Improving DTN Routing Performance Using Many-to-Many Communication: A Performance Modeling Study
Giridhari Venkatadri, Mahendran Veeramani, Siva Ram Murthy C
aa r X i v : . [ c s . N I] M a r Improving DTN Routing Performance UsingMany-to-Many Communication: A PerformanceModeling Study
Giridhari Venkatadri, V. Mahendran, and C. Siva Ram Murthy
Department of Computer Science and EngineeringIndian Institute of Technology Madras, IndiaEmail: [email protected], [email protected], and [email protected]
Abstract —Delay-Tolerant Networks (DTNs) have emerged asan exciting research area with a number of useful applica-tions. Most of these applications would benefit greatly by areduction in the message delivery delay experienced in thenetwork. The delay performance of DTNs is adversely affectedby contention, especially severe in the presence of higher trafficrates and node densities. Many-to-Many (M2M) communicationcan handle this contention much better than traditional one-to-one communication employing CSMA. In this paper, for thefirst time, we analytically model the expected delivery delay of aDTN employing epidemic routing and M2M communication. Theaccuracy of our model is demonstrated by matching the analyticalresults against those from simulations. We also show usingsimulations that M2M communication significantly improves thedelay performance (with respect to one-to-one CSMA) for high-contention scenarios. We believe our work will enable the effectiveapplication of M2M communication to reduce delivery delays inDTNs.
I. I
NTRODUCTION
Delay-Tolerant Networks (DTNs) [1], [2] are challengednetworks where end-to-end connectivity does not always existbetween nodes. One reason this may occur is that links getdisrupted either due to interference or due to nodes movingout of range of each other. DTNs can be put to a variety ofapplications such as bringing Internet connectivity to villages,and reducing the pressure on cellular bandwidth by offloadingdelivery through WiFi and mobiles of people moving in thevicinity. Since an end-to-end path seldom exists betweensource and destination, most existing Internet protocols failand instead, the store-carry-forward [1] protocol is used totransfer messages from source to destination through a num-ber of intermediate relays. The lack of an end-to-end pathmeans message delivery delays are large. Hence, lowering themessage delivery delays would be critical in enhancing theperformance of most of these applications.It has been shown that contention adversely affects thedelay performance of DTNs [3]. This is because contentionimplies that transfer of a message is not always possiblewhen one node meets another, owing to two of the followingreasons: ( i ) the message contends with other messages in thebuffer of the node to be chosen for transmission, and ( ii ) thenode contends with other nodes in the vicinity for opportuni-ties to communicate. While contention has been recognized as an importantfactor that increases delivery delays, the problem of fightingcontention and thereby reducing delivery delays in DTNs hasnot been adequately considered. For a lot of DTN applicationscenarios, it is possible for a number of nodes to come inthe vicinity of each other. For example, people in a crowdedpublic square, or villagers near a road where a vehicle ispassing by. In these scenarios, as a result of contention, mostnodes would not be able to transmit the messages they carrysince existing access schemes such as Carrier Sense MultipleAccess (CSMA) are one-to-one communication schemes andwould allow only one pair of nodes within the region ofcontention to communicate at any instant of time.Many-to-Many (M2M) communication [4], [5] has beenproposed to improve the capacity of Mobile Ad hoc NET-works (MANETs) and enables each node in a bounded groupof nodes to simultaneously communicate with each other nodein the group. Hence when a number of nodes come in thevicinity of each other, this precious contact duration is usedmuch better, with more nodes getting a chance to communi-cate. Hence, M2M communication can be used to battle theeffects of contention and therefore reduce message delivery de-lays. M2M communication has become feasible since the state-of-the-art technology enables support for both access schemes(such as Code Division Multiple Access (CDMA)) and loca-tion services such as Global Positioning System (GPS)) on asingle IC chip [4].In this paper, we analytically model the expected deliv-ery delay of a DTN employing epidemic routing and M2Mcommunication. Despite the fact that the analysis has beendone for epidemic routing, the framework we develop is easilyextended to other routing protocols. The accuracy of ourmodel is demonstrated by matching the results from analysisagainst those from simulations. Also, simulations show that forhigh-contention scenarios, M2M communication significantlyoutperforms one-to-one communication employing CSMA interms of delivery delay.II. R ELATED W ORK
