1

An Improved Stochastic Modeling ofOpportunistic Routing in Vehicular CPS

Deze Zeng Member, IEEE, Song Guo, Senior Member, IEEE, Ahmed Barnawi, Member, IEEE,Shui Yu, Senior Member, IEEE and Ivan Stojmenovic, Fellow, IEEE

Abstract —Vehicular Cyber-Physical System (VCPS) provides CPS services via exploring the sensing, computing and communicationcapabilities on vehicles. VCPS is deeply influenced by the performance of the underlying vehicular network with intermittentconnections, which make existing routing solutions hardly to be applied directly. Epidemic routing, especially the one using randomlinear network coding, has been studied and proved as an efficient way in the consideration of delivery performance. Much pioneeringwork has tried to figure out how epidemic routing using network coding (ERNC) performs in VCPS, either by simulation or by analysis.However, none of them has been able to expose the potential of ERNC accurately. In this paper, we present a stochastic analyticalframework to study the performance of ERNC in VCPS with intermittent connections. By novelly modeling ERNC in VCPS using a token-bucket model, our framework can provide a much more accurate results than any existing work on the unicast delivery performanceanalysis of ERNC in VCPS. The correctness of our analytical results has also been confirmed by our extensive simulations.

Index Terms —Vehicular Cyber-Physical System, Epidemic Routing, Random Linear Network Coding, Stochastic Analysis

✦

1 INTRODUCTION

W ITH the recent rapid development on informa-tion and communication technologies, informa-

tion systems transit from the pure cyber space to a hy-brid cyber-physical space. Cyber-Physical System (CPS)will transform how people interact with things aroundby allowing direct observation, coordination and manip-ulation of the physical world via cyber technologies [1].Vehicular CPS (VCPS) integrates the sensing, computingand communication capabilities on vehicles to supporta diversity of applications such as road ads, safetyimprovement, intelligent transportation, environment es-timation, on-road infotainment, etc [2], [3]. Vehicle-to-vehicle (V2V) communication plays an important role inthese VCPS applications. For example, in vehicular par-ticipatory sensing, the collected data shall be deliveredto the “sink node”, e.g., road-side unit (RSU), for post-processing [4]–[6].

The major challenge in the development of VCPS isthe intermittent network connectivity that a transmissionhappens only when two vehicles opportunistically comeinto the communication range of each other. To caterfor such characteristic, the communication in VCPS isconducted in a “store-and-forward” manner [7], i.e.,

• D. Zeng is with the School of Computer Science, China University ofGeosciences (Wuhan), China. E-mail: dazzae@gmail.com

• S. Guo is with the School of Computer Science and Engineering, TheUniversity of Aizu, Japan. E-mail: sguo@u-aizu.ac.jp

• A. Barnawi is with King Abdulaziz University, Jeddah, Saudi Arabia.E-mail: ambarnawi@kau.edu.sa

• S. Yu is with School of Information Technology, Deakin University,Australia

• I. Stojmenovic is with SIT, Deakin University, Melbourne, Australia; KingAbdulaziz University, Jeddah, Saudi Arabia; and SEECS, University ofOttawa, Canada. E-mail: Stojmenovic@gmail.com

a packet is stored at one node and then forwardedto another using V2V communication when there is atransmission opportunity, without relying on an instantnetwork connectivity. Such new scheme has been inten-sively studied and adopted by epidemic routing [7]–[9]as an efficient way to tackle the routing issue in VCPS.

The basic idea of epidemic routing is that a packetis greedily disseminated at each transmission opportu-nity in a hope that at least one copy will succeed inreaching its destination. Several variants (e.g., PRoPHET[10], MaxProp [8], RAPID [11], etc.) have been proposedwith different goals, such as to maximize delivery rate,or minimize delivery delay and resources (e.g., buffer,energy) consumption. A major design challenging issueis that when disseminating multiple packets, a relaynode can hardly make an optimal decision on whichpacket should be replicated and forwarded to achievethe best delivery performance in a network with un-expected connections. Fortunately, it has been foundthat network coding can tackle this issue and improvethe performance significantly [12]–[15]. Using networkcoding, a node only needs to simply forward a ran-dom linear combination of packets it has received usingrandom linear network coding technique upon eachtransmission opportunity. When the destination receivesenough number of linearly independent coded packets,the original packets can be recovered. Epidemic routingusing network coding (ERNC) can achieve much higherdelivery performance than the non-coding schemes asshown by both simulations [12], [15] and analysis [13].

The existing theoretical work [13] is based on anassumption that each received coded packet at the desti-nation is innovative with high probability. However, oursimulation study shows that this assumption is not true

2

and the resulting analysis only roughly approximatesthe reality. This motivates us to develop an improvedstochastic model to analyze the delivery performance ofERNC in terms of delivery delay of a group of packets.In particular, we propose a token bucket model that canachieve a significantly improved accuracy for checkingthe independency of each received coded packet. Toanalyze such token-bucket based stochastic process, atwo-dimensional time-heterogeneous Markov chain isdeveloped, and then the analytical performance of ERNCis derived using ordinary differential equations (ODEs).The correctness and accuracy of our theoretical analysisis validated by extensive simulations. Thanks to theaccurate capturing and modeling the network codingbehaviours, our analysis shows much higher accuracy,compared to existing theoretical work.

The rest of this paper is organized as follows. Section 2gives the system model and problem statement. Section3 presents stochastic modeling and analysis on ERNC.Section 4 shows our performance evaluation results. Sec-tion 5 provides a brief overview of related work. Finally,Section 6 concludes our work. For the conveniences ofthe readers, the major notations used in this paper arelisted in Table 1.

