INFORMATION DISSEMINATION DELAY IN VEHICLE-TO-VEHICLE COMMUNICATION NETWORKS IN A TRAFFIC STREAM

INFORMATION DISSEMINATION DELAY IN VEHICLE-TO-VEHICLE COMMUNICATION NETWORKS IN A TRAFFIC STREAM LiLi Du Depatment of Civil, Achitectual, and Envionmental Engineeing Illinois Institute of Technology 3300 South Fedeal Steet, Chicago, IL 6066, USA ldu3@iit.edu Hoang Dao Depatment of Civil, Achitectual, and Envionmental Engineeing Illinois Institute of Technology 3300 South Fedeal Steet, Chicago, IL 6066, USA hdao@hawk.iit.edu Xiang-Yang Li Depatment of Compute Science Illinois Institute of Technology 0 West 3 st Steet, Chicago, IL 6066, USA xli@cs.iit.edu Abstact Vehicle-to-vehicle (VV) communication netwoks, as one of the coe components in connected vehicle systems, have been ganted many pomising applications to addess taffic mobility, safety and sustainability. Howeve, only a limited amount of wok has been completed to undestand the fundamental popeties of infomation popagation in such systems, while compehensively consides taffic and communication eality. Motivated by this view, this poposed eseach develops analytical fomulations to estimate infomation popagation time delay via a VV communication netwok fomed on a one-way o two-way oad segment with multiple lanes. Distinguished to pevious effots, the poposed study caefully involves seveal citical communication and taffic flow featues in eality, such as wieless communication intefeence, intemittent infomation tansmission, and dynamic taffic flow. Moeove, this study elaboately analyzes the inteactions between infomation and taffic flow unde spase and congested taffic flow conditions. The numeical expeiments based on Next-Geneation Simulation (NGSIM) field data illustate that the poposed analytical fomulations ae able to povide vey good estimation, with the elative eo less than 5%, fo the infomation popagation time delay on a one-way o two-way oad segment unde vaious taffic conditions. The poposed wok can be futhe extended to chaacteize infomation popagation time delay and coveage ove local tanspotation netwoks. Index Tems vehicle-to-vehicle communication, time delay, dynamic taffic flow I. INTRODUCTION Connected vehicle systems (CVS), as a new geneation of intelligent tanspotation system, is consisted of smat vehicles and oadside infastuctue equipped with wieless communication facilities, which enable vehicle-to-vehicle (VV) and vehicle-to-infastuctue taffic infomation exchange. As one of the key components, VV communication netwoks (i.e. vehicula ad hoc netwoks (VANET) in liteatue) have been ganted many pomising applications in taffic safety, mobility, and sustainability. Fo example: [][][3] demonstate that VV will allow dives to awae of othe cas speed, acceleation o deceleation so that keep away fom accidents such as oad depatue and collisions. Thus, VV communication netwoks can be used to develop dive assistance systems which avoid taffic cash and impove diving safety. Alsabaan et al. [4] popose employing vehicle-to-vehicle as well as taffic-light-signal-to-vehicle communication technologies to enable dives to adaptively adjust thei diving speed so that the objectives to pomote fuel consevation and emission eduction within tanspotation systems can be achieved. Yamaha, et al [5] shows that vehicleto-vehicle communication netwok may detect oad condition such as oad suface status in snowing weathe, and then infoms taffic management cente to adjust taffic contol stategies and impove taffic mobility. In the field of taffic outing, VV infoms dives taffic condition infomation such as wok zone, accident ahead, closing lane, etc; so they can change oute to avoid the waiting time. The community based online navigation system developed by Waze [6] has indicated a geat potential of developing on-line outing guidance based on VV communication technologies. At the meantime, many national and intenational pojects such as PATH [7]; CaTalk [8]; FleetNet [9], have been dedicated to test the applicability of VV technologies in vaious tanspotation scenaios. Even though many damatic applications have been poposed, it is noticed that the availability of eal-time taffic infomation popagating via VV communication is one of the citical bottlenecks to limit thei implementation in pactice. Fo example, fowad collision waning applications based on VV need the infomation egading how fast a waning can be popagated to a vehicle; accident waning applications need the infomation to each as many vehicles as possible in the local tanspotation netwok. Without knowing the infomation popagation chaacteistics, such as connectivity, tansmission distance, time delay, coveage, it is had to successfully implement those applications based on VV technologies. This equiement has spued plenty of studies in both tanspotation and wieless communication communities. Reseach fom wieless communication field mainly focuses on developing

