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Virtual-MIMO-Boosted Information Propagation onHighways

Zhaoyang Zhang,Member, IEEE, Hui Wu, Huazi Zhang,Member, IEEE,Huaiyu Dai,Senior Member, IEEE, and Nei Kato,Fellow, IEEE

Abstract—In vehicular communications, traffic-related infor-mation should be spread over the network as quickly as possibleto maintain a safer transportation system. This motivates us todevelop more efficient information propagation schemes. Inthispaper, we propose a novel virtual-MIMO-enabled informationdissemination scheme, in which the vehicles opportunisticallyform virtual antenna arrays to boost the transmission rangeandtherefore accelerate information propagation along the highway.We model the information propagation process as a renewalreward process and investigate in detail theInformation Prop-agation Speed (IPS) of the proposed scheme. The correspondingclosed-form IPS is derived, which shows that the IPS increasescubically with the vehicle density but will ultimately converge toa constant upper bound. Moreover, increased mobility also facili-tates the information spreading by offering more communicationopportunities. However, the limited network density essentiallydetermines the bottleneck in information spreading. Extensivesimulations are carried out to verify our analysis. We also showthat the proposed scheme exhibits a significant IPS gain overitsconventional counterpart.

Index Terms—Information propagation, vehicular communi-cations, virtual MIMO, mobility

I. I NTRODUCTION

A. Motivation

Vehicular ad hoc networks (VANET) is undergoing ex-tensive study in recent years [1]–[5]. Maintaining a safertransportation system is of top priority. One safety measure isto enable inter-vehicle communication, especially in highwayswithout any fixed road-side infrastructure. Information ontraffic-related events, including accident, traffic jam, closedroad, etc, needs to be relayed to nearby vehicles in a multi-hop fashion. For instance, if an accident occurs on a highwayand induces temporary congestion, it is vital to ensure thatall vehicles in this region be informed as quickly as possible,so that some of them can make early detours before gettingtrapped. Therefore, we are interested in a new performancemetric that describes how fast a message can propagate alongthe route. This new metric,Information Propagation Speed

This work was supported in part by National Key Basic Research Pro-gram of China (No. 2012CB316104), National Hi-Tech R&D Program (No.2014AA01A702), Zhejiang Provincial Natural Science Foundation of China(No. LR12F01002), National Natural Science Foundation of China (Nos.61371094, 61401388) and the China Postdoctoral Science Foundation FundedProject (No. 2014M551736).

Z. Zhang, H. Wu, and H. Zhang (E-mails:{ning_ming,3090102503, hzhang17}@zju.edu.cn) are with the Department ofInformation Science and Electronic Engineering, ZhejiangUniversity, China.H. Dai (Email: hdai@ncsu.edu) is with the ECE Department, NC StateUniversity, USA. N. Kato (Email:kato@it.ecei.tohoku.ac.jp) iswith the Graduate School of Information Sciences, Tohoku University, Japan.

(IPS) [21], defined as the distance the information spreadswithin unit time, has drawn increasing attention recently.

B. Related Work

The allocation of a 75 MHz licensed band at 5.9 GHzfor Dedicated Short Range Communications (DSRC) [7],which was later standardized by IEEE 802.11p in vehicularenvironments [8], has renewed interest from both industryand academia in VANET, which usually supports high andyet regular and predictable mobility, due to road and traf-fic constraints (e.g. vehicles traveling on highways). In thiscontext, early works [1]–[3], [11], [12] focused on modelingvarious aspects of VANET. In [11], the authors studied vehicletraces and concluded that vehicles are very close to beingexponentially distributed in highways. Further, measurementsin [12] showed that vehicles traveling in different lanes (e.g.bus lane or heavy truck lane) have different speed distributions.

Information spreading in mobile networks has become aresearch hot spot recently. On the one hand, some interestingscaling properties have emerged, concerning mobile ad hocnetworks (MANET). For instance, [9], [10] analyzed the effectof mobility on information spreading in large-scale mobilenetworks. On the other hand, in VANET, information spread-ing also has wide applications, among which the emergencymessage dissemination is of major importance. On the otherhand, in VANET, information spreading also has wide appli-cations, among which the emergency message disseminationis of major importance. Different applications are likely tohave different requirements for the corresponding informationflow. For instance, safety applications are expected to be lowdata-rate, confined to a small neighbourhood, typically ofthe order of several hundred meters, but with strict latencyconstraints (of the order of few milliseconds). This motivatesworks dedicated to reliable emergency message exchange [13],[14]. Comparatively, other traffic related services such astraffic monitoring etc., tends to have more relaxed latencyconstraints, and require communication within neighborhoodspanning several or tens of kilometers. In this case, continuousand complete connectivity may be unavailable due to thefrequent-changing network topology, thus a store-and-forwardscheme is usually adopted to allow messages to be stored in thememory of a node and forwarded when possible and necessary.Therefore, in such delay tolerant vehicular networks, manyresearchers are more interested in the evaluation of IPS.Agarwal et al. [15], [16] first obtained upper and lower boundsfor the IPS in an 1-D VANET, where vehicles are Poissonly

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distributed and move at the same, constant speed in opposingdirections. Subsequently, Baccelli et al. derived more fine-grained results on the limit [17], under the same setting of [16].The authors complemented their work by taking into accountradio communication range variations at the MAC layer, andcharacterized conditions for the phase transition [18].

Other than traffic density, the impact of vehicle speed onIPS has also been extensively investigated. In [19], Wu etal. studied the IPS assuming uniformly distributed, but time-invariant vehicle speeds, and obtained analytical resultsinextremely sparse and extremely dense cases. [20] showed thatthe time-variation of vehicle speed is also closely relatedtoIPS. When extended to a multi-lane scenario, [21] obtainedsimilar results. Note that in all the above works, each vehicleplays the role of an individual relay, which communicateswith only one other vehicle at a time, and no inter-vehiclecooperation has been considered.