2 SYSTEM MODEL AND PROBLEM STATE-MENT

2.1 Network Model

We consider a VCPS with N + 1 vehicles, i.e., mobilenodes. Two nodes are able to communicate only whenthey come into the reciprocal radio range and we definethis as a “contact” between them. Let pairwise inter-contact interval between a pair of nodes denote theduration of time from the time when they go out of trans-mission range of each other to the next time they comeinto each other. To improve the existing analytical resultson the performance of ERNC in VCPS, we adopt thesame mobility model, in which the pairwise encounterinterval satisfies the exponential distribution with thesame rate λ. This model has been widely accepted inthe literature [13], [16], [17] because it is validated as agood approximation for the inter-contact interval in anumber of realistic vehicular networks [16], [18].

The packet size is the maximum data that can betransferred from one node to another during one contact.In other words, at most one packet is forwarded ateach transmission opportunity. Similar to the modelsused in [13], [17], we assume a single-packet memoryat each relay node and at most one packet is allowedto be carried by a relay node during its movement. Weconsider a unicast application session with K packetsoriginated from the source node and destined to the soledestination node via the help of relay nodes.

2.2 Routing Scheme

Since network partition happens in VCPS, all transmis-sions are performed in a “store-and-forward” manner.

TABLE 1Notations

N number of nodes in the networkλ pair-wise contact rateK the total number of packets to deliverFq Galois field with field size q

Iphy(t) The number of nodes that are infected with codedpackets

pphy(k, t) The probability that the destination node receivesk coded packets

P phy(K, t) The probability that the destination node receivesat least K coded packets

Ivir(t) The number of nodes that are infected with onetoken

pa(t) The probability that the destination node receivesone token

pvird

(k, t) The probability that the destination node receivesk tokens

P vird

(K, t) The probability that the destination node receivesat least K tokens

λ(t) The packet arrival rate in the token bucket modelµ(t) The token arrival rate in the token bucket modelX(Np, Nv, t) The probability that the destination node receives

Np packets and Nv tokensE[T ] The expected delivery delay of K packets using

ERNC

A packet is stored in the local buffer of its carrier untila transmission opportunity arises. ERNC scheme hasbeen shown with much improved delivery performanceover the traditional non-coding ones by both empiri-cal study [15] and theoretic analysis [13]. In ERNC, acoded packet pcoded is a linear combination of nativepackets pnative1 , pnative2 , · · · , pnativeK in the form: pcoded =∑K

i=1 αipnativei , where αi, i = 1, . . . ,K , are called coding

coefficients and are randomly chosen from a Galois Filed(Fq) [12]. The source node generates such a coded packetby randomly encoding all the native packets and for-wards it to the encountered node, which shall reserve thecoded packet in its local buffer. Suppose two relay nodesa and b encounter and a coded packet pcodeda is forwardedfrom node a to node b. Once received, node b updates itsown encoded packet pcodedb as pcodedb = αpcodedb + βpcodeda ,where α and β are also randomly chosen from Fq . In thisway, single buffer that holds one coded packet at eachrelay node can achieve almost the same performance asthe multiple buffer case as discovered in [13]. After thedestination node collects an enough number of linearlyindependent coded packets, it is able to retrieve all theoriginal native packets by Gaussian Elimination.

Let us use the example shown in Fig. 1 to illustrate theERNC process, in which source node s has two packetsp1 and p2 to be delivered to destination node d.

t1: Source s encounters relay r1 and transmits alinearly coded packet x = 1p1 + 2p2 with coef-ficients randomly selected from F256. Supposethe buffer of r1 is empty, its received codedpacket is stored in its original form.

t2: Relay r1 encounters destination d and forwardsx in its buffer.

3

Fig. 1. ERNC process illustration

t3: Source s encounters relay r2 and transmits anew coded packet y = 2p1 + 3p2.

t4: Relay r1 encounters relay r2 and forwards x inits buffer. By linearly combining the receivedpacket x and its stored packet y, relay r2 up-dates its buffer as z = 2x+ y = 4p1 + 7p2.

t5: Relay r2 encounters destination d and forwardsz in its buffer. Eventually, destination d obtainstwo linearly coded packets x and z with codingcoefficient matrix

(

1 24 7

)

with a rank of 2. There-fore, the original p1 and p2 can be recovered.

2.3 Problem Statement

As all communications in VCPS rely on the unpre-dictable transmission opportunities, it is desired to quan-titatively study the stochastic transmission process ofERNC, which is characterized in this paper by an impor-tant metric: the decoding probability at the destinationby any time. Later, the delivery delay of a group ofpackets from the generation to the reception of the entiregroup at the destination will be studied.

Existing modeling and analysis (e.g., [13]) of ERNCare based on a strong assumption: one relay node cantransmit an innovative coded packet with probability 1to another node such that a coded packet received bythe destination node from any encountered node can bealways regarded as innovative.

We also verify that this assumption is not valid inmost cases by experiments. For example, we simulatedelivering 100 native packets and count the number ofcoded packets received at the destination until all nativeones could be decoded in a network with N = 200 andλ = 0.005. The results in Fig. 2(a) show that in most cases(i.e., in 83%) the destination node must obtain more than100 coded packets before a successful decoding becomespossible. This number sometimes goes to as high as 148.

In other words, a significant portion of received codedpackets are actually dependent, i.e., non-innovative. Thisfurther incurs inaccuracy of the analysis on deliverydelay. In Fig. 2(b), we show both simulation and an-alytical results (according to [13]) of delivery delay incumulative distribution function (CDF). We observe thatthe analytical result is quite optimistic compared to thesimulated one. For example, the decoding is expectedat time 110 with probability 70% by analysis, whilethe true probability is only around 30%. Therefore, thekey challenging factor to the performance evaluation ofERNC in VCPS is that an accurate stochastic modelingof both data transmission and decoding is required. Thismotivates us to find an accurate analytical model thatcan truly reveal the successful decoding probability at agiven time.