advanced communication potocols to ensue efficient infomation tansmission among smat vehicles once the physical connections between smat vehicles ae built, which is not the focus on this study. The poposed study shaes common inteests with pevious studies in tanspotation community, which seeks to captue the infomation popagation benchmaks associated with taffic steam featues. Accodingly, the assumptions on successful wieless tansmission conditions and dynamic taffic flow along with thei methodologies shaped the chaacteistics of most pevious studies. Thus, this study eviews the existing liteatue fom these two aspects, which also diffeentiate the poposed study and highlight ou main contibutions. Plenty of eseach exploes instantaneous infomation popagation in a taffic steam, consideing that infomation popagation is instantaneous compaed to vehicle movement. Usually, a successful communication is simply identified by the condition stating the geometic distance between two devices (equipped on smat vehicles o oadside sensos) less than a pe-defined tansmission ange (with kilomete being the maximum). Repesentative eseach in this categoy studied the topics coveing connectivity, inte-vehicle communication system, infomation tansmission distance, and pobability of success fo infomation popagation. Fo example, Yang and Recke [0] built a simulation famewok to test the infomation dissemination efficiency (coveage, speed) ove vaious taffic conditions as infomation popagating though inte-vehicle communication; Jin and Recke [] computed the pobability of a successful instantaneous infomation tansmission between two vehicles in unifom and geneal taffic steams; Wang [] povided the mean and vaiance of infomation popagation distance consideing equipped vehicle density and tansmission ange. Both Ukkusui and Du [3] and Jin and Recke [4] developed analytical fomulations to pedict multi-hop connectivity of inte-vehicle communication netwok assuming stationay taffic steam, but diffeent mathematical models ae used. Chen et al. [5] evaluated the pefomance of multi-hop boadcast communication (infomation popagation distance, and thoughput of infomation package to be eceived fo a given distance) with vehicles following shockwave mobility patten which mixes fee flow and congested flow taffic. Wang et al. [6] and Yin et al. [7] estimated the expectation, vaiance and pobability distibution of instantaneous infomation popagation distance assuming that vehicles headway follows Gamma, Poisson, o Log-nomal distibution. Clealy, the deficiencies of this goup of eseach ae in two aspects: ) taffic flow dynamics is not fully consideed. Since infomation tansmission time is omitted, and message popagation is studied at a snapshot, taffic flow is teated as static flow; ) ovesimplify the wieless communication constains, ignoing backgound noise and intefeence. These two deficiencies will be addessed in the poposed study. Some eseaches ecognized that infomation popagation in VV communication is significantly impacted by taffic dynamics. Fo example Schonhof et al. [8] consideed dynamic communication link in a dynamic taffic flow on a two-way feeway taffic steam, and then investigated how the smat vehicle density impacts infomation popagation speed and efficiency. Agawal [9] studied delay toleant message popagation in VV and developed uppe and lowe bounds fo infomation popagation ate as the functions of taffic density, vehicle speed and tansmission ange. Wu et al. [0] indicated that infomation popagation distance and speed depend on elative vehicle movement and othe taffic chaacteistics such as vehicle density and aveage vehicle speed; both one- and two-way vehicle taffic scenaios ae consideed. Wang et al. [] used taffic flow theoies such as ca-following models to captue the vehicle mobility and applied a Monte Calo simulation model to evaluate the impacts of taffic flow, tansmission ange on the thoughput of a VANET. Wang [] modeled infomation popagation in VANET as a elay pocess, and povided the mean and vaiance of infomation popagation distance as well as its distibution in VANET, but consideing the pesence of equipped vehicles follows an independent homogeneous Poisson pocess, which is usually denied in actual taffic flow condition. Utilizing the infomation popagation model poposed in Wang [] to evaluate infomation tavel times on the individual acs, Ng and Walle [3] povided the lowe and uppe bounds of infomation popagation delay between two nodes in a netwok, whee taffic flow chaacteistics ae evaluated by a static taffic assignment model. Wang et al. [4] poposed an analytical model to estimate the expected infomation popagation speed in the ealy stage of deploying VV communication netwok, which implies vey low smat vehicle penetation. Du and Ukkusui [5] modeled infomation