Initial analytical models developed for DTN routing per-formance study [6], [7], [8] worked under the assumptionthat whenever two nodes are in contact with each other, allessages could always be successfully transferred from onenode to the other ( i.e., they assumed both buffer capacityand bandwidth to be infinite). While papers such as [9] havemodeled DTN performance with bounded buffer capacity, theyhave assumed that infinite bandwidth is available and hencethat there is no contention. The motivation for not consideringcontention has been that DTNs are sparse networks and suchsparsity yields negligible contention. However, this conjecturehas been disproved with the help of simulations in workssuch as [7], [8]. The authors in [3] show via simulationsthat, irrespective of whether the network is sparse or dense,the contention is substantial for high traffic rates; and alsothat the contention increases with an increase in networkdensity. Realizing the importance of contention in the routingperformance, the authors have attempted to include contentionin the analysis of routing [3], [10]. Their analysis assumes theone-to-one CSMA communication scheme.In our paper, however, we propose that in scenarios wherethere are areas of high node density ( i.e., a number of nodesin the vicinity of each other), nodes must co-operate using anM2M communication scheme such as the one proposed in [4]and not compete as assumed by [3]. This would mean thatinstead of just one pair of nodes in such an area of highernode density communicating, we can have a larger set ofnodes communicating with each other at the same time. Thiswould significantly increase the probability that two nodeswhich come into contact with each other get to exchangemessages, and thereby reduce the routing delay. The beneficialeffect of using such an M2M communication scheme wouldbe amplified in the presence of high traffic loads.III. A
NALYTICAL M ODEL
In this section, we develop an analytical model for theexpected delivery delay of a DTN employing M2M commu-nication. We follow the delivery of a particular message fromsource node to the destination node, and model the evolution ofthe state of the system with respect to this message. We firstdescribe the Medium Access Control (MAC) protocol usedfor achieving M2M communication which logically dividesthe network into square cells (or) tiles. We then derive themobility statistics for the Random Direction (RD) mobilitymodel, where the network is logically divided into square tilesas described in the MAC protocol. Using these statistics, wemodel the evolution of the network as a stochastic process andthereby derive the expected delivery delay.
A. MAC Protocol for M2M Communication
We adopt the scheme proposed by Moraes et. al., in [4] forM2M communication in MANETs. While they have proposedthis scheme to increase the capacity of MANETs, we adoptthe scheme to reduce delivery delays in DTNs. For the sake ofcompleteness, we briefly describe the scheme in this section.The network is logically divided into a number of squaretiles. Each node, with the help of GPS, is aware of itscurrent location (thereby the current tile it is in). GPS alsohelps in time synchronization among the nodes. Therefore, the communication in each tile is synchronized. In eachtile, communication is performed in the form of sessionsthat have two phases. The first phase, called the discoveryphase, is divided into slots. Each node in the grid that wantsto communicate randomly picks one slot and broadcasts itspresence to the other nodes in that slot, along with someother control plane information. The first bunch of nodes (say α -nodes) that successfully do so enter the second phase,which is the communication phase. The parameter α is chosenaccording to different factors such as the allowable receivercomplexity. We assume there are sufficient slots to make theprobability of collision negligible thereby ensuring that if thereare lesser than α nodes in a particular cell, then all of themare chosen for communication with a high probability. Also,this means that when there are at least α nodes in a particularcell, the MAC scheme effectively chooses a random subset of α -nodes from the set of nodes in that cell. Each node thatenters the communication phase is assigned a code on whichit transmits data and a frequency on which it receives data.Using narrowband FDMA/CDMA, all α nodes are able tocommunicate with each other simultaneously ( i.e., each nodeparticipating in the communication session can send up to b BW messages to every other node participating in the session,where b BW depends on the bandwidth available for M2Mcommunication). B. Mobility Statistics1) Intermeeting Time (IMT):