3 STOCHASTIC MODELING AND ANALYSIS ONERNCIn this section, we first present a logical view of the de-coding process and describe it by a token-bucket model.We then develop a two-dimensional Markov chain basedon the token-bucket model to derive the probability ofdecoding K native packets at the destination node.

3.1 Decoding Process Modeling

By ERNC, each coded packet in Pcoded =

[pcoded1 , · · · , pcodedK ] is a linear combination of Knative packets P

native = [pnative1 , · · · , pnativeK ]. Thedestination node will successfully decode all nativepackets when it is able to solve a linear equationsystem P

coded = CPnative, where P

native are unknownvariables and C is the coefficient matrix generated bythe ERNC process. As long as K innovative codedpackets are received, all K native packets are decodable.Conventionally, to determine the innovativeness of acoded packet is done by checking whether the receivedcoded packet can increase the rank of all received onesso far. However, it is difficult to directly model thecoefficient matrix because P

native is randomly encodedinto P

coded by relay nodes during contacts.We observe that at any time, the rank of all coded

packets in the network is equal to the number of nativepackets that have been injected into in the network bythe source. The rank of all received coded packets by thedestination can thus be obtained by checking the numberof “received” unknown variables that have shown in thecorresponding linear equation system. Specifically, thereception of an unknown variable implies that the corre-sponding coefficient is non-zero in at least one equationreceived so far at the destination. Therefore we definesuch arrival of “unknown variable” as virtual arrival todistinguish from the physical arrival of coded packets.This leads to a simple way to check the innovativenessof a coded packet as follows:

Lemma 1: If a coded packet makes the number ofequations exceed the number of unknown variables at

4

0 200 400 600 800 1000100

110

120

130

140

150

Simulation Instances

Num

ber

of P

acke

ts R

ecei

ved

(a) CDF of the actual number of coded packetsreceived before 100 native packets are successfullydecoded

80 100 120 140 1600

0.2

0.4

0.6

0.8

1

Delivery Delay

CD

F

SimulationAnalysis

(b) Comparison of the simulation results and theanalytical results according to [13]

Fig. 2. The experimental results of a rough existing analytical model

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

Miss Ratio

CD

F

Lemma 1Analysis

Fig. 3. Miss ratio of dependency checking

the destination node, then such coded packet is non-innovative.

The conclusion is straightforward because the rowrank (i.e., the number of received unknown variables)and column rank (i.e., the number of equations main-tained) of the coefficient matrix must be the same. If acoded packet satisfies the above rule, it can be regardedas non-innovative definitely. For example, suppose thesource node only disseminates two linearly independentcoded packets into the network. No matter how manydifferent linearly coded packets are generated during thecontacts between relay nodes and how many packetsare received by the destination node, it is impossiblefor the coefficient matrix at the destination to exceed 2.Lemma 1 is a necessary but not sufficient condition fordecoding. Our simulation study shows that a processonly applying the rule in Lemma 1 can approximate therealistic decoding process with a very high accuracy.

To validate the proposed method, we conduct simu-lation experiments with the following rules for innova-tiveness checking.

• Conventional rule: Upon receiving a coded packetfrom an encountered node, the destination nodechecks if its coefficient vector is linearly indepen-

dent to the coefficient matrix of other packets thathave been already received (i.e., rank checking). Ifand only if it is innovative, it is accepted. Otherwise,it is discarded.

• Optimistic rule: According to [13], it is assumedthat the received coded packet is always innovative.As a result, without any independency checking,any coded packet is always accepted uncondition-ally.

• Lemma-1 based rule: A coded packet is accepted ifand only if the total number of received unknownvariables exceeds the number of coded packets re-ceived so far.

Compared with the conventional rule, if both rules re-gard a coded packet as non-innovative, we say our rulehits while if a non-innovative coded packet is treated asinnovative by our rule, it is a miss. The simulation ofdelivering 100 packets is conducted in a network withN = 200 and λ = 0.005. The statistics of hit ratio andmiss ratio are obtained by running it 5000 times. Wenotice that our rule can achieve 100% hit ratio and verylow miss ratio, which is plotted in CDF as shown in Fig.3. For example, most cases by Lemma 1 have a missratio of 0% and 97.7% of them are with a miss ratio lessthan 5% while only 65.4% of optimistic model achievessuch miss ratio. This validates the high accuracy of ourinnovativeness checking rule.

Our experimental results also show that the processesof physical arrival and virtual arrival can be approxi-mately considered as independent since their Pearsonproduct-moment correlation coefficient [19] and mutualinformation [20] are as low as -0.039 and 0.0154, re-spectively, by running simulations 10000 times usingdifferent random seeds. These discoveries further inspireus to describe the decoding process by a token bucketmodel with two independent arrival processes as shownin Fig. 4. It works as follows.

1) A token arrival is equivalent to a virtual reception.Tokens are collected in an unlimited buffer.

2) A physical (packet) arrival can be accepted, called

5

Fig. 4. Token bucket model

a physical reception, if and only if there existsat least one token in the buffer. Each physicalreception consumes one token. Otherwise, it willbe discarded.

In this token bucket model, to accept a packet isthus equivalent to obtain an additional innovative codedpacket. By analyzing the probability that K packets areaccepted in the token bucket model at time t, we couldobtain the decoding probability at t. To this end, we shallanalyze the two arrival processes first.

