A Stochastic Geometry Approach to the Modeling of DSRC for Vehicular Safety Communication

A Stochstic Geometry Approch to the Modeling of DSRC for Vehiculr Sfety Communiction Zhen Tong, Student Member, IEEE, Hongsheng Lu 2, Mrtin Henggi, Fellow, IEEE, nd Christin Poellbuer 2, Senior Member, IEEE Abstrct Vehicle-to-vehicle sfety communictions bsed on the dedicted short rnge communiction (DSRC) technology hve the potentil to enble set of pplictions tht help void trffic ccidents. The performnce of these pplictions, lrgely ffected by the relibility of communiction links, stringently ties bck to the MAC nd PHY lyer design which hs been stndrdized s IEEE 82.p. The link relibilities depend on the signl-to-interference-plus-noise rtio (SINR), which, in turn, depend on the loctions nd trnsmit powers of the trnsmitting nodes. Hence n ccurte network model needs to tke into ccount the network geometry. For such geometric models, however, there is lck of mthemticl understnding of the chrcteristics nd performnce of IEEE 82.p. Importnt questions such s the sclbility performnce of IEEE 82.p hve to be nswered by simultions, which cn be very timeconsuming nd provide limited insights to future protocol design. In this pper, we investigte the performnce of IEEE 82.p by proposing novel mthemticl model bsed on queueing theory nd stochstic geometry. In prticulr, we extend the Mtern hrd-core type II process with discrete nd nonuniform distribution, which is used to derive the temporl sttes of bck-off counters. By doing so, concurrent trnsmissions from nodes within the crrier sensing rnges of ech other re tken into ccount, leding to more ccurte pproximtion to rel network dynmics. A comprison with ns2 simultions shows tht our model chieves good pproximtion in networks with different densities. Index Terms IEEE 82.p, Vehiculr Ad Hoc Networks, Queueing Theory, Poisson Point Process, Mtern Hrd-core Point Process of nodes increses. It mens plunging delivery rtio of sfetyrelted messges to recipients, leding to deteriorted trcking ccurcy. Mny efforts hve focused on improving the performnce of IEEE 82.p, most of which re bsed on simultions [], [2]. However, it is extremely time-consuming, if not impossible, to simulte every cse or combintion of system prmeters. A mthemticl understnding is needed, to help sve computtionl cost, to determine the fundmentl performnce limits nd to provide guidnce on the design of novel solutions. The performnce depends criticlly on the signlto-interference-plus-noise rtios (SINRs) t the receivers, nd the signl strength nd interference powers re functions of the distnces between the nodes [3]. Relistic vlues for the internode distnces nd hence SINR vlues re obtined from geometric network models where nodes re plced on line or one the plne ccording to some rndom process. Our gol in this pper is to devise such geometric models tht cn ccurtely cpture the temporl nd sptil behvior of this CSMA-bsed protocol for different network configurtions. e use both stochstic geometry nd queueing theory for modeling nd nlysis of IEEE 82.p. The model is exploited to predict the system performnce in more efficient mnner thn Network Simultor 2 (ns2) nd to better understnd the chrcteristics of IEEE 82.p in V2V sfety communictions. A. Motivtion I. INTRODUCTION Vehicle-to-vehicle (V2V) sfety communictions bsed on DSRC t 5.9 GHz shows promising potentil to improve driving sfety on the rod. ith DSRC, every node (i.e., vehicle) brodcsts up to sfety-relted messges every second, where ech messge contins GPS informtion (i.e., vehicle s loction, speed, nd heding). Vehicles tht receive such messges re ble to trck the senders, which therefore helps void vehiculr collisions. At the lower lyers, IEEE 82.p provides the medi ccess control nd physicl lyer solution. However, prior studies hve shown tht IEEE 82.p suffers significnt performnce degrdtion when the number Zhen Tong nd Mrtin Henggi re with the Deprtment of Electricl Engineering, University of Notre Dme, Notre Dme, IN 46556, USA, E- mil: {ztong,mhenggi}@nd.edu. 2 Hongsheng Lu nd Christin Poellbuer re with the Deprtment of Computer Science nd Engineering, University of Notre Dme, Notre Dme, IN 46556, USA, E-mil: {hlu,cpoellb}@nd.edu. B. Relted ork A rich body of literture on the performnce nlysis of CSMA-bsed networks cn be found in the reserch community, mong which most works re bsed on queueing theory. For exmple, the performnce of the crrier-sense multiple ccess with collision voidnce (CSMA/CA) scheme hs been nlyzed in [4], [5] using discrete Mrkov chin model. The uthors focused on one-hop networks (where ll the nodes re within ech other s crrier sensing rnge). A closed-form expression of the network throughput ws developed with n ssumption of sturted dt trffic. Lter, reserchers pplied the Mrkov chin pproch to broder set of cses. [6] investigted the impct of non-idel chnnels nd cpturing techniques on the throughput of the IEEE 82. protocol with non-sturted dt trffic. Sttes for the trnsmission filures nd sttes representing the cse where no pckets exist in the buffer re dded into the model used in [4]. The throughput s function of severl prmeters, such s pcket size, is clculted. Authors in [7] chrcterized the impct of different

