Lemma 4.2). With probability M −1 r M , the packet copy is received by a node which itself is rnot the destination but belongs to the destination’s community.This node does a direct transmission to the destination whichtakes an additional time whose expected value is equal to f sf + (1−psuccesss1 )E[Mcomm,same ]E[Mcomm,same ] + f sf . (This is derived psuccesss1in a manner similar to the derivation of Lemma 5.1.)