3.2 Analysis on the Physical Arrival Process

According to the process of ERNC, a packet arrival isequivalent to a contact between the destination node anda node with coded packet. Under the infectious diseasemodel, if a node carries a packet in its local buffer, it isregarded as “infected”. Initially, only the source nodeis infected and all the other nodes are “susceptible”to be infected. Once the transmission starts, the sourcenode begins to infect the other nodes by the contactopportunities during its movement. The infected nodes(excluding the destination) are then able to infect othersusceptible nodes and so on. Therefore, the number ofinfected nodes at time t, denoted as Iphy(t), can bedescribed by the following ordinary differential equation(ODE):

dIphy(t)

dt= λIphy(t)(N − Iphy(t)), (1)

where λ is the average pairwise contact rate.

Solving (1) under the initial condition Iphy(0) = 1,(i.e., at time 0, only the source node can be regardedas infected), we get

Iphy(t) =N

1 + (N − 1)e−λNt. (2)

Next, we can derive the expected number of infectednodes that have contacted the destination node by timet as:

g(t) =

∫ t

0

λIphy(τ)dτ

= lneλNt +N − 1

N.

In our VCPS model, the contact process between anytwo nodes is a Poisson process since their contact inter-val is exponentially distributed. The contact process ofmeeting a infected node by the destination node is thesum of the Poisson processes with each destination nodeand is also a Poisson process with rate equals to thesePoisson processes’ rate. In other words, the destinationnode contacts an infected node in an exponentially dis-tributed interval. As a result, the probability that thereare exactly k (k = 0, 1, 2, · · · ) contacts with the infectednodes by time t can be written as

pphy(k, t) =g(t)k

k!e−g(t). (3)

Therefore, the probability that there are at least Kpacket arrivals at the destination node at time t is:

P phy(K, t) = 1−

K−1∑

k=0

g(t)ke−g(t)

k!

= 1−N

eλNt +N − 1

K−1∑

k=0

(ln eλNt+N−1N

)k

k!.

(4)It shows that the packet arrival process is a non-homogeneous Poisson process.

3.3 Analysis on the Virtual Arrival Process

A token, say ti, emerges in the network whenever thesource node delivers a native packet (pnativei ) to anothernode encountered. In ERNC, any relay node is “infected”by this token as long as the coded packet in its bufferhas a non-zero coefficient of the component relating topnativei . The destination node detects the arrival of tokenti when it, by the first time, encounters a relay nodeinfected by ti. Let Ivir(t) be the number of relay nodesthat have been infected by token ti at time t with astarting point t = 0 when the source node deliverspnativei . A fundamental difference from the last sectionis that after t = 0, the source node is get ready forthe delivery of next token and only the encounterednode that receives ti can be regarded as infected, i.e.,Ivir(0) = 1. Therefore, we have

dIvir(t)

dt= λIvir(t)(N − 1− Ivir(t)). (5)

Solving (5) under the initial condition Ivir(0) = 1yields

Ivir(t) =N − 1

1 + (N − 2)e−λ(N−1)t. (6)

Let T virp be the propagation delay of token ti, which

is defined as the time from the source delivers pnativei

to the time when the destination detects its arrival, i.e.,encounters a relay node infected by ti by the first time.Since the propagation of ti is epidemic, the CDF of T vir

p ,denoted as P vir

p (t) = Pr(T virp < t), can thus be obtained

by solving the following ODE developed in [21]:

dP virp (t)

dt= λIvir(t)(1 − P vir

p (t)). (7)

6

Notice that in our model, packet pnativei (equivalentlytoken ti) could be directly delivered to the destination atthe very beginning with probability 1/N , i.e., P vir

p (0) =1/N .

By solving the ODE in (7), we have:

P virp (t) = 1− C1e

−

∫λIvir(t)dt, (8)

where∫

λIvir(t)dt can be derived as follows:∫

λIvir(t)dt =

∫

λ(N − 1)

1 + (N − 2)e−λ(N−1)tdt

=

∫

deλ(N−1)t

N − 2 + eλ(N−1)t

= ln(N − 2 + eλ(N−1)t) + C2.

(9)

Let C3 = C1e−C2 , P vir

p (t) can be written as:

P virp (t) = 1− C3

1

N − 2 + eλ(N−1)t. (10)

Solving P virp (0) = 1/N yields:

C3 =(N − 1)2

N. (11)

Taking (11) into (10), we finally obtain the solution ofP virp (t) from (7):

P virp (t) = 1−

(N − 1)2

N(N − 2 + eλ(N−1)t). (12)

Now we consider the system starting time t = 0that begins when the source node is just ready fordata delivery. Each token goes into network with anexponentially distributed interval at rate λN accordingto the routing protocol. Thus the probability that the k-th token arises in the network at time t is equivalent tothe probability that total k tokens have been injected, asa Poisson process, into the network at time t:

ps(k, t) =(λNt)k

k!e−λNt. (13)

Then the k-th token propagates to the destination nodeindependently in an epidemic manner. The CDF of itsarrival time, denoted as P vir

a (k, t), can be calculated as

P vira (k, t) =

∫ t

0

ps(k, τ)Pvirp (t− τ)dτ. (14)

Simulation based study is also conducted to verify thecorrectness of (14). In our experiments, we first specifyany token, and then trace and check the time when acoded packet including such token is first received bythe destination. We run the simulation several roundswith different random seeds to obtain the CDFs of arrivaltime for the 50th token and 100th token. Both resultsfrom simulation and (14) are shown in Fig. 5. Theymatch quite well in both cases. This confirms our under-standing that each token can be regarded as propagatingindependently and individually by epidemic routing, asif no other tokens exist.