2 messge genertion rtes nd trnsmission power levels on the network cpcity in presence of of hidden terminls. Under some simplifying ssumptions, the pper nlyzed the hidden nodes nd showed tht the chnnel occupncy or busy rtio cn be used s feedbck mesure tht quntifies the success of informtion dissemintion. The work presented in [8] focuses on developing n ccurte model for chrcterizing the impct of hidden terminls on the network performnce. The uthors pointed out the limittions of using renewl theory with vrible time slot in the literture nd proposed new model with fixed-length chnnel slot. ith simple network topology, the model developed shows good mtch with the simultions. In [9], [], the uthors investigted the performnce of IEEE 82.p with enhnced distributed chnnel ccess cpbility where pplictions with different priorities re divided into four ccess ctegories (ACs) ccording to their criticlities for the vehicle s sfety. Since ech AC hs seprte bck-off process, AC cn be viewed s "virtul" node. Ech virtul node competes with other virtul nodes s well s rel nodes to get ccess to chnnel. The uthors ssumed tht ll the nodes observe the sme chnnel to compute the throughput nd dely using queueing theory, which constitutes n extension of [4], [5]. However, ll the resulting models were either more complicted or only pplicble to simplified network topologies. On the other hnd, stochstic geometry, in prticulr point process theory, hs been widely used in the lst decde to provide models nd methods to nlyze wireless networks, see [] [5] nd references therein. Stochstic geometry provides nturl wy of defining nd computing criticl performnce metrics of the networks, such s the interference distribution, outge probbility nd so forth, by tking into ccount ll potentil geometricl ptterns for the nodes, in the sme wy queueing theory provides response times or congestion, considering ll potentil rrivl ptterns. To our best knowledge, the first pper using stochstic geometry to model the relibility of IEEE 82.p protocol is [6]. However, it is limited to the nlysis of dense vehiculr networks using to pproximte the CSMAbsed MAC protocol. Similrly, [7], [8] only focus on the performnce nlysis of vehiculr networks using s the MAC scheme. The connectivity of the vehiculr networks in urbn environments hs been studied in [9]. A model bsed on stochstic geometry hs been developed to obtin the probbility of node to be connected to the origin nd the men number of connected nodes for given set of system prmeters. However, the model does not explicitly tke into ccount the temporl dynmics specific to vehiculr networks, which re cused by fctors like periodic trffic nd the bckoff process in IEEE 82.p. The Mtern hrd-core process of type II [3, Chpter 3] 3 hs been used in [2] to nlyze dense IEEE 82. networks, in which the nodes loctions re plced ccording to Poisson point process nd the rndom 3 Strting with bsic uniform Poisson point process (PPP) Φ b with intensity λ b, dd to ech point x n independent rndom vrible m(x), clled mrk, uniformly distributed in [, ]. All points tht hve neighbor within distnce r with smller mrk re flgged. The remining (non-flgged) points form the Mtern hrd-core process of type II. bck-off counter is modeled s n independent mrk tht is ssocited with ech node. However, the mrk is ssumed to be uniform between [, ] which is not the cse for IEEE 82.p since the bck-off counter tkes only integer vlues from to, where is the mximl vlue of the bck-off counter ( typicl vlue for is 5). C. Contributions Our min contributions re: ) e investigte the trnsmission behvior of V2V sfety communictions with non-sturted dt trffic using continuous-time Mrkov chin model tht is bsed on novel combintion of queueing theory nd point process theory. e show tht the loction distribution of the trnsmitting nodes is well modeled rnging from hrdcore processes to PPPs s the network density increses, in contrst to [6], where only the dense cse is studied. 2) A novel Mtern hrd-core process is proposed to cpture the hrd-core effect of CSMA nd concurrent trnsmissions occurring within the sme crrier sensing rnge. In prticulr, modified Mtern hrd-core process of type II (clled Mtern-II-discrete process for the rest of the pper, wheres the originl Mtern hrd-core process of type II [2] is henceforth clled Mtern-II-continuous process) is used with discrete nd non-uniform mrk distribution to model the temporl informtion of the bck-off counter nd the sptil loctions of the trnsmitting nodes. In this wy, nodes with the sme bckoff counter vlue cn trnsmit t the sme time with nonzero probbility even if they re within the crrier sensing rnge of ech other, which mkes the model more relistic. 3) e vlidte the effectiveness of our model by compring it to other mthemticl models nd simultions from ns2, which hve been clibrted for vehicle-to-vehicle communictions by [], [2] nd will be used to generte ground truth in this study. The comprisons show tht our model cn be pplied to networks of different densities with good ccurcy. D. Orgniztion The rest of the pper is orgnized s follows: Section II introduces the system model. In Section III, queueing theory nd stochstic geometry re used to investigte the distribution of the trnsmitting nodes nd the bck-off counter distribution in networks with different densities. Section IV proposes nd describes the Mtern-II-discrete process in detil. The system performnce is evluted nd compred with other models in Section V. e conclude our work in Section VI. A. Trffic Model II. SYSTEM MODEL e focus on V2V sfety communictions using DSRC t 5.9 GHz in the highwy scenrio where the trnsmission rnges of the vehicles re lrger thn the width of the rod. Therefore, ll the nodes cn be ssumed to be plced on