Notice that most existing literature mentioned above focuson the evaluation of IPS under various analytical models[15]- [21], where the routing schemes are usually based onthe simplest point-to-point transmission, i.e., each vehicleonly communicates with one other vehicle within its owntransmission range [19]- [21]. Particularly, when consideringa typical bidirectional setting, [15]- [17] adopted a routingscheme which uses reverse traffic opportunistically to maintaina transient connectivity of the whole network. To be morespecific, upstream messages are forwarded by vehicles trav-eling in the same direction, until a partition in the upstreamtraffic is encountered. Then the downstream traffic to utilizedto bridge traffic clusters (i.e., maximal sequence of vehiclessuch that two consecutive cars are within communicationrange) and forward the message to the next upstream cluster.Note that in all above works, each vehicle plays the roleof an individual relay, and no inter-vehicle cooperation hasbeen considered. Innovatively, our previous work, based onasimilar bi-directional model, proposed a cluster-based coop-erative message forwarding scheme, which involves multipletransmitters and multiple receivers for each cluster-to-clustertransmission, and dramatically enhances the IPS by exploit-ing the power and diversity gain offered by virtual MIMOarrays. However, [22] assumes identical vehicle densitiesandinstantaneous intra-cluster transmission, which are somewhatsimplistic. Therefore, we are motivated to build a modelwith more realistic assumptions and develop more efficientinformation propagation schemes.

C. Summary of Contributions

In this paper, we study the information propagation in abidirectional highway scenario. We propose a virtual-MIMO-enabled information dissemination scheme and analyze thecorresponding IPS. We show that the proposed scheme yieldsfaster information propagation than conventional ones. Ourmain contributions are summarized below.

1) Compared with the existing highway information prop-agation schemes, the novelty of ours lies in the follow-ing aspects: (i) Instead of the widely-adopted unicastmechanism, we alternatively adopt an ACK-free epi-demic broadcast protocol, which will not only reduce

communication overheads but also avoid unpredictabledelays (particularly in some error-prone conditions). (ii)We allow signals to travel between traffic lanes, sothat more communication opportunities can be exploitedto accelerate the dissemination process. (iii) To furtherimprove information propagation performance, we allowan individual vehicle to aggressively combine as manysignals as it can detect, which can increase the proba-bility of successful decoding.

2) We model the information propagation process as arenewal reward process and evaluate in detail the IPS ofthe proposed scheme. Closed-form results are obtained,and further analysis reveals that the IPS increases cubi-cally with the vehicle density but will ultimately con-verge to a fixed upper bound. Meanwhile, mobility canalso facilitate information propagation by offering morecommunication opportunities. Most importantly, it canbe observed that our scheme significantly outperformsothers, exhibiting a remarkable IPS gain.

The rest of the paper is organized as follows - In Sec-tion II, we present the system model and the main idea ofour proposed virtual-MIMO-enabled propagation scheme. InSection III, we derive the closed-form expression of IPS forthe proposed scheme and its asymptotic properties will beaddressed in Section IV. Simulations results are given inSection V. The paper is finally concluded in Section VI.

II. SYSTEM MODEL AND MAIN IDEA

A. Network Model

As depicted in Fig. 1, we consider a highway scenariowhere vehicles in two lanes move in opposite directions ata constant speedv. We assume that both eastbound andwestbound vehicles are Poisson distributed, which is showntobe a reasonable approximation on non-congested highways [6].Without loss of generality, we focus on the problem of unidi-rectional dissemination of single-piece small-sized information(the beacon) which deserves being exclusively disseminatedin the highway network, e.g., a piece of emergency message,issued by a westbound vehicle and to be propagated eastward.In the following, the westbound lane is referred to as thereverse lanewhere vehicles travel in the opposite directionof the beacon, while the eastbound lane is referred to as theforward lane. Let λr andλf denote the Poisson densities ofthe two lanes respectively.

It is noteworthy that synchronization is crucial to the studiedinformation propagation scheme. In this work, by assumingthe vehicles are equipped with some global timing facilitieslike GPS (Global Positioning System) as widely deployed inmany systems nowadays, it is easy to achieve network-widetime synchronization. In case there is no global timing facilityavailable, some practical algorithms based on time-stampexchange and update [23] can also be used to achieve thisgoal. In addition, more precise frame-level and symbol-levelsynchronization can be accomplished by employing algorithmsdeveloped for distributed MIMO systems based on the spe-cially designed preambles as elaborated in [24], [25], in whichthe preambles are designated to be capable of distinguishing

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the starting point of each node’s frame signal that experiencesuncertain propagation delay and channel realization. For thosenewly-joined vehicles, we can allow a a warm-up period forthem to learn the sync pattern incumbent transmissions, andonly those well-synchronized cars (which pass a sync test)can join the communication. Therefore, hereafter, we assumesynchronization is achieved.

v

v

a b

c e

R

d

f

RiH

(a) At the beginning of theith slot, the current head isb.

c ed

v

v

a b f

a,b,c,d

a,b,c

(b) During theith slot, e and f successfully receive the message.

c edv

v

a b f

1iH +

(c) At the beginning of the(i+1)th slot, f becomes the new head.

Fig. 1. Illustration of the information propagation process over a highwayvehicular network.

tt (s)

packet

preamble data

Fig. 2. Time is equally divided into slots ofτ seconds.

We examine the dissemination problem over discrete timesteps, as shown in Fig. 2. We assume that a vehicle withthe emergency message, referred to as theinformed vehicle,will repeatedly broadcast the message at each time slot ofduration τ and with equal fixed powerPt. To facilitate ourfurther analysis, it is further assumed that the message packetof each vehicle contains a specific preamble segment used foridentification, synchronization and channel estimation, and alltransmission links work in the half-duplex mode. Additionally,it is worth mentioning that the time slot duration is setsufficiently long enough to not only cover the transmissiondelay and propagation delay of the short-sized beacon/packet

(usually a few bits is enough to indicate critical informationsuch as time and location), but also enable essential signalprocessing such as synchronization, interference elimination,combining and decoding, etc.