40 60 80 100 120 1400

0.2

0.4

0.6

0.8

1

Arrival Time

CD

F

SimulationAnalysis

k=50

k=100

Fig. 5. CDFs of arrival time for the 50th token and 100thtoken in a VCPS with N = 200 and λ = 0.005.

We notice that the arrival of the k-th token at thedestination node does not mean the arrival of at leastk tokens because their propagations are independent.According to our token bucket model, we are interestedin the probability that at least k tokens arrive at thedestination node by time t. Recall that the emergenceof tokens is a Poisson process. By the uniformity propertyof Poisson process, the time that a token appears in agiven period is uniformly distributed.

Supposing the time that such token appears in thenetwork is τ and its arrival time at the destination is t(i.e., the propagation delay is t−τ ), we apply the uniformproperty to derive the probability that each token hasarrived at the destination node at time t as:

pa(t) =

(∫ t

0

1−(N − 1)2

N(eλ(N−1)(t−τ) +N − 2)dτ

)

/t

= 1−

(∫ t

0

(N − 1)2dτ

N(eλ(N−1)(t−τ) +N − 2)

)

/t

= 1−N − 1

(N − 2)λNt

∫ t

0

deλ(N−1)τ

eλ(N−1)τ + eλ(N−1)t/(N − 2)

= 1−N − 1

(N − 2)λNtln

eλ(N−1)t + eλ(N−1)t/(N − 2)

1 + eλ(N−1)t/(N − 2)

= 1−N − 1

(N − 2)λNtln

N − 1

(N − 2)e−λ(N−1)t + 1.

(15)

Lemma 2: The probability that there are exactly k tokenarrivals by time t is

pvird (k, t) =(λNtpa(t))

k

k!e−λNtpa(t). (16)

Proof: If M(M ≥ k) tokens have emerged duringinterval (0, t), the probability that k of them have arrivedat the destination node by time t is

(

M

k

)

pa(t)k(1− pa(t))

M−k , 0 ≤ k ≤ M, (17)

where pa(t) is the probability that a token received bythe destination node at time t, as defined in (17).

7

Note that the destination node can obtain k packetsonly when there are at least k tokens emerging in thenetwork. In other words, M must be larger or equal thank (i.e., M = k, k + 1, k + 2, ...). As we have known, thetokens are sent out by the source node to the networkfollowing a Poisson process. The probability that thereare M tokens in the network is ps(M, t) as defined in(13). Therefore, we have:

pvird (k, t)

=

∞∑

M=k

(λNt)M

M !e−λNt

(

M

k

)

pa(t)k(1− pa(t))

M−k

=

∞∑

M=k

(λNt)M

M !e−λNt M !

(M − k)!k!pa(t)

k(1 − pa(t))M−k

= e−λNt

∞∑

M=k

(λNt)Mpa(t)k(1− pa(t))

M−k

(M − k)!k!

= e−λNt

∞∑

M=k

(λNt(1 − pa(t))M−k

(M − k)!·(λNtpa(t))

k

k!

(18)Let Y = M − k, pvird (k, t) can be then rewritten as:

pvird (k, t)

=(λNtpa(t))

k

k!· e−λNt

∞∑

Y=0

(λNt(1 − pa(t))Y

Y !

=(λNtpa(t))

k

k!· e−λNt · eλNt(1−pa(t))

=(λNtpa(t))

k

k!e−λNtpa(t).

(19)

From Lemma 2, we can see that the token arrivalprocess is a heterogeneous Poisson process. This imme-diately leads to the following lemma.

Lemma 3: The probability that there are at least Ktoken arrivals is:

P vird (K, t) = 1−

K−1∑

i=0

pvird (i, t). (20)

3.4 Analysis on the Decoding Probability

According to our analysis, both token and packet arrivalprocesses are heterogeneous Poisson process with arrivalrate

λ(t) =d(λNtpa(t))

dt

= λN −λ+ λN(N − 2)

N − 2 + e−λ(N−1)t

(21)

and

µ(t) =λN

1 + (N − 1)e−λNt, (22)

respectively.We denote the total number of accepted packets as Np

and the number of token arrivals (including the tokens inthe buffer and the ones that have been consumed) as Nv,

Fig. 6. State transition of the token bucket model

respectively. The state transition process can be describedby a two-dimensional Markov chain shown in Fig. 6,where each state is represented by (Np, Nv) and Np ≤ Nv

holds. By denoting X(Np, Nv, t) as the probability thatthe Markov chain stays at (Np, Nv) at time t, we candescribe the transition process by a group of ODEs asfollows. According to the transition characteristics, thewhole figure can be divided into four regions.

Region I: all states in the first line except (0,K). InRegion-I, besides self-transition, state (0, 0) transits onlyto (0, 1) while all others except (0, 0) transits to twoneighboring states. These can be described as:

X ′(0, 0, t) = −λ(t)X(0, 0, t)

X ′(0, i, t) = λ(t)X(0, i− 1, t)− (λ(t) + µ(t))X(0, i, t),

∀i = [1, 2, · · · ,K − 1](23)

Region II: all states in the diagonal line except (0, 0)and (K,K). No packet arrival can be accepted any moreif the system is in a state of Region-II as this is equivalentto the condition that all tokens in the token buffer havebeen used up already. Therefore, each state (i, j) inRegion-II depends on state (i − 1, j) and would transitto (i, j + 1). We then have

X ′(i, i, t) = µ(t)X(i− 1, i, t)− λ(t)X(i, i, t),

∀i = [1, 2, · · · ,K − 1](24)

Region III: all states in the last column. In a similarway, we can derive ODEs for the states in Region-III as:

X ′(0,K, t) = λ(t)X(0,K − 1, t)− µ(t)X(0,K, t),

X ′(i,K, t) = λ(t)X(i,K − 1, t) + µ(t)X(i− 1,K, t),

− µ(t)X(i,K, t), ∀i = [0, 2, · · · ,K − 1],

X ′(K,K, t) = µ(t)X(K − 1,K, t).(25)

Region IV: all remaining states in the central region.Each state (i, j) in Region-IV relies on its two neighbor-ing states (i−1, j) and (i, j−1) on the upper and left, and

8

40 60 80 100 120 1400

0.2

0.4

0.6

0.8

1

Delivery Delay

CD

F

SimulationAnalysis

K=60

K=80

K=100

(a) On the effect of packet number

50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Delivery Delay

CD

F

SimulationAnalysis

N=300 N=200 N=100

(b) On the effect of network size

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

Delivery Delay

CD

F

SimulationAnalysis

λ=0.004

λ=0.008

λ=0.01

(c) On the effect of pairwise contact rate

0 20 40 60 80 1000

50

100

150

200

250

Number of Packets

Ave

rage

Del

iver

y D

elay

SimulationAnalysis

λ=0.05, N=200

λ=0.008, N=100

λ=0.01, N=100

(d) Average delivery delay

Fig. 7. Performance evaluation on the accuracy of the analysis

also would transit to the other two neighboring states(i + 1, j) and (i, j + 1) on the lower one and right:

X ′(i, j, t) = λ(t)X(i, j − 1, t) + µ(t)X(i − 1, j, t)

− (λ(t) + µ(t))X(i, j, t),

∀i = 1, 2, · · · ,K − 1, j = i+ 1, · · · ,K − 1.

(26)

Numerical method can be applied to solve the ODEsin (23)-(26) under the boundary condition:

X(i, j, 0) =

{

1, (i, j) = (0, 0),

0, otherwise.(27)

Note that, if the system transits into state (K,K), weconclude that all K native packets are decodable now.Therefore, X(K,K, t) represents the probability that allK packets have been successfully received by time t. Inother words, the value of X(K,K, t) can be regarded asthe CDF of decoding probability for K packets at timet.

Theorem 1: The expected delivery delay of K packetsby ERNC in VCPS is

E[T ] =

∫

∞

0

(1−X(K,K, τ))dτ, (28)

where X(K,K, τ) is obtained by jointly solving (21)-(27).

4 PERFORMANCE EVALUATION

In this section, we present our evaluation on the accuracyof the delivery delay analysis by comparing the simula-tion results to the analysis results obtained by numericalmethod. The simulation results reported in this paper areobtained through a self-developed discrete-event simu-lator that strictly follows the system model presented inSection 2. For practical network coding operations, Ga-lois field F256 is adopted. Numerical results are acquiredusing Matlab ODE solver ode45.

4.1 On the Accuracy of Our Analysis

According to our analysis, it can be inferred that thedelivery probability of a content in VCPS is determinedby its content size K (i.e., the number of packets),network size N and contact rate λ. In order to thoroughlyshow the accuracy of our analysis, we conduct threegroups of simulations with different values of K , Nand λ, respectively. In each group, in order to producesmooth CDF curve on the successful delivery probabilityon time t, 1000 rounds of simulations with differentrandom seeds are performed. Fig. 7 shows the evaluationresults in terms of CDFs of delivery delays and averagedelivery delay.

The parameter settings for our experiments are asfollows: N = 200 and λ = 0.005 in Fig. 7(a), K = 100 andλ = 0.005 in Fig. 7(b), and N = 100 and K = 50 in Fig.7(c). For all cases, the analytical and simulation results

9

are almost overlapped. The correctness high accuracy ofour analysis is validated. This validates the correctnessand accuracy of our stochastic modeling and analysisincluding the token bucket model for decoding processand the physical/virtual arrival process characterized bya time-dependent two-dimensional Markov chain.

To validate the accuracy of our analysis on averagedelivery delay as shown in Theorem 1, simulation s-tudies under various network settings by differing thevalues of N and λ (i.e., (N , λ)=(100, 0.005), (100, 0.008)and (200, 0.005)) are conducted. Fig. 7(d) shows theaverage deliver delay E[T ] as a function of the number ofpackets (K) to deliver under each network setting. It canbe obviously noticed that E[T ] is a linearly increasingfunction of K . Furthermore, we also see that the analysisresults always tightly match with the simulation results.This validates the high accuracy of our derivation inTheorem 1.

4.2 On the Significance of Our Analysis

To show the significance of our accurate analysis, weconsider a scenario where the content to be delivered iswith a predetermined lifetime Tl set according to the de-sired delivery probability, i.e., Tl = argmint Prob(t) ≥ P ,where Prob(t) is the probability that the content has beendecoded by time t and P is the desired delivery prob-ability within the content’s lifetime. An outage happenswhen a content cannot be successfully decoded at thedestination within its lifetime. We simulate the unicastof a content with 100 packets (i.e., K = 100) in a networkwith N = 200 and λ = 0.005. We vary the desired de-livery probability and set the content lifetime accordingto analysis in [13] and ours (i.e., Prob(t) = X(K,K, t)),respectively. For each case, 1000 simulation instances areconducted to obtain the outage probability. Fig. 8 showsthe simulation results. Obviously, we can see that theinaccurate analysis on the decoding probability leads toa high outage probability. For example, when the desireddelivery probability is 0.80, the outage probability is ashigh as 0.613 if the message lifetime is set according to[13]. Thanks to the accurate analysis on the decodingprobability, our analysis can always make sure that thedesired delivery probability is achieved.