3 line ccording to PPP with intensity λ. This set of nodes is denoted s Φ. Ech node genertes messges every second s suggested in [22], which is equivlent to the requirement to trnsmit messge every ms. These messges go to the sme ccess ctegory (AC), nd trffic in other ACs [23] is out of the scope of this pper. A messge is dropped if it hs not been sent out before new one rrives. Since the messge is beneficil to ll nerby vehicles, brodcst communiction is used, which mens no RTS-CTS is exchnged before trnsmissions nd no ACK is trnsmitted fter trnsmission. More specificlly, when messge is vilble t node, if the chnnel is idle, the messge is brodcst immeditely. Otherwise, bckoff counter in [ ] {,,, } is drwn uniformly t rndom. + is clled the contention window size; it is fixed since no ACK feedbck is used. A node hs to go through bck-off process if it cnnot send the messge immeditely, i.e., if its bck-off counter is set. The bck-off process executes in EDCA (Enhnced Distributed Chnnel Access) mnner where the decrement of the bck-off counter vlue hppens t the beginning. As shown in Fig., when the chnnel becomes idle, the node decreses its bck-off counter vlue immeditely by one. It keeps doing so fterwrds ech time time intervl δ elpses (δ is clled the slot time), nd it freezes the count-down when the chnnel becomes busy gin. hen the vlue of the bck-off counter reches zero, if the chnnel stys idle within the following δ time (i.e., slot time), the node strts trnsmitting right fter. Otherwise, the node wits nd the trnsmission will occur immeditely fter the chnnel turns idle next time. B. Nottion From the bove description, one cn see tht nodes my sty in different sttes (e.g., trnsmitting, bck-off, idle) t given time. These sttes (equivlently, clled mrks in point processes) need to be crefully understood nd noted becuse they ffect the distribution of interference cross the network, which contributes to the relibility of communictions. Hence, we list key nottions used for point processes in Tble I. C. Chnnel Model The power received t loction y from node locted t point x, denoted by P (x, y), is given by where Bckoff Counter Vlue P (x, y) = P S x y l(x, y), () Time Busy chnnel Trnsmitting Figure : Bck-off process in IEEE 82.p. P is the trnsmit power, which is ssumed to be the sme for ech node; l(x, y) is the pth loss or similr between x nd y. Similr to [6], it is modeled s follows: l(x, y) = A min { r α, d(x, y) α}, where d(x, y) is the Eucliden distnce between x nd y nd α is the pth loss exponent; we lso use the pth loss function in this form: l(r) = A min { r α, r α}, ( ) α where A = λ 4πr is dimensionless constnt in the pth loss lw determined by the wvelength λ nd the reference distnce r for the ntenn fr field. Sy x is rndom vrible denoting the fding between ( points x nd y, nd y, the rndom vribles S x y, Sy x2, ) re ssumed to be i.i.d. exponentilly distributed with men one, which mens the chnnels re Ryleigh fding. For the non-fding cse, S x y. In this pper, we consider both the fding nd non-fding cses. The neighborhood of node x (denoted s V (x)) is defined s the rndom set of nodes in its contention domin, nmely the set of nodes whose messges this node receives with power lrger thn some detection threshold P, i.e., V (x) = { y Φ [ ] : P S y xl(y, x) > P }, x Φ[ ]. (2) Equivlently, where V (x) = { y Φ [ ] : d(y, x) < R y x}, x Φ[ ]. (3) ( P AS Rx y y = x P ) α is clled the crrier sensing rnge, which is rndom vrible ( ) relted to Sx. y In the non-fding cse, R = Rx y α P A = P is deterministic. The impct of the vehicle mobility nd direction is neglected since the node is lmost sttionry within one pcket trnsmission durtion, which is usully less thn ms. III. TEMPORAL CHARACTERISTICS OF V2V COMMUNICATIONS In this section, we will show tht the vehiculr networks dynmics cn be explined using queueing theory. A. Dense Networks In [6], the uthors climed tht when the network is dense tht ) fter every busy period node cn decrese the vlue of its bck-off counter only by one; 2) if lmost ll the sections of the rod re covered by trnsmissions, the system performnce under the CSMA-bsed protocol is similr to tht of. In other words, this mens tht the performnce of such networks t high node density cn be nlyzed by pproximting the slotted-csma network s slotted- network where the set of concurrently trnsmitting nodes Φ tx forms PPP. In [6], slotted-csma (4)

4 Nottion Description [ ] {,,, }, the set of bckoff counter vlues Φ Entire PPP of intensity λ Φ B Φ B, B R. Nodes within B s x (t) Stte (or mrk) of node x Φ t time t S A set of sttes of the nodes. S = [ ] {idle} {tx}. i [ ] is the stte where the bck-off counter is i nd there is pcket witing in the buffer to be trnsmitted. idle is the stte where no pckets re in the buffer. tx mens tht pcket is in trnsmission. Φ S (t) Subset of nodes with stte s(t) S S. 5 Φ i (t) Nodes tht hve pcket in the buffer nd reduce the bck-off counter to i from different vlue by time t Φ i (t) Φ i (t)\φ i (t), i [ ]. Nodes with pcket in the buffer tht select i s bck-off counter vlue nd hve not decresed it by time t. Φ tx Set of concurrent trnsmitting nodes selected by Mtern-II-discrete process Set of concurrent trnsmitting nodes selected by Mtern-II-continuous process Φ tx Tble I: Nottion 5 E.g., Φ tx(t) is the set of nodes tht trnsmit t time t, Φ 3 (t) is the set of nodes with bck-off counter vlue 3 t time t, nd Φ [ ] (t) is the set of nodes hving pcket witing to be trnsmitted t time t. model is ssumed to simplify the nlysis. This ssumption my introduce inccurcies in the model but is n cceptble pproximtion in dense networks. e nlyze the distribution of Φ tx by studying wht hppens fter the trnsmissions of some nodes in Φ tx finish. The ssumption we mke is tht Φ idle follows PPP. If subset of Φ tx finish their trnsmissions t time t 4 _, they leve sections of the rod (intervls of the rel line) in which the chnnel is sensed idle. e pick ny of the sections nd cll it s. Nodes in Φ s (t) will strt trnsmission immeditely. Using queueing theory, we rgue tht Φ s (t) forms PPP. Define T p s