The IMT of any two nodes isdefined as the time elapsed between two successive meetingsof the nodes. Since we logically tile the terrain to make theMAC easier, the IMT is expected to increase since somecommunication opportunities are lost as a result of tilingthe terrain. Hence existing derivations [11] do not hold assuch. We adopt the methodology used in [12] to derive thedistribution of the IMT, show it to be exponential and deriveits mean. We define the area covered by a node while movingalong a path as the total area that came under the node’scoverage when it moved along that path. It must be notedthat a node’s coverage area at any point is the cell it is in atthat point. a x y θ v Figure 1: Node performing RD motion
Lemma 1:
The average area A covered during an epoch Under RD mobility, a node chooses its mobility parameters such as speedand angle of direction, and travels with the chosen mobility parameters for aduration of time drawn from an exponential distribution (this duration of timeis called an epoch). After that, the node halts for a random amount of time,following which it chooses a new set of values for its mobility parametersand the process repeats. = Z ∞ t =0 Z π θ =0 Z V max v = V min Z ay =0 Z ax =0 (cid:18) vt ( cosθ + sinθ ) a + x + ya (cid:19) (cid:18) a (cid:19) dx (cid:18) a (cid:19) dy (cid:18) V max − V min (cid:19) dv (cid:18) π (cid:19) dθ (cid:18) vL (cid:19) e − ( vL ) t dt (1)under RD mobility over a square-tiled terrain is given by A = 2 a + Laπ , where a is the length of a side of a cell,and L is the average distance covered in an epoch of the RDmodel. Proof:
Consider a node starting an epoch within a cellat the coordinate ( x, y ) as illustrated in Fig 1. Since the RDmodel has a uniform stationary distribution, these coordinates x and y are drawn from a uniform distribution over [0 , a ] . Also,according to the RD model, the node chooses an angle θ drawnfrom a uniform distribution over [0 , π ] , velocity v drawn froma uniform distribution over [ V min , V max ] , duration of epoch t drawn from an exponential distribution with average Lv , anda halting time at the end of the epoch drawn from a uniformdistribution with average T s . Let the number of vertical wallsand horizontal walls crossed during this epoch be η v and η h ,respectively. We have for θ ∈ [0 , π ] , η v = (cid:24) vt cos( θ ) − ( a − x ) a (cid:25) and η h = (cid:24) vt sin( θ ) − ( a − y ) a (cid:25) . We now derive the area covered in the epoch, by countingthe number of cells that are traversed in that epoch. For moststarting configurations, the node does not pass through corner-points of cells during the epoch. Hence, it is reasonable toassume that every wall crossed during the epoch is intersectedsomewhere along the side and therefore that each intersectionrepresents crossover of the node into a new cell. Hence thenumber of cells n covered in the epoch for θ ∈ (cid:2) , π (cid:3) is givenby, n = 1 + η v + η h ≤ vt cos( θ ) − ( a − x ) a + vt sin( θ ) − ( a − y ) a = 1 + vt (cos( θ ) + sin( θ )) a + x + ya . Because of symmetry, the average number of cells coveredin an epoch for θ ∈ (cid:2) , π (cid:3) is the same as the average for θ ∈ [0 , π ] . Hence, the average number of cells n covered inan epoch is given by Eq. 1, which on solving gives, n = 2 + 4 Lπa . (2) Hence we have A = na = 2 a + 4 Laπ .
Having estimated the average area covered during eachepoch, we can proceed as in [12] to show that the expectedhitting time T h is as follows: T h = N a + Laπ (cid:18) Lv + T s (cid:19) (3)In a similar way, as done in [12], we can show that the meetingtime T m is exponentially distributed with an average given by T m = T h p m v rd + 2(1 − p m ) (4)where v rd is a constant ( . ) for the RD model such that theexpected relative speed between two nodes v rel is given by v rel = v rd v, (5)and p m is the probability that a node is moving at any pointin time given by, p m = TT + T s . (6)Finally we can show that the intermeeting time is also ap-proximately exponentially distributed such that its average T im equals T m .