5 RELATED WORK

5.1 Vehicular Cyber-Physical System

VCPS has emerged as a cutting edge technology forthe next-generation intelligent transportation system. Inthe literature, various VCPS architectures have beenproposed. Kim et al. [22] propose a CPS applicationframework that is able to provide a generic service torepresent, manipulate, and share knowledge across thenetwork without persistent network connectivity, e.g.,Delay-Tolerant networks (DTNs). Jia et al. [3] proposea platoon-based VCPS architecture and investigate theexpected time of safety message delivery among pla-toons with the consideration of traffic flow, velocity,

0.80 0.85 0.90 0.95 0.98 0.990

0.2

0.4

0.6

0.8

Desired Delivery Probability

Out

age

Prob

abili

ty

Analysis [12]Our Analysis

Fig. 8. Simulation results on the outage probability

platoon size and transmission range. Ni et al. [23] ex-plore a distributed cyber-physical solution by integrat-ing networked and embedded sensing, computationalintelligence and real-time vehicular communications toincrease the safety and efficiency of transportation sys-tems.

Besides the architecture issue, a diversity of algorithmshave been studied towards different VCPS applicationoptimizations. Miloslavov and Veeraraghavan [2] pro-pose a sensor data fusion algorithm for calculating aver-age per-division (i.e., a small segment of a route) speedand average route travel time in VCPS. Li et al. [24]devise a human factor aware service delivery schemefor VCPS that can transmit multiple messages with time-dependent utility to a subset of intended drivers so as tominimize the network-wide service utility loss. Later, Liet al. [25] study the targeted on-road ad delivery problemwith the joint consideration of message scheduling andAP bandwidth allocation.

5.2 Content Delivery in DTNs

Since the underlying network of VCPS is of intermittentconnectivity, the communications must be robust againstdelays and disruptions due to vehicle mobility. Mucheffort has been devoted to the performance analysis ofDTN routing protocols.

For the delivery of a single packet, flooding-basedepidemic routing, firstly proposed by Vahdat et al. [7],achieves the shortest delay [26]. Zhang et al. [21] de-rive the closed-form expression of delivery delay forsingle-packet unicast using epidemic routing in DTNsby an ODE-based approach. Subramanian et al. [27]analyze the efficiency of multihop routing over two-hop routing for multiple unicasts in DTNs with finite-sized buffer. Gunasekaran et al. [28], [29] apply QueuingPetri-Net and derive the end-to-end delivery delay fromthe modeled semi-Markov process. Khouzani et al. [30]investigate an energy-aware optimal epidemic routingin DTNs by a threshold-based scheme, where the for-warding probability is determined by relay node’s cur-rent remaining energy. Recently, Kim et al. [31] presenta distributed cross-layer monitoring and optimization

10

method for secure content delivery toward decentralizedcontent-based mobile ad hoc networking. Abdrabou etal. [6] analyze the message delivery delay in a two-laneroad where vehicles moving in one direction send theirsensing data to vehicles in the opposite direction in orderto deliver the packets to the nearest road-side unit.

For the multi-packet delivery, network coding hasbeen proved as a promising way to improve the deliveryperformance. There are many different network codingtechniques available in the literature. Rateless codes, likeLT code and Raptor code, require a predefined codingstructure (e.g., Soliton distribution) and hence are notsuitable for multihop forwarding. As a result, the studieson rateless codes, such as [17], [32]–[34], usually limit totwo-hop routing (i.e., source-relay-destination), withoutthe relay-to-relay transmissions.

When relay-to-relay transmission is considered, ran-dom linear network coding has shown its compellingstrength in the delivery performance. Widmer et al.[12] compare the performance of random linear networkcoding based forwarding to alternative protocols andfind out that it significantly outperforms probabilis-tic routing, particularly in challenging scenarios whereconnectivity is rare. In [15], Zhang et al. show thatthe network coding scheme achieves slightly smalleraverage block delay than non-coded schemes underunconstrained buffer case, but significant benefits underconstrained buffer case. To optimize the network codingperformance in DTNs, Ali et al. [35] propose a contentdelivery mechanism by combining network coding andglobal selective acknowledgement. The applicability ofnetwork coding to DTNs has been also validated bypractical testbed. In [36], Joy et al. develop a networkarchitecture using Android phones that exploits partialcaches by utilizing network coding to deliver large files(e.g. images).

6 CONCLUSION

In this paper, we have developed a framework to eval-uate the delivery performance of opportunistic routingby ERNC in VCPS. To accurately analyze the deliverydelay of a group of packets using ERNC, we first novellypartition the decoding process at the destination nodeinto two different arrival processes, namely physicalarrival process and virtual arrival process, which arethen translated into packet arrival and token arrivalprocesses in a token-bucket model, respectively. The twoarrival processes to the token bucket are stochasticallyanalyzed and described using ODEs. Then, based on atwo-dimensional Markov chain, the decoding probabil-ity at any time and the average delivery delay of allpackets are derived. The correctness and accuracy of ouranalysis has been extensively validated by simulations.Our work provides the first accurate performance anal-ysis of ERNC in VCPS.

ACKNOWLEDGEMENT

This research is supported by the Fundamental ResearchFunds for the Central Universities, China Universityof Geosciences (Wuhan) (Grant No. CUG140615). Thisresearch of Ivan Stojmenovic is partially supported byNSERC Discovery grant (Canada), and TR32016, Ser-bian Ministry of Science and Education (associated withSingidunum University, Serbia).