the trnsmission time of pcket followed by two slot times (s required by IEEE 82.p). Φ s (t) = Φs (t) Φs (t). e clim tht Φ s (t) forms PPP on s since the pcket rrivls t ech node re independent nd Φ s (t) cn be viewed s n independent thinning of Φ idle ((t T p ) + ) where the thinning probbility is the probbility tht node hs pcket rriving between (t T p ) + nd t nd chooses its bck-off counter to be. However, to understnd the distribution of Φ s (t), we need to highlight tht the bck-off processes of the nodes in Φ s [ ] re synchronized. In other words, they observe the sme chnnel sttes (idle or busy) nd strt or stop bck-off counters simultneously. Synchroniztion cnnot generlly be ssumed cross nodes in CSMA-bsed multi-hop networks due to hidden terminls. As consequence, we need to understnd how it occurs for nodes in Φ s [ ]. To do so, snpshots of the nodes sttuses re recorded in Fig. 2 on section of the rod t different times. Initilly (s shown by the snpshot t T ), we ssume none of the nodes is trnsmitting. hen the first pcket rrives, it is trnsmitted without bck-off process (since the chnnel is sensed idle). The following pckets re held if their nodes re within the crrier sensing rnges of ny ongoing trnsmissions. Nodes within the sme crrier sensing rnge my synchronize the next trnsmission. 4 e use t _ to note the time just before t nd t + for the time just fter t Figure 2: Snpshots of sttuses of nodes on section of rod tken t four different times, T, T i, T j nd T k with T < T i < T j < T k where Υ denotes the (union of) crrier sensing rnges of trnsmitting nodes. Assume tht x j nd x k from the snpshot t T i re two of those nodes nd they re within the crrier sensing rnge of x i (denoted s Υ i ). They both hve their bck-off counter vlues set to zero nd strt trnsmissions immeditely fter x i finishes. The union of x j s nd x k s crrier sensing rnges constitutes Υ jk (shown in the snpshot t T j ). In this wy, the region on which trnsmissions my be synchronized will increse from Υ i to Υ jk. The sme process hppens on other sections of the rod, until lmost the whole rod is covered by trnsmissions (s shown by the snpshot t T k ). The system now reches point where it cn be viewed s collection of regions on which nodes bck-off processes re synchronized nd gps between these synchronized regions. e cn think of s s ny one of such regions where the bck-off processes of nodes in Φ s [ ] re synchronized. As

5 time Mrkov chin [24], we hve New Arrivls!!! pk! X qki = i=! X pi qik, (6) i= where the non-zero trnsition rtes re qk,k = λ k λ + for k [ ] nd q,k = + for k [ ] (their vlues lbeled in Fig. 3). Therefore, for the dense cse s in [6], we cn compute the stedy stte probbility pk s! ( " )!! Nodes with no pckets pk = 2 -! Nodes with pckets Figure 3: Mrkov chin model for the dense cse where the nodes bck-off counter vlues re modeled s the sttes of the chin (from to ). The trnsition rtes lbeled between the sttes stnd for the number of nodes per second per unit length of rod tht re ble to chnge their bck-off counter vlues. 2 ( k) ( + ) (7) P using queueing theory, combining (6) nd k= pk =. Expression (7) is equivlent to tht in the one-hop communiction networks in [5]. The difference is tht discrete-time Mrkov chin with vrible slot ssumption is used in [5] to obtin the stedy stte probbility pk while our continuous-time Mrkov chin model more nturlly cptures the stedy stte in ny given time instnt. B. Sprse Networks 8 consequence, Φs (t) cn be written s the union of Φs (t Tp ) nd Φs (t Tp ). Applying the sme logic itertively to Φs (t Tp ) yields = [ Φsi (t 8 Φs (t) itp ). New Arrivls ' ' ' ' ' (5) i= Since Φsi (t itp ), i on s, Φs (t), s the ' Figure 4: Mrkov chin model for the sprse cse where the nodes bck-off counter vlues re modeled s the sttes of the chin (from to ). The trnsition rtes lbeled between the sttes stnd for the number of nodes per second per unit length of the rod tht re ble to chnge their bck-off counter vlues. The bove process cn be pproximted using queueing theory where the counter vlues k for given node cn be considered s the sttes in continuous-time Mrkov chin. As illustrted in Fig. 3, the density of nodes with new rriving pckets cn be considered s the men rrivl rte per unit length, nd the density of trnsmitting nodes s the men service rte per unit length. Assume tht t stedy stte, the men rrivl rte per unit length (denoted s λ ) is equl to the men service rte per unit length (denoted s µ ), tht is, λ = µ. The system cts s if the nodes hve sturted dt trffic, i.e., the nodes finishing their trnsmissions will generte new pcket nd join the queue tht consists of the nodes with bck-off counters immeditely. Define qki s the trnsition rte t which the nodes mke trnsition from counter k to counter i. pk represents the stedy-stte probbility where the counter vlue is k. By the globl blnce equtions for continuous- Inspired by the work in [6], we explore the distribution of Φtx in sprse networks. According to IEEE 82.p, pcket is trnsmitted immeditely upon rrivl if the chnnel is sensed idle. In sprse networks, limited number of nodes exist. The cumultive chnnel lod consumes only smll portion of the chnnel cpcity, leving the chnnel idle most of the time s observed in [25], which implies tht most nodes will send out their pckets without going through ny bck-off process (s indicted by λ in Fig. 4). Furthermore, s there is finite number of nodes within the sme crrier sensing rnge nd the pcket rrivl process is continuous, the probbility tht two nodes within the crrier sensing rnges of ech other hve pckets rrive t exctly the sme time is zero. Therefore, we will hve no pirs of nodes within distnce less thn crrier sensing rnge tht strt trnsmissions t the sme time, which cn be modeled by the hrd-core process described 8 {, 2,, }, re independent PPPs union of them, is lso PPP [3]. However, it does not men Φtx follows PPP on ll the sections in the system which is ssumed in [6]. Assume tht fter nodes in Φs (t) strt trnsmitting, the trnsmitters in n djcent region to s, nmed s2, finish trnsmissions. Although Φs2 (t2 ) forms PPP where t2 > t, there cnnot be ny trnsmitters in s s2. In other words, there my be sections where the trnsmitters form PPP, interleved with sections without trnsmitters. Nevertheless, the trnsmitters in the dense cse cn be well pproximted s PPP s shown in the simultion lter nd [6]. ' ( ) Nodes with no pckets ' ' 2 - Nodes with pckets