2) Contact Time (CT):
The contact time between two nodesis defined as the time elapsed between when the two nodescome within communication range of each other and whenthey go out of the communication range of each other. In thissection, we derive the expression for the expected contact time( T con ). We assume that the probability of contact between twonodes lasting over multiple cells is negligible. When a contactoccurs, two cases arise: either one node could be stationary orboth nodes could be moving. Let the expected contact time inthe first case be T scon and in the second case be T mcon . Lemma 2: T con = (1 − p m ) T scon + p m T mcon Proof:
The probability that both nodes are moving giventhat one node is moving is p m and the probability that onenode is stationary given that one node is moving is − p m .The result follows from this. The hitting time is the time taken for a node starting from the stationarydistribution to hit a randomly chosen static target. emma 3: T scon = 2 I aw + I ow where I aw = Ka log (cid:18) V max V min (cid:19) h log (cid:16) √ (cid:17) + ( √ − i and I ow = 2 Ka log (cid:18) V max V min (cid:19) h log (cid:16) √ (cid:17) + (1 − √ i for K given by K = 1 aπ ( V max − V min ) Proof:
When one node is stationary, the contact beginswhen the other node enters the cell and ends when the othernode leaves (our assumption that contacts last only over a cellis especially valid in this case). Hence the expected contacttime in this case T scon is given by the expected time elapsedbetween the moving node entering and leaving the cell. Weassume the moving node enters through any particular wallof the cell (since the problem is symmetric w.r.t the differentwalls of the cell) with a uniform distribution on point of entry,velocity, and angle of entry. Since epoch lengths are assumedlarge, it is assumed that with high probability, the movementof the node through the cell is part of one epoch and hence astraight line. The node can leave either through one of the twoadjacent walls or the opposite wall. I aw is the product of theexpected time taken to leave through a particular adjacent wallwith the probability that the node leaves through that adjacentwall. Similarly, I ow is the product of the expected time takento leave through the opposite wall with the probability that thenode leaves through the opposite wall. Hence, T scon = I aw + I aw + I ow = 2 I aw + I ow . (7) I aw and I ow can be derived using geometric argumentswhich we do not present here due to space constraints.Now we approximate the contact time distribution with onenode stationary as an exponential distribution with expectedvalue T scon . Hence the contact time for both nodes moving isgiven by the minimum of the times spent by each node inthe cell and is hence approximately exponential with expectedvalue T scon . Therefore T mcon = T scon . IV. M
ODELING C ONTENTION UNDER
M2MC
OMMUNICATION
In this section we develop a framework for modelingcontention in a DTN employing M2M communication. Wefirst derive the distribution for the number of neighbors ofany node and then derive the expected number of messages inthe buffer of a node. These results are then used to developthe contention model.
A. Distribution of Number of Neighbors
We have established that the IMT and CT are approximatelyexponential with means T im and T con , respectively. We hencemodel the process of nodes arriving within communicationrange of a particular node ( i.e., entering the same cell asthe node), and moving out of communication range of thenode as a discrete time Markov chain, where the states aregiven by the number of neighbors at the beginning of eachcommunication session. We assume that the probability ofhaving more than N max neighbors is negligible. We alsoassume that since the duration of a communication session ( τ )is small, the probability of more than one arrival or departurein the duration of a communication session is negligible.Hence the transition probability P ( n + 1 | n ) of the Markovchain is the probability of one arrival and no departure and isgiven by, P ( n + 1 | n ) = (cid:18) λ effn τ e − λ effn τ (cid:19) (cid:18) − µ effn τ e − µ effn τ (cid:19) where λ effn = ( n tot − n ) T im and µ effn = nT con . In a similar way, P ( n − | n ) is given by P ( n + 1 | n ) = (cid:18) − λ effn τ e − λ effn τ (cid:19) (cid:18) µ effn τ e − µ effn τ (cid:19) and P ( n | n ) is given by, P ( n | n ) = 1 − P ( n + 1 | n ) − P ( n − | n ) . The steady state distribution of this Markov chain gives usthe probability distribution of neighbors. We assume in thesubsequent sections that the probability that the number ofneighbors N nhb equals j is denoted by v j . B. Expected Number of Messages in Buffer
In order to derive the expression for the expected numberof messages in buffer ( E [ B ] ), we note that by Little’s Law,we have, E [ B ] = λE [ W ] (8)where λ is the total traffic arrival rate at each node (assumedPoisson) and E [ W ] is the expected waiting time for a messagein a node’s buffer.We first derive the expression for the arrival rate λ . We thenderive E [ W ] in the next section. λ is given by λ = λ gen + λ rel where λ gen is the rate of generation of messages at that nodeand λ rel is the rate at which relay messages (messages forwhich this node is going to act as relay) arrive at the node.To calculate λ rel , we assume that any given message spreadsto nearly the entire network before being removed by theecovery mechanism. This is valid for the direct recoverymechanism we assume is used, which makes minimal effort atrecovery. Hence, we assume that any message in the networkis replicated at a node with a high probability before all itscopies are erased from the network. Now, working under thisassumption, we treat the network as a queue with messagesgenerated at other nodes considered as arrivals (with an aver-age rate λ gen ( n tot − ), and messages delivered to a particularnode under consideration as service completions (with anaverage rate λ rel ). Clearly, for the number of messages inthe network to be stable, the service rate λ rel must be at least λ gen ( n tot − ). We approximate the value of λ rel by thislower bound.