B. Virtual-MIMO-Enabled Information Propagation

Before elaborating the proposed scheme, let us first lookinto the transmission process of a single pair of transmitter andreceiver. Denote byx the transmitted beacon, then the signalreceived by an uninformed vehicle from one of its neighboringinformed vehicles is

y = hx+ w, (1)

whereh is the channel coefficient between the vehicles, andw is the received additive white gaussian noise (AWGN) withpowerN0. Considering the open space nature of most vehicu-lar environments, we adopt a free space model to characterizeall the inter-vehicle channels, i.e.,|h|2 ∝ 1

d2, whered is the

transmitter-receiver distance. Hence, the received signal-to-noise ratio (SNR) is

γ =αPt

N0d2, (2)

whereα is a constant associated with antenna characteristics.Define two fixed SNR thresholds: thedecoding threshold

γdec and thedetecting thresholdγdet (γdet ≤ γdec), respec-tively, which correspond to two different ranges, namely, thetransmission ranger and thedetection rangeR (R > r),respectively, given the transmit power and the received noise.The transmission ranger means the maximum distance be-tween two communicating vehicles when the beacon can besuccessfully decoded (orγ ≥ γdec), while the detection rangeR means the maximum distance for the receiving vehicle tosuccessfully detect the signal from the transmitting vehicle (orγ ≥ γdet) (see Fig.1(b)). As a matter of fact, vehicles that canmake meaningful contacts and thus essentially contribute to theinformation dissemination lie within a distance of no fartherthan R from each other. The reason is that, an uninformedvehicle beyond a distanceR from the informed vehicles hasno chance of detecting the beacon, let alone decoding it, whilean informed vehicle beyondR from the uninformed vehicleshas no chance of being detected at all.

Simply put, the main idea of our proposed virtual-MIMO-enabled information propagation is to allow individual vehiclesto aggressively combine as many signals as they can detectwithin their own detection ranges, so as to achieve successfuldecoding as early as possible. The process is illustrated inFig.1. At the beginning of a certain time slot, all the informedvehicles (such asa, b, c, d in Fig.1(a)) independently andsimultaneously broadcast the beacon. The nearby uninformedvehicles within the detection range (such ase andf in Fig.1(a))are able to overhear some of these signals. Takef for example.As illustrated in Fig.1(b), it can hear three signals{hix}simultaneously sent froma,b andc, respectively, where{hi}are the respective channel coefficients which can be viewedas independent as long as the distances between vehicles aremuch longer than the wavelength. That means, the signal

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received byf is

yf =∑

i∈{a,b,c}

hix+ w. (3)

By using the dedicated preamble of each branch signal,festimates the channel coefficients{hi}. Then in a way similarto the traditional maximal-ratio combining (MRC) [26], itmultiplies the received signal with weights{ h

∗

i∑

i

|hi|2} and then

linearly combines them together, which yields

ȳ =∑

i

h∗i∑

i

|hi|2yf . (4)

The resultant average SNR ofȳ is just γ =αPt

∑

i

|hi|2

N0. If it

exceedsγdec, f will be informed. Note that following a similarprocedure, other uninformed vehicles (such ase) may alsoyield successful receptions at the same time, thus making itavirtual MIMO scenario.

Define head as the rightmost informed node at each timeslot. Fig. 1 shows an example of a single-slot propagation: letHi be the head at the beginning of theith slot (i.e.,b). Duringthe slot,e andf are able to decode the message packet throughthe above virtual MIMO enhanced approach. Thus, when thenext time slot begins,f becomes the new head, denoted byHi+1. Therefore, the information propagation process can becaptured by the evolvement of the head coordinate, which willbe discussed in detail in Section III.

Apparently, the scheme can disseminate the informationfaster, by allowing vehicles beyondr to opportunisticallydecode the message as well. Besides, it is also practicalfor one-dimensional spatially-distributed vehicular networks,without any complex signal processing techniques (e.g. space-time block code (STBC)) in most virtual-MIMO-based sys-tems. Instead, vehicles work simply and independently, eitherbroadcasting or receiving and combining signals, yet stillbenefit from virtual antenna arrays. Moreover, to the best ofour knowledge, it is the first time that the both lanes are giventhe same priority to be informed of the emergency, concerninga bidirectional highway network, which better exploits thedynamics of the network topology to improve the systemperformance.

III. I NFORMATION PROPAGATION SPEEDEVALUATION

We now analyze the IPS, the key performance metric of ourproposed scheme, by establishing a tractable analytical modelfor the information dissemination process.

A. Analytical Model

As mentioned above, the propagation progress can be essen-tially captured by the evolving coordinate of the head. Thiskeyobservation leads to an effective way to evaluate the IPS. Tofacilitate the analysis, vehicles in the reverse lane are viewed asmotionless nodes, while those in the forward lane are movingat a speed of2v. Our main goal is to quantify the relative IPSwith respect to the westbound vehicles.

As in Fig. 3(a), the beacon keeps propagating forward in anew time slot, if the gap between the head and the nearby

v

v

e

f ga b

cd

(a) The beacon keeps propagating forward.

v

v

e

f

g

a b

cd

(b) The beacon is blocked in the forward lane.

v

v

e

fa b

cd

(c) The beacon is blocked in the reverse lane.

Fig. 3. The information propagation pattern.

uninformed vehicles can be bridged by our virtual MIMOapproach. The rightmost newly informed vehicle, as indicatedby the upright dotted line, thus becomes the new head of thecurrent time slot. Otherwise, if a large gap is encountered,thebeacon will be blocked and thus have to wait in the currenthead, either in the forward lane (see Fig. 3(b)) or the reverselane (see Fig. 3(c)), until connectivity is restored. Concerningthe relative IPS with respect to the reverse traffic, we can seethat only when the beacon is blocked in the reverse lane shallit be considered a halt while in other cases it will remainpropagating. Therefore, we can model the whole propagationprocess as a renewal reward process with two alternatingstates, i.e., the PROPAGATE state and the STOP state, asillustrated in Fig. 4. Here the reward of each cycle can beinterpreted as the relative distance the beacon traverses acrossthe heads.

STOP PROPAGATE

reward>0 reward=0t

STOP STOPPROPAGATE PROPAGATE

Fig. 4. Renewal reward process.