6 in [2]. However, beyond the spce dependence introduced by the hrd-core process, there is time dependence on the loctions of trnsmitters. In fct, some nodes my hve pckets rrive when the chnnel is sensed busy, which my result in concurrent trnsmissions within the sme crrier sensing rnge. This ffects the ccurcy of the hrd-core process. C. Networks with Intermedite Density A relevnt question is wht the right model is for networks with intermedite densities. On the one hnd, compred with sprse networks, more trnsmissions re delyed, incresing the probbilities of trnsmission collisions from synchronized bck-off processes (but not s mny s in dense networks). On the other hnd, the trnsmissions my not cover lmost ll the sections of the rod, leding to distribution of trnsmitters different from the cse of dense networks. For exmple, it could hppen tht on some sections of the rod the chnnel is sensed idle but no nodes hve pckets nd thus no trnsmissions tke plce. As consequence, the distribution of the trnsmitters for intermedite networks cnnot purely be modeled or pproximted by hrd-core processes or PPPs. Insted, it should be hybrid process between hrd-core nd Poisson. D. Observtions Bsed on the observtions bove, it is generlly true tht ll the nodes with bck-off counter t t j hve chnces to prticipte in trnsmissions t t i where t i > t j if the chnnels turn idle before t i. From stochstic geometry perspective, the concurrent trnsmitters form thinned process like the Mtern- II-continuous process. However, to ccount for the concurrent trnsmitters within the sme crrier sensing rnge, we need to discretize the mrks in the Mtern-II-continuous process, nd ny two nodes hving the sme mrk should not silence ech other. This discrete choice for the mrks mkes sense since the bck-off counter in IEEE 82.p tkes integer vlues from to only nd hs concurrent trnsmissions if the counters of two nodes within ech other s crrier sensing rnge hit zero simultneously. To obtin the non-uniform distribution from which mrks re drwn, we smple the number of nodes hving different bck-off counter vlues from ns2 simultions. The empiricl probbility mss functions (PMFs) for scenrios with different densities λ re shown in Fig. 5. A few observtions cn be esily mde from this plot. First, the PMF is not uniform. It is skewed towrds smll counter vlues. The mximlly skewed curve is obtined from (7). Second, the empiricl PMF for different densities cn be pproximted by n ffine function of k. Assume tht it follows the form p k = b k, (8) where b. Since k= p k =, we hve p k = + + k. (9) 2 2 where ( +). should be function of the density λ. To obtin concrete results of (λ ), we cn estimte it from ns2 simultions or using queueing theory. Here, we proceed with the former method. In the following section, we will use it s the counter distribution or mrk distribution for the Mtern- II-discrete process. Estimted Prob Mss Function.2..8.6.4.2 Sprse Medium Dense Uniform Mximlly Skewed 5 5 Counter Figure 5: Estimted probbility mss function of the nodes bckoff counter compred with the uniform PMF nd mximlly skewed PMF: node density λ =.33,.66,.32 for the sprse, intermedite nd dense cses, respectively. IV. MATERN-II-DISCRETE PROCESS Section III discussed the temporl chrcteristics of vehiculr networks from queueing theory perspective where the distribution of bckoff counter is derived nd the concurrently trnsmitting nodes re nlyzed. Bsed on the nlysis, we propose the Mtern-II-discrete process to pproximte the distribution of the concurrent trnsmitters of IEEE 82.p in vehiculr networks in this section. A. Model Description As mentioned in the system model, Φ [ ] is the set of nodes tht hve pcket witing to be trnsmitted. Assume tht it is one-dimensionl homogeneous PPP with density λ < λ, i.e., n independent thinning of Φ. Denote by Φ tx the set of nodes selected by the CSMA-bsed brodcst protocol to trnsmit t given time. Φtx is dependent thinning of Φ [ ] built s follows: ech point of Φ [ ] is ttributed n independent mrk which is discrete non-uniformly distributed in [ ]. The discrete mrk mimics the discrete bck-off counter vlues. A point x of Φ [ ] is selected in the Mtern-II-continuous process if its mrk is smller thn or equl to tht of ny other point of Φ [ ] in its neighborhood V (x). Hence, Φ tx is defined by Φ tx = { x Φ [ ] : m(x) m(y) for ll y V (x) }, () where m(x), denoting the mrk of point x, models the bckoff counter of the node nd hs the PMF given in (9), i.e., P (m(x) = k) = p k. This model cptures the fct tht CSMA will grnt trnsmission opportunity to given node if this node hs the