1) Expected Waiting Time for Epidemic Routing:
The ex-pected waiting time depends on several factors such as thechosen mobility model, routing protocol, and the recoverymechanism. In order to derive the expected waiting time forepidemic routing, we assume direct recovery is used, i.e., anode will only drop a message after meeting the destinationand either transmitting the message to the destination itselfor learning from the destination node that the destinationnode has already received the message. We also assume thatthe scheduler at each node works such that whenever thedestination of some messages is encountered, those messagesare given top priority. We also assume that if contact isestablished with the destination of some messages in thebuffer, the bandwidth available is sufficient to transmit all themessages to the destination. This is valid since we assumesymmetric traffic generation, i.e, for any source node, there isan equal amount of traffic generated destined for every othernode and hence a limited amount of traffic destined for anyparticular destination. Under these assumptions we consider aparticular message in the buffer of a particular node (call it thecurrent node) and derive its expected waiting time for epidemicrouting. Let p succ be the probability of successfully commu-nicating with the destination of the message given that thecurrent node meets the destination (successful communicationmay not happen because of contention with other neighboringnodes). The number of meetings with the destination beforethe current node is successfully able to communicate with itis geometrically distributed with mean p succ and the averagetime elapsed between meetings is T im . Hence, E [ W ] is givenby E [ W ] = T im p succ (9)where p succ is given by p succ = 1 − (cid:16) − p ′ succ (cid:17) Tconτ where T con τ is the average number of communication sessionsfor which the current node remains in contact with thedestination, and p ′ succ is the probability that the current nodesuccessfully communicates with the destination in a particular communication session. p ′ succ is given by p ′ succ = ∞ X j =1 P ( N nhb = j | N nhb ≥
1) ( p jcomm ) where p jcomm is the probability that a particular node gets tocommunicate in a session given that it has j neighbors. p jcomm depends upon the MAC protocol used to resolve contention.As described in Section III-A, the MAC protocol effectivelychooses all nodes in a cell if the number of nodes is atmost α , else it chooses a random α -sized subset of the nodes in thecell. Hence, p ′ succ = ∞ X j =1 v j ( P ∞ k =1 v k ) min (cid:18) αj + 1 , (cid:19) min (cid:18) αj + 1 , (cid:19) C. Contention Model
We model contention in terms of three different factors asfollows: ( i ) the probability of increase in number of copies ofa particular message from i to i ′ , namely p i ′ ,i , ( ii ) the prob-ability of delivery to the destination of a particular messagegiven that there are i copies of the message in the network,namely p d,i , and ( iii ) the expected time lapse between thenumber of copies reaching i and either a rise in the numberof copies or delivery to the destination, namely E [ D i ] . Weassume that with a high probability, in any communicationsession, only at most one of the i nodes carrying a copy ofthe message attempts to transmit the message and hence p i ′ ,i is zero for i ′ ≥ i + α .For i < i ′ < i + α , since we assume that at most one nodetransmits this message in any communication session, p i ′ ,i = i α − X j = i ′ − i P ( N nhb = j ) × min (cid:18) αj + 1 , (cid:19) × ( p tx ( i )) i ′ − i × (1 − p tx ( i )) j − ( i ′ − i ) where P ( N nhb = j ) has already been derived in Section IV-A,and p tx ( i ) is the probability that a message is successfullytransmitted to another node given they are in communicationwith each other and that there are i copies in the network. Theprobability p tx ( i ) is given by, p tx ( i ) = (cid:18) − i − n tot − (cid:19) (cid:18) b BW E [ B ] (cid:19) . To compute p d,i , we assume that at most one of the i nodescarrying a copy of the message meet the destination at a givenpoint in time. Hence p d,i is the probability that any one ofthese nodes meets the destination and successfully transfersthe message to the destination. And p d,i is given by, p d,i = ∞ X j =1 P ( N nhb = j ) × jn tot − × min (cid:18) αj + 1 , (cid:19) × min (cid:18) αj + 1 , (cid:19) . ) E [ D i ] Computation:
To derive E [ D i ] , we first estimatethe average time elapsed before a particular node out of the i nodes carrying the message spreads a copy/copies of themessage. We jointly model the process of arrival and departureof neighbors and the attempted transfer of the message as astochastic process with states j = 0 , , · · · n max representingthe event that at the beginning of a communication session,the node has j neighbors and still has not spread copies of themessage. The stochastic process also has a state C representingthat the node has successfully spread one or more copies ofthe message and that therefore the number of message copiesin the network has changed. This has been illustrated in Fig 2. n max n−system Figure 2: Digraph representing the stochastic process used tocompute E [ D i ] We define the n -system as the subset of states n, n +1 · · · , n max . Let E [ t n ] be the expected dwell time in the state n , p sn,n +1 ( i ) be the probability that the number of neighborsof the node under consideration increase from n to n + 1 in a communication session with no message copies spread,given there are i copies of the message, n neighbors for thatparticular node, and that some change in the state of thestochastic process happens at the end of the communicationsession. Also, let p ss ( n +1) ,n ( i ) be the probability that a transi-tion happens from the ( n + 1) -system to the state n given thata transition from the ( n + 1) -system occurs. The followinglemma gives the expected dwell time E [ t n ] of the process inthe n -system. Lemma 4: E [ t n ] = E [ t n ] + p sn,n +1 ( i ) × E [ t n +1 ]1 − p sn,n +1 ( i ) × p ss ( n +1) ,n ( i ) Proof:
We replace the dwell time distribution at each state n by its expected value E [ t n ] . Then the expected time spentin the n -system is given by, E [ t n ] = E [ t n ]+ p sn,n +1 ( i ) (cid:16) E [ t n +1 ] + p ss ( n +1) ,n ( i ) × ( E [ t n ] + · · · ) (cid:17) which gives us E [ t n ] = E [ t n ] + p sn,n +1 ( i ) (cid:16) E [ t n +1 ] + p ss ( n +1) ,n ( i ) × E [ t n ] (cid:17) which can be rearranged to give us the result.To terminate the recursion, we have E [ t n max ] = E [ t n max ] . To compute E [ t n ] we first compute p sn,n ( i ) which is theprobability that the number of neighbors remains n at the endof a communication session with no copies of the messagetransferred, given that there are i copies of the message in thenetwork. For n ≤ α − it is easy to derive p sn,n ( i ) = P ( n | n ) × p sn ( i ) where P ( n | n ) is as derived in the aforementioned Sec-tion IV-A and p sn ( i ) is the probability that there is no suc-cessful communication of the message given that there are i copies of the message and n neighbors. For n ≤ α − , p sn ( i ) is given by, p sn ( i ) = (1 − p tx ( i )) n . And for n > α − , p sn ( i ) is given by, p sn ( i ) = (cid:16) − αn + αn × (1 − p tx ( i )) n (cid:17) . The number of communication sessions for which the systemstays in state n is then a geometric random variable withprobability of success − p sn,n ( i ) . Hence, the expected dwelltime in state n is given by, E [ t n ] = τ − p sn,n ( i ) . Also, it must be noted that since the session time is small,the expected dwell times E [ t n ] can be approximated bycontinuous exponential distributions. Hence, if we considered i nodes in state n instead of one and computed the expecteddwell time (the dwell time in this case would be the minimumof the dwell times of the i nodes), the expected dwell timewould approximately be E [ t n ] i .The transition probability p sn,n +1 ( i ) is given by, p sn,n +1 ( i ) = p sn ( i ) × P ( n + 1 | n )1 − p sn ( i ) × P ( n | n ) . To compute the other transition probability p ss ( n +1) ,n ( i ) , weassume that the dwell time within the ( n + 1) -system issufficient for the stationary distribution within the ( n + 1) -system to be