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[30] M. Khouzani, S. Eshghi, S. Sarkar, N. B. Shroff, and S. S.Venkatesh, “Optimal Energy-aware Epidemic Routing in DTNs,”in Proceedings of the 13th ACM International Symposium on MobileAd Hoc Networking and Computing (MOBIHOC). ACM, 2012, pp.175–182.

[31] M. Kim, A. Gehani, J.-M. Kim, D. Tariq, M.-O. Stehr, and J.-s. Kim,“Maximizing Availability of Content in Disruptive Environmentsby Cross-layer Optimization,” in Proceedings of the 28th AnnualACM Symposium on Applied Computing (SAC). ACM, 2013, pp.447–454.

[32] E. Altman and F. De Pellegrini, “Forward Correction and Fountaincodes in Delay Tolerant Networks,” in Proceedings of The 28thAnnual IEEE International Conference on Computer Communications(INFOCOM). IEEE, Apr. 2009, pp. 1899–1907.

[33] E. Altman, F. De Pellegrini, and L. Sassatelli, “Dynamic Controlof Coding in Delay Tolerant Networks,” in Proceedings of The 29thAnnual IEEE International Conference on Computer Communications(INFOCOM), vol. 1, no. 4. IEEE, Mar. 2010, pp. 1–5.

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[35] A. Ali, M. Panda, T. Chahed, and E. Altman, “Improving theTransport Performance in Delay Tolerant Networks by RandomLinear Network Coding and Global Acknowledgments,” Ad HocNetw., vol. 11, no. 8, pp. 2567–2587, Nov. 2013.

[36] J. Joy, Y.-T. Yu, M. Gerla, S. Wood, J. Mathewson, and M.-O. Stehr,“Network Coding for Content-based Intermittently ConnectedEmergency Networks,” in Proceedings of the 19th Annual Interna-tional Conference on Mobile Computing & Networking (MobiCom).New York, NY, USA: ACM, 2013, pp. 123–126.

Deze Zeng received his Ph.D. and M.S. degreesin computer science from University of Aizu,Aizu-Wakamatsu, Japan, in 2013 and 2009, re-spectively. He received his B.S. degree fromSchool of Computer Science and Technology,Huazhong University of Science and Technolo-gy, China in 2007. He is currently an associateprofessor in School of Computer Science, ChinaUniversity of Geosciences (Wuhan), China. Hiscurrent research interests include: cloud com-puting, software-defined sensor networks, data

center networking, networking protocol design and analysis. He is amember of IEEE.

Song Guo (M’02-SM’11) received the PhD de-gree in computer science from University of Ot-tawa, Canada. He is currently a Full Professorat School of Computer Science and Engineer-ing, the University of Aizu, Japan. His researchinterests are mainly in the areas of protocoldesign and performance analysis for computerand telecommunication networks. He receivedthe Best Paper Awards at ACM IMCOM 2014,IEEE CSE 2011, and IEEE HPCC 2008. Dr. Guocurrently serves as Associate Editor of IEEE

Transactions on Parallel and Distributed Systems and IEEE Transac-tions on Emerging Topics in Computing. He is in the editorial boards ofACM/Springer Wireless Networks, Wireless Communications and Mo-bile Computing, and many others. He has also been in organizing andtechnical committees of numerous international conferences, includingserving as a General Co-Chair of MobiQuitous 2013. Dr. Guo is a seniormember of the IEEE and the ACM.

Ahmed Barnawi is an associate professor atthe Faculty of Computing and IT, King Abdu-laziz University, Jeddah, Saudi Arabia, wherehe works since 2007. He received PhD at theUniversity of Bradford, UK in 2006. He wasvisiting professor at the University of Calgary in2009. His research areas are cellular and mo-bile communications, mobile ad hoc and sensornetworks, cognitive radio networks and security.He received three strategic research grants andregistered two patents in the US. He is IEEE

member.

Shui Yu received the B.Eng. and M.Eng. de-grees from University of Electronic Science andTechnology of China, Chengdu, P. R. China, in1993 and 1999, respectively, and the Ph.D. de-gree from Deakin University, Victoria, Australia,in 2004. He is currently a Senior Lecturer withthe School of Information Technology, Deakin U-niversity, Melbourne, Australia. He has publishedaround 100 peer review papers; many of themare in top journals and top conferences, such asIEEE TPDS, IEEE TIFS, IEEE TFS, IEEE TMC,

and IEEE INFOCOM. His research interests include networking theory,network security, and mathematical modeling. Dr. Yu actively serveshis research communities in various roles, which include the editorialboards of IEEE Transactions on Parallel and Distributed Systems, IEEECommunications Surveys and Tutorials, and three other internationaljournals, TPC of IEEE INFOCOM, symposium co-chairs of IEEE ICC2014, IEEE ICNC 2013 and 2104, and many different roles of inter-national conference organizing committees. He is a senior member ofIEEE, and a member of AAAS.

12

Ivan Stojmenovic was editor-in-chief of IEEETransactions on Parallel and Distributed System-s (2010-3), is Associate Editor-in-Chief of Ts-inghua Journal of Science and Technology, andis founder of four journals (Journal of MultipleValued Logic and Soft Computing, Ad Hoc &Sensor Wireless Networks, International Jour-nal of Parallel, Emergent and Distributed Sys-tems, Cyber Physical Systems). He is editor ofIEEE Transactions on Computers, IEEE Net-work, IEEE Transactions on Cloud Computing

etc. Stojmenovic has top h-index in Canada for mathematics and statis-tics, and has >16000 citations and h=63. He received five best paperawards. He is Fellow of IEEE, Canadian Academy of Engineering andAcademia Europaea. He received Humboldt Research Award in Ger-many and Royal Society Research Merit Award in UK. He is Tsinghua1000 Plan Distinguished Professor.