7 miniml bck-off counter mong ll the nodes in its crrier sensing rnge nd the fct tht node will be kept from trnsmitting if nother node in its crrier sensing rnge lredy trnsmits. This is similr to the Mtern-II-continuous process. The difference is tht the mrks hve discrete nd nonuniform distribution insted of continuous nd uniform distribution, nd hence this model cn lso include the concurrent trnsmissions since the probbility of two nodes with the sme mrk is not equl to zero, i.e., P (m(x) = m(y)). This is more ccurte ssumption in IEEE 82.p for V2V communictions s discussed in Section III-C. B. Retining Probbility Let p = P { Φ tx } be the Plm probbility of retining the typicl point of Φ [ ] in the thinning defining Φ tx. p cn be rewritten s ( p = P x x Φ ) tx m(x) = k p k. () k= Similr to the rgument in [2], the following theorem cn be obtined: Theorem. Given the probbility mss function p k of the bck-off counter nd Ryleigh fding, the probbility for typicl node to be retined in the thinning from Φ [ ] to Φ tx is p = exp ( λf X (k) c) p k (2) with F X (k) = c = 2π k= { k i= p i, if k >, if k =. + (3) e K mx(r,r)α rdr, (4) where K = P /P A. For α = 2, c = πe Kr2 ( K + ) r2. The proof is omitted since it is similr to the cse with uniform nd continuous counter in [2]. The following corollries give the retining probbility for the two specil cses of the bck-off counter distribution. One is the uniform discrete distribution corresponding to the sprse cse, nd the other the discrete distribution corresponding to the dense cse. Corollry. For p k = +, the retining probbility is p = + e λc, (5) e λc/( +) where c is given in (4). Proof: Insert p k = + into (2), nd it is strightforwrd to obtin the result. As λ, p, which mens ll nodes will trnsmit with probbility one if the system is extremely sprse. Also, note tht s, p e λc λc, which is the probbility for node to be grnted trnsmission in the Mtern-II-continuous process [2]. Hence, our mrk distribution ssumption generlizes the uniform mrk distribution. Corollry 2. For p k = p = k= 2( k) ( + ) exp where c is given in (4). Proof: Inserting p k = 2( k) ( +), the retining probbility is ( λc ) k (2 + k), (6) ( + ) 2( k) ( +) into (2), we obtin (6). As λ, p 2 +, which is the probbility of the bck-off counter to be zero. It mens tht when the system is extremely dense, only the nodes with bck-off counter zero hve chnce of trnsmitting. This is intuitive. From Theorem nd Corollries nd 2, it is esy to see tht the retining probbility is lower bounded by p nd upper bounded by p. V. PERFORMANCE EVALUATION Bsed on the newly proposed Mtern-II-discrete process, we cn evlute the performnce metric of interest for vehiculr networks. It is well ccepted tht pcketized trnsmission is considered successful if the signl-to-interference-plus-noise rtio (SINR) is greter thn some threshold [3]. So we define the trnsmission success probbility s follows. Definition. The trnsmission success probbility is the probbility of successful trnsmission from node x to node y t distnce r = x y, p (r, T, α) P (SINR T ), (7) where SINR = P Sx y (r) l(r) I(y)+N, I(y) is the interference t the receiver y, nd N is the noise power. It is one of the most importnt metrics in evluting the performnce of vehiculr networks. e will nlyze the trnsmission success probbility for nd CSMA nd compre the trnsmission success probbilities for different models in the next subsection. A. Trnsmission Success Probbility for First, we define the thinning probbility in the MAC scheme. At ny given time, the probbility tht node is trnsmitting cn be computed s p = T p /τ, where τ =. s is the pcket genertion period. For comprison, the trnsmission success probbility of is given by the following theorem: Theorem 2. The trnsmission success probbility with pth loss exponent α = 2 nd distnce r is ( p (r, T, 2) = exp λ p ) πt r exp ( NT r 2 /P A ) (8) in the Ryleigh fding cse, nd it is ( λ p ) π p(r, T, 2) = erf /T r2 N/P A in the non-fding cse. (9)

8 Proof. (8) is directly from [6]. For the non-fding cse, the probbility density function of the interference is [6] Since f I (y) = λ p y 3 2 e λ /P A 2 p2 πp A y. (2) ( ) λ pπ 2 f I (y) dy = erf, (2) /P A it follows tht ( ) P A p (r, T, 2) = P T (I + N) (22) r2 ( λ p ) π = erf. (23) /T r2 N/P A B. Trnsmission Success Probbility for CSMA Since it is difficult to derive the trnsmission success probbility for the Mtern-II-discrete process model theoreticlly, n estimtor similr to tht in [2] is used to estimte the trnsmission success probbility of the new model. First, the loctions of the nodes with pckets re smpled ccording to PPP on the intervl [, L]. The density of this PPP is determined by the density of nodes with bck-off counter t ny given time instnt, which include those from the previous time nd the new rrivls. The power fding coefficient from ech trnsmitting node to ny other loction is exponentil with men one (Ryleigh fding). The interference is evluted s the sum of the powers of ll other concurrent trnsmitting nodes. The Mtern-II-discrete process Φ tx with discrete nd non-uniform counter is simulted using (). To get rid of the border effect, the intervl [, L] is considered s circulr. The counter distribution is given by (9) with slope estimted from simultion dt. The trnsmission success probbility is clculted using the estimtor ˆp (r, T, 2) = 2 E [ x Φ [,L] B + C ] tx [,L] Φ [,L] Φ tx, tx { (24) P AS x } { x+r where B = r 2 I(x+r)+N T P AS x x r nd C = r 2 I(x r)+n }. T B (or C) is the event tht for given node x Φ tx the SINR t distnce r right (or left) from x is greter thn or equl to the threshold T. Φ [,L] tx indictes the number of nodes in Φ tx in the intervl [, L]. The Mtern-II-continuous process is formed by dependent thinning with the following definition Φ tx = { x Φ [ ] : m b (x) < m b (y) for ll y V (x) }, (25) where m b (x) is the mrk of x Φ [ ], which is uniformly distributed on [, ] [2]. The estimtor of the trnsmission success probbility of the Mtern-II-continuous process is given in similr wy: ˆp b (r, T, 2) = 