reached. Hence we have, p ss ( n +1) ,n ( i ) = v n +1 × P ( n | n + 1) v n +1 × P ( n | n + 1) + P n max k = n +1 v k × (1 − p sk ( i )) . The expected time elapsed before a particular node carryingthe message spreads copies of the message is finally given by E [ t ] . However, E [ D i ] is given by the expected time elapsedbefore any one of the i nodes carrying the message spreadscopies of the message. Since each E [ t n ] is approximatelyinversely proportional to the number of nodes we considerfor the stochastic process, and E [ t ] is a linear combinationof E [ t n ] , we can approximately say E [ D i ] = E [ t ] i . . R OUTING D ELAY
In this section we derive the routing delay for epidemicrouting in a DTN which uses M2M communication. We modelthe evolution of the network with respect to a specific message,from the generation of the message to the spreading of copiesof the message to finally the delivery of the message, as astochastic process, represented as a digraph in Fig 3. Thisstochastic process has a set of states i = 1 , · · · , n tot − wherestate i corresponds to the presence of i copies of the messagein the network, and n tot refers to the total number of nodesin the network. The stochastic process also has an absorbingstate D which represents successful delivery of the messageto the destination. The network spends an average time of E [ D i ] in state i before moving on to one of the next possiblestates. Since we assume no message drops ( i.e., buffers areadequately sized), the probability of transition p i ′ ,i from state i to i ′ is zero for i ′ < i . When the network is in state i , we n tot α D1 2
Figure 3: Digraph representing the stochastic process used tocompute the routing delayassume that the probability that more than one of the nodescarrying a particular message try to transmit the message atthe same time is negligible. Hence, when the network is instate i , at most one of the i nodes will transmit the message.This can be transmitted to at most α − other nodes (sinceat most α nodes can participate in an M2M communicationsession). Hence, the network can evolve from state i only tostates i +1 , · · · , i + α − . We call the set of states reachable ina single transition from state i as Ch ( i ) . This corresponds tothe children of node i in the digraph representing the stochasticprocess (assuming edges corresponding to zero-probabilitytransitions are deleted). Similarly, a state i can be entered onlyfrom states i − , · · · , i − α + 1 . We call the set of states fromwhich state i is reachable in a single transition as P a ( i ) . Thiscorresponds to the parents of node i in the digraph.We then derive the expected delay E [ D ] of epidemic routingby averaging the delay over all the possible paths Φ on thedigraph starting from the starting state and ending in state D . It must be noted such a path corresponds to a sequence ofstates s , s , · · · , s m where s = 1 and s m = D . E [ D ] = X Φ P (Φ) D (Φ)= X (1 , ··· ,D ) ( P ( s | · · · P ( D | s m − )) × ( E [ D ] + · · · E [ D m − ]) . Table I: Simulation settings
Parameters Values
Terrain size (in m × m) × Grid size (in m) × M2M communication session time (in seconds) . M2M α -value Message size (in Kb) Message generation rate Message TTL ∞ Buffer size ∞ Scheduling FCFSSimulation time (in seconds)
This expression can be re-written as follows: E [ D ] = X s X p ∈ P a ( s ) X c ∈ Ch ( s ) ∪{ D } P ( P ath (1 , p )) .p s,p .p c,s .P ( P ath ( c, D )) .E [ D s ] where P ( P ath ( i, j )) represents the total probability of allpaths starting from state i and ending at state j , given thatthe system is initially in state i . P ( P ath (1 , i )) is given by thefollowing recursion P ( P ath (1 , i )) = X p ∈ P a ( i ) P ( P ath (1 , p )) .p i,p . And this can be calculated for all states i efficiently usingdynamic programming. Similarly, P ( P ath ( i, D )) is given bythe recursion P ( P ath ( i, D )) = X c ∈ Ch ( i ) ( p c,i .P ( P ath ( c, D ))) + p D,i . And this can also be calculated efficiently for all i usingdynamic programming.VI. P ERFORMANCE E VALUATION