Due to the linear mobility model and the Poisson trafficmodel, the network can be viewed as stationary. Thus, acontinuous Markov chain with each state lasting for a certainnumber of time slots, as shown in Fig. 5, can be adoptedto characterize the whole information propagation process.The state machine of the chain can be determined from thefollowing observations. First, the PROPAGATE state in Fig.4can be viewed as a combination of two sub-states, denoted asPROP I and PROPII, respectively, with the former indicatingthat the beacon keeps propagating along and/or across thelanes, and the latter indicating that it is blocked in the forwardlane but still maintains a positive propagating speed of 2v.

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Second, once the system leaves the PROPI state due to alarge gap, it will transit either to the PROPII state with someprobabilityσ1 or to the STOP state with some probabilityσ2,depending on whether the beacon is blocked in the forwardlane or the reverse lane, respectively. Third, the PROPII statewill always evolve to the PROPI state (with probability 1),since the moving head will eventually approach some reverselane vehicle ahead and pass on the beacon to it. Similarly, theSTOP state will always evolve to the PROPII state, since areverse lane head will eventually be overtaken by some movingvehicle in the forward lane, which then serves as the new head.

PROP_

PROP_

1s

2s

STOP

Fig. 5. The whole propagation can be modeled by a continuous Markovchain.

bp

BLOCK

rp

rp fp

fp

bp

PROP_

_REV

PROP_

_FOR

Fig. 6. The PROPI state is modeled by a discrete Markov chain.

Moreover, in the PROPI state, the head position evolvesslot by slot, depending on the distribution of the currentinformed vehicles and the nearby uninformed ones within thedetection ranges. Therefore, this state can be further charac-terized by another discrete Markov chain as depicted in Fig.6, where its two substates, PROPI REV and PROPI FOR,denote that after each time slot the new head lies in thereverse lane and the forward lane, respectively, withpf andpr denoting the corresponding transition probabilities. BLOCKdenotes an absorbing state in which the beacon is blocked by agap and thus the system temporally leaves the current PROPIstate. Since the nearby uninformed vehicles are independentlyPoisson distributed in the two lanes given the locations of thecurrent informed ones, the probability of the next head lying ineither lane is proportional to the corresponding vehicle density.

Hence, there ispr+pf = 1−pb andpr : pf = λr : λf , whichyields

pf = (1− pb) ·λf

λr + λf, (5)

pr = (1− pb) ·λr

λr + λf, (6)

where pb, the probability of being blocked, is given by thefollowing lemma:

Lemma 1. The probabilitypb that the beacon is blocked, i.e.,it cannot propagate any further, is:

pb = e−λrΦ

( 1r2

− µσ

)

= Θ(

e−βλ)

, (7)

where Φ (·) denotes the cumulative distribution function(CDF) of the standard normal distribution,λ = λr + λf ,

µ = λ(

1r− 1

R

)

, σ =√

13λ

(

1r3

− 1R3

)

, and β > r is somepositive constant.

Proof: See Appendix A.Now, let us return to the continuous Markov chain in Fig.

5, which further implies that the transition probabilityσ1 (σ2)is identical to the conditional probability that given the initialstate PROPI FOR, the last state before absorption in Fig. 6 isPROP I REV (resp. PROPI FOR). To computeσ1 andσ2,we first look intoσ1m, the probability that afterm steps fromthe initial state PROPI FOR the system arrives the absorbingstate via PROPI REV, which is

σ1m = (1− pb)m−2 prpb, m = 2, 3, ..., (8)where (1− pb)m−2 accounts for the first(m − 2) steps notbeing absorbed, whilepr and pb are for the last two statesPROP I REV and BLOCK, respectively. Thus we have

σ1 =

+∞∑

m=2

σ1m =

+∞∑

m=2

(1− pb)m−2prpb

= pr =λr

λr + λf− λr

λr + λfpb,

(9)

σ2 = 1− σ1 =λf

λr + λf+

λr

λr + λfpb. (10)

B. Evaluation of IPS

By the virtue of renewal reward process, the IPS can bedefined as the average distance traversed within a time unit(i.e., the long-run average reward):

Vprop VMIMO =E [D]

E [T ]=

E[Dprop]

E[Tprop] + E[Tstop], (11)

where E[Dprop] is the expected distance traversed in eachcycle, and E[Tprop] and E[Tstop] are the time durationsof the PROPAGATE state and the STOP state, respectively.Therefore, the remaining effort is to separately evaluate theabove three expectations.

Lemma 2. The expected duration of the stop state is givenby:

E [Tstop] =1

2vλ, (12)

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whereλ = λr + λf .

Proof: To ease the analysis, we only consider the worstcase where the stop state will last untile is overtaken bythe nearest informed vehicle. Thus the average distance tobe covered by the bidirectional traffic is1

λ, i.e., the expected

inter-vehicle distance. Straightforwardly, the expecteddurationof the stop state, i.e., the average time needed for the catch-upis given by:

E [Tstop] =1

2vλ. (13)

Lemma 3. The expected duration and the distance traversedby the beacon of the PROPII state are given by:

E [Tprop ii] =1

2vλ(14)

E [Dprop ii] =1

λ, (15)

whereλ = λr + λf .

Proof: In the PROPII state, an eastbound head will storethe beacon, until it approaches the nearest vehicle ahead andpass it on. Thus, the expected distance to be covered equalsE[d|d > r] − r, yielding Eq. (15). Thus Eq. (14) is obtainedby E[Tprop ii] =

E[Dprop ii]2v .

Lemma 4. The expected duration and the distance traversedof the PROPAGATE state are given by:

E[Tprop] =

(

σ2

σ1+ σ1

)

× τpb

+σ2

σ1· 12vλ

(16)

E[Dprop] =

(

σ2

σ1+ σ1

)

×1

1

λr2+ 1

R

pb+

σ2

σ1· 1λ, (17)

whereσ1 andσ2 are given by Eq.(9) and Eq.(10) respectively.pb is given by Eq.(7).

Proof: Apparently, for the discrete Markov chain in Fig.6, the following equation holds:

TrTfTb

=

110

+Π ·

TrTf0

, (18)

where Π is the transition matrix, andTr, Tf , Tb denotethe expected absorbing time given that the initial state isPROP I REV, PROPI FOR and the absorbing state BLOCK,respectively.