2 E x Φ [,L] tx C. Performnce Comprison [ B + C ] Φ [,L] tx Φ [,L] tx. (26) e compre the trnsmission success probbilities for, Mtern-II-discrete process nd Mtern-II-continuous process, while using simultions from ns2 s the bseline. ns2 hs been widely ccepted by the reserch community due to its cpbility to ccurtely simulte communictions in lrge vriety of wireless environments. For vehicle-to-vehicle brodcst service in highwy scenrios, the rticles [], [2] describe work crried out to further improve the relism of simultions nd lrge number of publictions therefore used ns2 to provide ground truth dt (e.g., [6]). In this pper, we rely on the ns2.34 simultor. e implemented the EDCA bck-off process nd clibrted it with work in [22]. e set up circulr rod of L = km nd rndomly plce the nodes. The speed of the vehicles is ignored becuse they brely move in the trnsmission time of one single pcket nd only reltive positions mtter. All other simultion prmeters re summrized in Tble II. Prmeter Vlue crrier frequency 5.9 GHz pcket size 44 Byte noise floor 99 dbm trnsmit power dbm brodcsting contention window ( + ) 6 periodicity ms slot time δ 3µs modultion BPSK brodcst rte 6 Mbps pth loss exponent α 2 pth loss constnt A 7.86 dbm reference distnce r SINR threshold T 7 db simultion length L km Tble II: Simultion Prmeters Fig. 6 shows the trnsmission success probbilities of the vrious models for the non-fding cse under different node densities λ. Fig. 6() vlidtes tht the concurrent trnsmitters in IEEE 82.p form hrd-core process in the sprse cse s discussed in Section III-B. For the ner distnce (less thn 3 m), the trnsmission success probbility of the ns2 simultions is very close to tht of two Mtern type II models. There is little difference between the Mtern-II-continuous nd Mtern-II-discrete processes since the probbility tht nodes within ech other s crrier sensing rnge re trnsmitting simultneously is zero in the sprse cse. For the intermedite-density cse, the Mtern-II-discrete process produces lower trnsmission success probbility thn the Mtern-II-continuous process s it llows for nodes to

9 hve the sme mrks. Although none of the three models cn mtch ns2 simultions very well due to the complexity of networks with intermedite densities, the Mtern-II-discrete process provides the closest pproximtion within -2 m, which is the most criticl rnge to vehiculr sfety [2], [26], [27]. In the dense cse, the Mtern-II-discrete process mtches the ns2 simultion precisely. The trnsmission success probbility for is lso close to tht of the ns2 simultion in the dense cse s climed in [6] while it seems to be very loose lower bound for the sprse nd intermedite cses. The trnsmission success probbility for the Mtern-II-continuous process for the intermedite nd dense cses look like becuse there is sturtion phenomenon in its intensity λ b, i.e., λ b is upper bounded by λ b,mx = 2R, where R is the crrier sensing rnge in (4). For the non-fding cse, R is fixed, nd hence the mximum verge number of retined nodes on rod of length L from the Mtern-II-continuous process is upper bounded by L 2R, which is independent of λ. For the fding cse, the performnce of the Mtern-IIdiscrete process shows the sme trend s in the non-fding cse. However, seems to hve good performnce s well. This cn be explined by the fct tht fding rndomizes the interference nd therefore the ctul network is perceived s n equivlent Poisson network. A similr observtion ws mde in [28] in the cse of cellulr networks. In other words, fding dmpens the hrd-core effect nd mkes the trnsmitter distribution look more like PPP to the receivers. In ddition, the lck of RTS/CTS my further reduce the hrd-core effect. VI. CONCLUSIONS In this pper, we explored the geometric modeling of DSRC for V2V sfety communictions by using tools from stochstic geometry nd queueing theory. Firstly, we nlyzed the distribution of trnsmitters for networks with different densities. e found tht without considering fding, by incresing the network density, the trnsmitter distribution chnges from hrd-core model to PPP model. ith fding, the rndomness of interference increses, which mkes the networks pper s n equivlent Poisson network. In other words, the trnsmitters behve like PPP from the receivers perspective. Secondly, we proposed to use non-uniform discrete distribution to replce the uniform distribution for the mrks in the Mtern- II-continuous process. The resulting Mtern-II-discrete process therefore retins concurrent trnsmitters within the sme crrier sensing rnge nd thus pproximtes the network dynmics more precisely thn the Mtern-II-continuous process. Thirdly, we compred our models with simultions from ns2. The results show tht our model performs well in wide rnge of network densities. REFERENCES [] Q. Chen, F. Schmidt-Eisenlohr, D. Jing, M. Torrent-Moreno, L. Delgrossi, nd H. Hrtenstein, Overhul of IEEE 82. modeling nd simultion in ns-2, in the th Interntionl Symposium on Modeling Anlysis nd Simultion of ireless nd Mobile Systems, MSiM 27, Chni, Crete Islnd, Greece, October 22 26, 27. ACM, 27, pp. 59 68. Trnsmission Success Probbility Trnsmission Success Probbility Trnsmission Success Probbility.9.8.7.6.5.4.3.2. Mtern II Continuous Mtern II Discrete 2 4 6 8 Distnce /m.9.8.7.6.5.4.3.2. () Sprse cse with density λ =.33 Mtern II Continuous Mtern II Discrete 2 4 6 8 Distnce /m.9.8.7.6.5.4.3.2. (b) Intermedite-density cse with density λ =.66 Mtern II Continuous Mtern II Discrete 2 4 6 8 Distnce /m (c) Dense cse with density λ =.32 Figure 6: Trnsmission success probbility of different models for the non-fding cse: the trnsmission success probbilities for the two Mtern type II processes re verged over 4 reliztions.