The simulations were done using the ONE simulator [13],which we custom modified to simulate one-to-one CSMAand M2M communication schemes. The simulation settingsare shown in Table I. Each experiment was run for simulation seconds. The system was verified to have reachedthe stability region within this time. As mentioned earlier,traffic generation is symmetric, i.e., every message generatedis given a randomly chosen destination node.Figure 4 compares the routing delay values obtained fromanalysis and simulations for and nodes, and for twodifferent link bandwidths, namely kbps and kbps. Foreach configuration, the comparison is done over average nodespeeds ( v ) ranging from m/sec to m/sec. These delay valueswere observed in the stability region of the experiments andthe delivery ratios ( i.e., the fraction of messages deliveredsuccessfully) were observed to be close to . . While we havemade a number of approximations in developing our analytical D e li v e r y de l a y ( i n s e c ond s ) Time (in seconds)
Analytical delay for M2M
CSMAM2M 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000 2500 D e li v e r y r a t i o Time (in seconds)CSMAM2M 0 50 100 150 200 250 300 350 0 500 1000 1500 2000 2500 D e li v e r y de l a y ( i n s e c ond s ) Time (in seconds)
Analytical delay for M2M
CSMAM2M 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000 2500 D e li v e r y r a t i o Time (in seconds)CSMAM2M
Figure 5: Routing performance of CSMA and M2M communication schemes for nodes with two different average nodespeeds ( v ), namely and m/sec D e li v e r y de l a y ( i n s e c ond s ) Speed (in metres/second)Analysis-50kbpsSimulation-50kbpsAnalysis-15kbpsSimulation-15kbps 0 20 40 60 80 100 120 140 160 5 5.5 6 6.5 7 7.5 8 D e li v e r y de l a y ( i n s e c ond s ) Speed (in metres/second)Analysis-50kbpsSimulation-50kbpsAnalysis-15kbpsSimulation-15kbps
Figure 4: Delivery delay performance over two differentnumber of nodes (namely, and ) and across differentaverage speeds ( v ), for two different link bandwidthsmodel, we find that the delay values from analysis andsimulations match, thereby validating our model and showingthat the approximations were reasonable ones to make. As isexpected, it is observed that with an increase in node speedsthe routing delay decreases, as the mixing of nodes enablefaster delivery of messages. Moreover, an approximately three-fold increase in the link bandwidth proportionally reduces therouting delay, which is intuitive as messages are pumped threetimes faster for the same contact time scenarios.Figure 5 compares the routing performance of one-to-oneCSMA and M2M schemes. These experiments were run witha lower link bandwidth of kbps for M2M and kbps forone-to-one CSMA. CSMA is given a higher bandwidth since ituses the entire spectrum available while M2M communicationdivides the spectrum among the α ( = 4 ) nodes participatingin a communication session using FDMA. The M2M schemeclearly outperforms the one-to-one CSMA scheme. We findthat when the M2M scheme reaches its stability region, theCSMA scheme has not yet attained stability, and the differ-ences between the two schemes are increasing with time. Thefact that M2M has reached its stability region while CSMA hasnot, is supported by the high delivery ratio values ( i.e., above . ) for M2M and lower delivery ratio values ( i.e., around . ) for CSMA. Also our M2M delay values stabilize nearthe corresponding analytical delay values.VII. C ONCLUSION
In this paper, we have shown for the first time how M2Mcommunication can be used to combat contention, and therebyreduce the routing delay in a DTN. A theoretical model hasbeen developed for the delivery delay of a DTN employingepidemic routing and M2M communication and has been validated against simulations. Using simulations, we have alsoshown that M2M communication significantly outperformscommunication that uses one-to-one CSMA in terms of deliv-ery delay. In the future, we plan to extend the model developedto work with more sophisticated mobility models.A
CKNOWLEDGEMENT
This work was supported by the Department of Science andTechnology (DST), New Delhi, India.R
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