By solving Eq. (18), we haveTr = Tf = 1pb . Note that theexpected absorbing steps is equivalent to the expected numberof time slots of the PROPI state, so we have:

E[Tprop i] =τ

pb. (19)

Accordingly, the expected distance traversed in the PROPIstate, is given by:

E[Dprop i] =E [Dmprop]

pb, (20)

where E [Dmprop] is the expected distance of a single-slotpropagation and can be determined by Lemma 5 in AppendixB.

Back to the continuous Markov chain in Fig. 5, we have:

E[Tprop] = σ1×E[Tprop i]+ (E[Tprop i]+E[Tprop ii])

×σ1σ2× limn→∞

n∑

i=1

iσ2i−1

=

(

σ2

σ1+ σ1

)

× E[Tprop i] +σ2

σ1E[Tprop ii]

(21)

E[Dprop] = σ1×E[Dprop i]+ (E[Dprop i]+E[Dprop ii])

×σ1σ2× limn→∞

n∑

i=1

iσ2i−1

=

(

σ2

σ1+ σ1

)

× E[Dprop i] +σ2

σ1E[Dprop ii].

(22)To complete the proof, substitute the Eq. (20), (14), (15)

and (19) into Eq. (21) and (22).

Theorem 1. The IPS achieved by the proposed virtual MIMOapproach is:

Vprop VMIMO =

(

σ2σ1

+ σ1

)

×1

1

λr2+ 1

R

pb+ σ2

σ1· 1λ

(

σ2σ1

+ σ1

)

× τpb

+ 1σ1

· 12vλ, (23)

whereσ1 andσ2 are given by Eq.(9) and Eq.(10) respectively.pb is given by Eq.(7).

Proof: The main steps of the proof are listed in Table Ifor explicitness and brevity.

TABLE IOVERVIEW OF THE PROOF

IPSVprop VMIMO : Eq.(11)

DistanceE[D]: Time E[T ]

Waiting timeE[Tprop]: E[Tstop]:

Eq.(17) Eq.(16) Eq.(13)

IV. A SYMPTOTIC ANALYSIS AND KEY OBSERVATIONS

Based on the closed-form IPS given by Eq. (23), in thefollowing, we will conduct further asymptotic analysis toreveal the impacting factors on IPS, such as network densityand moving speed, etc., so as to get more insight and clearerpicture on highway information propagation.

A. Impact of the Traffic Density

1) Low Density Regime:In the low density regime, we havethe following observations.

Corollary 1. When the traffic density is low, the IPS increasescubically with respect to the total vehicle density. Extremely,as λ → 0, Vprop VMIMO → v.

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Proof: According to Lemma 1, p−1b increases quasi-exponentially withλ, which makes it a dominant term in-fluencingVprop VMIMO. Therefore, by keeping the dominantterms, we can approximateVprop VMIMO to:

Vprop VMIMO ≈(λλf + λr

2) λr2R

λr2+Rp−1b + (λf − λr)λr λr

2Rλr2+R + λf

(λλf + λr2)τp−1b + (λf − λr)λrτ + λ2v

.(24)

Without loss of generality, we setλr = λf = λ2 . With somere-organization, we have:

Vprop VMIMO ≈3λp−1b

λr2Rλr2+R + 2

3λτp−1b +2v

. (25)

Then, based on the result ofLemma1, Eq. (25) becomes:

Vprop VMIMO ≈3λΘ

(

eλβ)

λr2Rλr2+R + 2

3λτΘ(eλβ) + 2v

. (26)

Whenλ is sufficiently small, Eq. (26) can be further approx-imated by:

Vprop VMIMO ≈3λ (ωλβ)λr2 + 2

3λ (ωλβ) τ + 2v

≈ 3ωβr2v

2λ3 + v,

(27)

where the first approximation follows from:λr2R

λr2+R =1

1

λr2+ 1

R

≈ λr2, andΘ(

eλβ)

≈ ωλβ, whereω is a constant,meanwhile the second approximation holds by omitting thefirst term in the denominator as2

vis the dominant term, in

other words, there exist a sufficient smallλ0 which satisfiesthat whenλ < λ0, 2v ≫ 3τωβλ2. Extremely, asλ → 0,obviously, there isVprop VMIMO → v.

Remarks:In a word, the cubical growth of IPS is causedby the positive feedback loop as follows: more senders leadto longer transmission range, which in turn includes moresenders as contributors. Specifically, in the very low vehicledensity (sparse) case, the IPS is determined by vehicle speedv because there is little packet forwarding. Intuitively, asthevehicle density starts to grow, the cubical growth of IPS comesfrom the following interrelating facts: 1) the average numberof vehicles that fall within the detection range R increasesas the vehicle density grows, 2) the resultant combined SNRgrows approximately linearly in the number of informedvehicles that form the virtual MIMO, and 3) as the combinedSNR increases, the effective propagation range (i.e., distancepropagated during one time slot) increases accordingly, whichin turn makes more vehicles informed within the detectionrangeR and thus increases the opportunities to form newvirtual MIMO. This way, the information propagation is spedup cubically.

2) High Density Regime:Correspondingly, in the highdensity regime, we have the following observation.

Corollary 2. When the traffic density is high, there is

Vprop VMIMO ≈1

τ

λr2R

λr2+R. (28)

Extremely, asλ → ∞, Vprop VMIMO → Rτ .

Proof: First, follow the same approximation line as inthe low density case, which also leads to Eq. (26). Asλ issufficiently large, we can safely omit the constant terms, andyield Eq. (28). Then,R

τdirectly follows asλ → ∞.

Remarks: As the vehicle density further increases, theforwarding process starts to dominate, i.e., the propagationis seldomly interrupted by large gaps. However, the packetcan propagate no further than the detection range per timeslot, since it cannot be detected by any vehicle beyondR.Therefore, the IPS will keep increasing until it reaches themaximum valueR

τ, which also implies that increased R, i.e.,

a stronger detection ability can also boost the informationpropagation by exploiting more power gain, exhibiting aslower converging process (by checking the derivative withrespect toλ) as well as a higher upper bound.

B. Impact of the Vehicle Speed

Apart from the vehicle density, we also examine the impactof mobility on IPS.