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[25] G. Bnsl, J. Kenney, nd A. einfield, Cross-vlidtion of DSRC rdio testbed nd ns-2 simultion pltform for vehiculr sfety communictions, in 2 IEEE Vehiculr Technology Conference (VTC Fll), 2, pp. 5. [26] Vehicle Sfety Communictions-Applictions (VSC-A) Finl Report: Appendix Volume. Ntionl Highwy Trffic Sfety Administrtion, 2. [27] X. Cheng, L. Yng, nd X. Shen, D2D for intelligent trnsporttion systems: A fesibility study, IEEE Trnsctions on Intelligent Trnsporttion Systems, vol. PP, no. 99, pp., 25. [28] B. Błszczyszyn, M. Krry, nd H. Keeler, Using Poisson processes to model lttice cellulr networks, in 23 IEEE INFOCOM, 23, pp. 773 78. Christin Poellbuer (S 97, M 4, SM 9) received his degree of Diplom Ingenieur in Computer Science from the Vienn University of Technology, Austri, in 998 nd his Ph.D. degree in Computer Science from the Georgi Institute of Technology, Atlnt, GA, in 24. He is currently n Associte Professor in the Deprtment of Computer Science nd Engineering t the University of Notre Dme. His reserch interests re in the res of wireless sensor networks, mobile computing, d-hoc networks, pervsive computing, nd mobile helthcre systems. He hs published more thn 9 ppers in these res nd he hs co-uthored textbook on ireless Sensor Networks. His reserch hs received funding through the Ntionl Science Foundtion (including CAREER wrd in 26), Ntionl Institutes of Helth, Army Reserch Office, Office of Nvl Reserch, IBM, Intel, Toyot, GE Helth, the Ntionl Footbll Legue, Serim Reserch Corportion, nd Motorol Lbs. Zhen Tong received the B.Eng. degree in utomtion from Hrbin Institute of Technology, Chin, in 29, nd the M.S. nd Ph.D. degrees in electricl engineering from the University of Notre Dme, Notre Dme, IN, USA, in 22 nd 25, respectively. He is currently Senior Engineer t Qulcomm Technologies Inc., Sn Diego, CA, USA, where he works on modem softwre design nd development. His reserch interests include power control nd energy efficient MAC design, nd nlysis nd implementtion of full-duplex wireless networks. Hongsheng Lu received the B.S. nd M.S. in Electricl Engineering from Beihng University, Beijing, Chin, in 26 nd 29, respectively. He strted working towrds Ph.D. degree in Computer Science nd Engineering from 29 in University of Notre Dme, Notre Dme, IN. His interests re congestion control in DSRC-bsed vehicle-to-vehicle sfety communictions. Mrtin Henggi (S 95, M 99, SM 4, F 4) is Professor of Electricl Engineering nd Concurrent Professor of Applied nd Computtionl Mthemtics nd Sttistics t the University of Notre Dme, Indin, USA. He received the Dipl.-Ing. (M.Sc.) nd Dr.sc.techn. (Ph.D.) degrees in electricl engineering from the Swiss Federl Institute of Technology in Zurich (ETH) in 995 nd 999, respectively. For both his M.Sc. nd Ph.D. theses, he ws wrded the ETH medl, nd he received CAREER wrd from the U.S. Ntionl Science Foundtion in 25 nd the 2 IEEE Communictions Society Best Tutoril Pper wrd. He served s n Associte Editor for five journls from 25-23 nd presently is the chir of the Executive Editoril Committee of the IEEE Trnsctions on ireless Communictions. He lso ws Distinguished Lecturer for the IEEE Circuits nd Systems Society in 25-26, TPC Co-chir of the Communiction Theory Symposium of the 22 IEEE Interntionl Conference on Communictions (ICC 2) nd of the 24 Interntionl Conference on ireless Communictions nd Signl Processing (CSP 4), Generl Co-chir of the 29 Interntionl orkshop on Sptil Stochstic Models for ireless Networks (SpSiN 9) nd the 22 DIMACS orkshop on Connectivity nd Resilience of Lrge- Scle Networks, nd Keynote Speker of SpSiN 3, CSP 4, nd the 24 IEEE orkshop on Heterogeneous nd Smll Cell Networks. He is co-uthor of the monogrph "Interference in Lrge ireless Networks" (NO Publishers, 29) nd the uthor of the textbook "Stochstic Geometry for ireless Networks" (Cmbridge University Press, 22). His scientific interests include networking nd wireless communictions, with n emphsis on cellulr, morphous, d hoc, cognitive, nd, sensor networks.