Corollary 3. The IPS grows with the vehicle speed. Extremely,we have:

v → 0 =⇒ Vprop VMIMO → 0 (29)

v → ∞ =⇒ Vprop VMIMO →1

τ

λr2R

λr2+R+

σ2σ1

· 1λ

(

σ2σ1

+ σ1

)

· τpb

.

(30)

Proof: As v → 0, 12vλσ1 in Eq. (23) tends to infinity,straightforwardly, we haveVprop VMIMO → 0. In contrast, asv → +∞, 12vλσ1 → 0 and thus can be omitted, which directlyleads to the result inCorollary 3.

Remarks:Intuitively, in the low mobility case, once thebeacon is blocked, the propagation process is terminated.Thereby the average IPS in the long-run can be viewed aszero. Asv increases, more dynamics are introduced into thenetwork topology, thus more communication opportunities arecreated to accelerate the information propagation. However,the performance is still constrained by the network density,with the limit indicating fastest possible spreading case wherethe latency for restoring connectivity is negligible due toinfinite v.

C. Comparison with the Conventional Flooding

In this subsection, we show the advantage of our proposedscheme by theoretically comparing it with the conventionalflooding scheme, which no longer adopts the signal-combiningmechanism as described in Section II-B. In other words, ateach time slot, only those who are inside the transmissionranger of an informed vehicle can successfully decode thepacket.

We may follow the same line of evaluating the IPS of thevirtual MIMO scheme. The differences are that, the expecteddistance covered in hop state is replaced byr, and the prob-ability of being blocked equalse−λr, which straightforwardlyleads us to:

8

Theorem 2. The IPS achieved by the conventional floodingScheme is:

Vprop conv =

(

σ2′

σ1′ + σ1

′

)

eλrr + σ2′

σ1′ · 1λ

(

σ2′

σ1′ + σ1

′

)

eλrτ + 1σ1

′ · 12vλ, (31)

whereσ′

1=λr

λr+λf− λr

λr+λfe−λr, σ

′

2=λf

λr+λf+ λr

λr+λfe−λr.

We denote byg the IPS gain, defined as the the ratiobetweenVprop VMIMO andVprop conv.

Corollary 4. In the high density regime,g ≈ λrRλr2+R . Ex-

tremely, we have:

λ → +∞, g −→ Rr. (32)

Proof: Thanks to the symmetry between the Eq. (31) and(23), we can similarly derive:

As λ → +∞, Vprop conv → rτ . Hence, with the resultin Corollary 2, simply taking the ratio of these two yieldsCorollary 4.

Remarks:Surprisingly, we find that the performance gainunder extreme dense case is determined by the ratio ofdetection rangeR, and the transmission ranger, which furthersuggests the potential benefits of a stronger detection ability.

V. SIMULATION RESULTS

In this section, intensive information propagation experi-ments are conducted to verify the correctness of the derivedtheoretical results. The simulations follow precisely themodeldescribed in Section II. We generate two opposing trafficwhich are Poisson distributed with densityλf and λr vehi-cles/m and measure the IPS by selecting a sufficient remotesource-destination pair, taking the ratio of the propagationdistance over the corresponding delay, and averaging over mul-tiple iterations of randomly generated traffic. The transmissionranger and the detection rangeR are set to 200m and 600mby default andτ is set to 25ms.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5x 10

4

λr × 1000

Vpr

op_V

MIM

O (

m/s

)

λf=80/1000,theoretical

λf=80/1000,simulated

λf=20/1000,theoretical

λf=20/1000,simulated

λf=5/1000,theoretical

λf=5/1000,simulated

Fig. 7. IPS with respect to vehicle density.

In Fig. 7, we varyλr, and plot the resulting IPS by theproposed virtual MIMO approach under severalλf settings.A close match between theoretical and simulation results can

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5x 10

4

λ*1000

Vpr

op_V

MIM

O (

m/s

)

theoreticalsimulatedapproximate

1/τ × (λ r2 R/(λ r2 +R))

Fig. 8. Approximation of IPS in the high density regime.

be observed, which confirms the analysis of IPS presentedin Section III. Fig. 8 shows that the IPS increases withthe total vehicle density, yet the curve tends to flatten out,consisting with the predicted approximation of1

τλr2Rλr2+R , and

finally approachRτ

, as expected inCorollary 2. It is alsoobserved that, during the primary stage, particularly in thecase of a smallλr, the sharp increase of the IPS consists witha “y = aλ3 + b” curve pattern, which confirms the cubicalgrowth in Corollary 1.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3x 10

4

λr× 1000

Vpr

op_V

MIM

O(m

/s)

R=800, theoreticalR=800, simulatedR=600, theoreticalR=600, simulatedR=400, theoreticalR=400, simulated

Fig. 9. IPS under differentR’s.

Furthermore, Fig. 9 demonstrates the impact of the detectionrangeR on the achieved IPS. As expected, largerR resultsin faster IPS increase as well as higher convergence value,indicating a great benefit of stronger detecting ability.

In Fig. 10 and Fig. 12, we compare the achieved IPS of ourvirtual-MIMO-enabled scheme with other baseline methods,interms of vehicle density and vehicle speed, respectively. Otherthan the conventional flooding scheme, a unicast-based routingscheme proposed by [17], referred to as “reverseaided” ap-proach, is also conducted here for comparison purpose. Unlikethe conventional flooding, the “reverseaided” will keep thebeacon in the head until a sufficiently large cluster is encoun-

9

tered to bridge the gap. In both experiments, our virtual MIMOapproach exhibits a remarkable IPS gain over others. Forinstance, when the traffic density is around 0.003 vehicles/m,the “reverseaided” approach can only achieve an IPS of about2000m/s, and the conventional flooding performs better withan IPS of 8000m/s, but already reaches its limit, while ourvirtual MIMO approach yields 16000m/s, and still has roomfor uprise; The IPS gain is further plotted semi-logarithmicallyin Fig. 11, where a sharp increase, which nearly coincideswith a straight dotted line, can be observed as the vehicledensities increases from zero, and the curve tends to flattenout as the traffic becomes extremely denser, approachingR

r,

which confirmsCorollary 4. Moreover, Fig. 11 also impliesthe existence of an optimalλ, which unfortunately, is difficultto be derived in closed-form, but can be numerically obtained.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5x 10

4

λr*1000

Vpr

op (

m/s

)

VMIMO, theoreticalVMIMO, simulatedconventional, theoreticalconventional, simulatedreverse_aided, simulated

Fig. 10. IPS Comparisons with respect to vehicle density.

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

3.5

4

λ × 1000

g (I

PS

gai

n)

R=600,r=200

R=800,r=200

Fig. 11. IPS gain over conventional flooding scheme with vehicle density.

Again, the proposed scheme is superior when varying thevehicle speed (e.g. a huge IPS enhancement from 1500m/s byconventional approach (or worse, 500m/s by “reverseaided”approach) to about 4500m/s by the virtual MIMO approach).Moreover, notice that Fig. 12 confirms remarks onCorollary3, indicating that increased mobility accelerates informationpropagation, though not unboundedly.

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

4000

5000

6000

7000

v(m/s)

Vpr

op (

m/s

)

Theoretical virtualmimoSimulated virtualmimoTheoretical conventional Simulated conventionalreverse_aided, simulated

Fig. 12. IPS Comparisons with respect to vehicle speed.

VI. CONCLUSIONS ANDFUTURE WORK

In this paper, we study the information propagation speedin the bidirectional highway scenario. We proposed a novelvirtual-MIMO-enabled information dissemination scheme,inwhich the vehicles opportunistically form virtual antennaarrays to simultaneously transmit signals while others inde-pendently and aggressively combine as many signals as theycan detect, to hopefully decode the emergency message at afurther distance. It is shown that the scheme effectively boostsone-hop transmission range and exhibits significant IPS gainover others. Both closed-form IPS results of the proposedscheme and the conventional flooding scheme are derived.Although our current work is still theoretical-based, and themodel is still too simplified for practical implementation,our theoretical results revealed some insightful properties ofthe derived IPS, which may help designing and evaluatingappropriate routing protocols in VANET. It is implied thatincreased vehicle density and network mobility will both aidin information propagation. In our future work, we intend todesign some MAC layer protocols and physical layer tech-niques which perfectly match our proposed scheme, to createa more completed and implementable cross-layer framework.Further, we hope to extend the work to a two-dimensionalscenario.

APPENDIX

A. Proof of Lemma 1

Assume that the nearest uninformed vehicleu (located atxu) is able to detect a random number ofN transmittingvehicles. Denote B1 the event that the propagation is blocked,B2 the event that there is no informed vehicle withinr distanceof u. Applying law of total probability, we writepb as follows:

pb = Pr{B1|B2} · Pr{B2}+ Pr{B1|B2} · Pr{B2}= Pr{B1|B2} · e−λr.

(33)

Therefore, our remaining effort is to quantify the blockprobability under the condition thatN informed vehicles arewithin range(xu −R, xu − r) whereN is Poisson distributedwith densityλN = λ (R− r) (note thatλ is the number of

10

vehicles per unit length). The received SNR atu is

γ =αPt

N0

N∑

i=1

1

d2i, (34)

wheredi (i = 1, 2, ..., N ) denotes the corresponding distancebetween each transmitter andu. Due to the Poisson distri-bution property, for adequately largeR, di is approximatelyuniformly distributed within(r, R), thus we haveE[ 1

d2i

] = 1rR

and the second momentE[(

1d2i

)2

] =λ(R3−r3)

3(R−r)R3r3 .Now applying the following result ( [27], page 1160,

Example i): if an RVY has the formY =N∑

i=1

Xi, whereN is

Poisson distributed with densityλ and Xi (i = 1, 2, ..., N )are i.i.d. RVs with meanE[X ] = a and second momentE[X2] = b2, thenY is asymptotically normal with meanλaand deviationb

√λ, we can conclude thatγ is approximately

Gaussian distributed with meanαPtN0

λNE[1d2i

] and deviation

αPtN0

√

λNE[(

1d2i

)2

]. Hence, the conditional block probability

can be directly derived from the distribution ofγ, which leadsto the first equation of Eq.(7) after some manipulation. Giventhe first equation holds, the second equation of Eq.(7), i.e.,pb = Θ

(

e−βλ)

, is obvious.

B. Lemma 5 and Its Proof

Lemma 5. For the proposed virtual MIMO-based scheme, theexpected distance that the beacon can maximally propagate inone time slot, is given by:

E [Dmprop] =1

1λr2

+ 1R

. (35)

Proof: First, we consider a general scene where thedistance between the head (located atx0) and the nearestuninformed vehicle ahead, referred to as ‘receiver’, isd, andthe receiver is able to detectn vehicles at the time. Denotedi(i = 1, 2, 3..., n) the distance between each detected vehicleand the receiver. Recalling the conditional distribution ofthe arrival times proposition in [28] (Page 336, Proposition5.4), we argue that,di (i=1, 2, ..., n− 1) are identically in-dependently distributed over the interval(x0 − (R− d) , x0),which directly gives the probability density function (pdf) ofzi =

1d2i

: fZi (z) =1

2(R−d)z− 3

2 . By taking the approximationn−1∑

i=1

1d2i

≈ (n− 1)E [Zi], we can approximate the receivedpower as:

Pr ≈ αPt(

1

d2+

n− 1dR

)

. (36)

Ideally, we assume that the beacon can maximally reacha point Dm-distance away. Based on Eq. (36), the receivedpowerPrm clearly satisfies:

Prm = N0γdec =αPt

r2

= αPt

(

1

D2m+

Nm − 1DmR

)

≈ αPtNm

DmR,

(37)

which gives:

Dm ≈Nmr

2

R. (38)

Taking the expectation of both sides withE [Nm] =λ (R− E [Dm]) further yields (with some necessary rear-rangement):

E [Dm] ≈λRr2

R+ λr2(39)

Note that the actual distanceDh traversed by the beaconcannot exceedDm. In fact, we have:

Dh =

Nm∑

i=1

ci|Nm = max (m : c1 + c2 + · · ·+ cm ≤ Dm) ,

(40)where ci(i = 1, 2, 3...) denote the i.i.d exponential inter-vehicle distances. Borrowing results from [28] (Chapter 7,page 456), we argue that:

E [Dh] = E

[

Nm∑

i=1

Ci

]

= E

[

E

[

Nm∑

i=1

Ci|Nm]]

= E [NmE [C]] = E [Nm]E [C]

= λE [Dm] ·1

λ= E [Dm] ,

(41)

where the fifth equality holds due to the exponentially-distributed inter-vehicle distance. Thus,E [Dh] is obtaineddirectly form Eq. (39).

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Zhaoyang Zhang(M’00) received his Ph.D. degreein communication and information systems fromZhejiang University, China, in 1998. He is cur-rently a full professor with the College of Infor-mation Science and Electronic Engineering, Zhe-jiang University. His research interests are mainlyfocused on information theory and coding theory,signal processing techniques, and their applicationsin wireless communications and networking. He hasco-authored more than 200 refereed internationaljournal and conference papers as well as two books

in the above areas. He was a co-recipient of three international conferenceBest Paper / Best Student Paper Awards. He is currently serving as Editorfor IEEE Transactions on Communications, IET Communicationsand someother international journals. He served as TPC Co-Chair or Symposium Co-Chair for many international conferences likeWCSP 2013andGlobecom 2014Wireless Communications Symposium, etc.

Hui Wu received the B.S. degree in electrical engi-neering from Zhejiang University, Hangzhou, Chinain 2013. She is currently a doctoral student at the De-partment of Information Science and Electronic En-gineering, Zhejiang University. Her recent researchinterests include mobile relay systems, informationpropagation, and distributed sensing networks.

Huazi Zhang (S’11-M’13) received the B.S. andPh.D. degrees in electrical engineering from Zhe-jiang University, Hangzhou, China, in 2008 and2013, respectively. From 2011 to 2013, he wasa Visiting Scholar with the Department of Elec-trical and Computer Engineering, North CarolinaState University, Raleigh, NC, USA. He is currentlywith the Department of Information Science andElectronic Engineering, Zhejiang University. His re-search interests include OFDM and MIMO systems,distributed clustering and compressive sensing, cod-

ing techniques and cognitive radio networks. His recent interests are mobilenetworking, social networking and Internet of Things (IoT).

Huaiyu Dai (M’03-SM’09) received the B.E. andM.S. degrees in electrical engineering from TsinghuaUniversity, Beijing, China, in 1996 and 1998, respec-tively, and the Ph.D. degree in electrical engineeringfrom Princeton University, Princeton, NJ in 2002.

He was with Bell Labs, Lucent Technologies,Holmdel, NJ, in summer 2000, and with AT&TLabs-Research, Middletown, NJ, in summer 2001.Currently he is an Associate Professor of Electricaland Computer Engineering at NC State University,Raleigh. His research interests are in the general

areas of communication systems and networks, advanced signal processing fordigital communications, and communication theory and information theory.His current research focuses on networked information processing and cross-layer design in wireless networks, cognitive radio networks, wireless security,and associated information-theoretic and computation-theoretic analysis.

He has served as an editor of IEEE Transactions on Communications, SignalProcessing, and Wireless Communications. He co-edited twospecial issuesof EURASIP journals on distributed signal processing techniques for wirelesssensor networks, and on multiuser information theory and related applications,respectively. He co-chairs the Signal Processing for Communications Sym-posium of IEEE Globecom 2013, the Communications Theory Symposiumof IEEE ICC 2014, and the Wireless Communications Symposiumof IEEEGlobecom 2014.

12

Nei Kato received his Bachelor Degree from Poly-technic University, Japan, in 1986, M.S. and Ph.D.Degrees in information engineering from TohokuUniversity, in 1988 and 1991 respectively. He joinedComputer Center of Tohoku University as an assis-tant professor in 1991, and was promoted to full pro-fessor position with Graduate School of InformationSciences, Tohoku University, in 2003. He becamea Strategic Adviser to the President of TohokuUniversity in 2013 and the Director of ResearchOrganization of Electrical Communication(ROEC),

Tohoku University in 2015. He has been engaged in research oncomputernetworking, wireless mobile communications, satellite communications, adhoc & sensor & mesh networks, smart grid, and pattern recognition. He haspublished more than 300 papers in peer-reviewed journals and conferenceproceedings. He currently serves as a Member-at-Large on the Board ofGovernors, IEEE Communications Society, the Chair of IEEE Ad Hoc &Sensor Networks Technical Committee, the Chair of IEEE ComSoc SendaiChapter, the Editor-in-Chief of IEEE Network Magazine, theAssociate Editor-in-Chief of IEEE Internet of Things Journal, an Area Editor of IEEE Transac-tions on Vehicular Technology. He has served as the Chair of IEEE ComSocSatellite and Space Communications Technical Committee(2010-2012), theChair of IEICE Satellite Communications Technical Committee(2011-2012).His awards include Minoru Ishida Foundation Research EncouragementPrize(2003), Distinguished Contributions to Satellite Communications Awardfrom the IEEE Communications Society, Satellite and Space CommunicationsTechnical Committee(2005), the FUNAI information ScienceAward(2007),the TELCOM System Technology Award from Foundation for ElectricalCommunications Diffusion(2008), the IEICE Network SystemResearchAward(2009), the IEICE Satellite Communications ResearchAward(2011), theKDDI Foundation Excellent Research Award(2012), IEICE CommunicationsSociety Distinguished Service Award(2012), Distinguished Contributions toDisaster-resilient Networks R&D Award from Ministry of Internal Affairsand Communications, Japan(2014), seven Best Paper Awards from IEEEGLOBECOM/WCNC/VTC, and IEICE Communications Society BestPaperAward(2012). Besides his academic activities, he also serves on the expertcommittee of Telecommunications Council, Ministry of Internal Affairs andCommunications, and as the chairperson of ITU-R SG4 and SG7,Japan.Nei Kato is a Distinguished Lecturer of IEEE CommunicationsSociety andVehicular Technology Society. He is a fellow of IEEE and IEICE.