popagation along a one-way oad segment as a time-expanded netwok and povided a closed-fom fomulation to captue the netwok connectivity ove a time peiod (eachability) fo VANET. The above eview noticed that many studies exploed infomation popagation distance o speed, but a limited amount of wok studies infomation delay (which is the focus of this study); in addition, all the afoementioned study applied simplified successful tansmission condition, which will be substituted by a moe compehensive condition elevant to communication intefeence. Oveall, the above bief suvey demonstates that pevious eseach has significantly pomoted the undestanding of infomation popagation in a taffic steam fom diffeent angles by eniching ealistic taffic flow featue in the modeling pocess, nevetheless, the consideation of the communication side is elatively weak, which degades the value of the eseach in pactical application. In addition, thee is not sufficient study focusing on the inteactions between taffic flow movement and infomation popagation unde vaious taffic conditions on two-way oads. Motivated by the above points, this eseach poposes mathematical desciption models to stengthen pevious eseach fom the following aspects. i) The poposed appoach identifies successfully wieless communication by Signal Intefeence plus Noise Ratio (SINR) condition athe than only factoing the tansmission ange and Euclidean distance between communication devices. SINR consides multiple impact factos in wieless communication such as tansmission powe and intefeence between concuent tansmissions. This

impovement will make ou desciption model moe ealistic fom wieless communication pespective. ii) This study takes account of infomation communication time to povide the applicability of the poposed fomulations to the eseach of netwok level infomation dissemination. It is ealized that the communication time is ignoable fo measuing the delay on a oad segment, but its accumulation effect is significant fo the infomation delay ove the netwok. iii) This eseach elaboately consides the inteaction of taffic flow and infomation popagation. Namely, vaious taffic condition (e.g. fee flow, mild congested taffic flow, congested taffic flow), diffeent moving diections of taffic flow and infomation flow (i.e. infomation and taffic flows in same o opposite diection on one-way o two-way oads) as well as thei combinations ae fully coveed. Theefoe, the poposed eseach compehensively consides the ealistic in both tanspotation and communication sides. It will impove ou undestanding of eal-time taffic infomation delay in VV communication and pomotes eliable applications in pactices. The whole of the pape is oganized by the stuctue below: section I intoduces eseach backgound and motivations; section II pesents the poblem fomulations including taffic flow model, infomation flow model and successful wieless communication condition; section III poposes ou methodology to develop mathematical estimation fo infomation popagation time delay in VV communication netwoks on a oad segment. Vaious taffic conditions ae consideed. The poposed fomulations ae validated by numeical expeiment tests pesented in section IV; and the conclusion of this study is given in section V. II. PROBLEM FORMULATIONS The poposed eseach is dedicated to exploing mathematical fomulations to estimate infomation popagation time delay in a VV communication netwok, unning on a oad segment. This study fist conduct analyses on taffic flow, infomation flow, and successful communication condition, which all togethe seviced as the basis to develop igoous analysis in the poposed study. A. Road Segment and Taffic Flow Without loss of geneality, this study woks on a oad segment (eithe one-way o a two-way) only with an exit and an entance at each end of the oad segment. Namely, no vehicles exit o ente in the middle of the oad segment. The oad segment is with length and its taffic steam is composed of vehicles including both smat vehicles and nonsmat vehicles moving on eithe same o opposite diections. Non-smat vehicles do not have wieless communication capability; they only impact oveall taffic flow featues and do not influence infomation popagation diectly, thus they ae not counted. Thoughout the pape, unless noted othewise vehicle and smat vehicle ae equally used. Smat vehicles ae numbeed fom left to ight accoding to thei position in a taffic steam. As shown in Fig., epesents the distance between smat vehicles and j. As, vehicles and ae consecutive; epesents the coesponding space headway. Taffic Flow 3 i x x 3 x ij Case (a) Case (b) Case (c) Case (d) Infomation Flow Fig.. Infomation popagation pocess B. Infomation Flow Model The infomation flow in this study is modeled by specifying the following aspects. (i) Infomation always popagates fom one vehicle to its fist neaest neighbo (fom vehicle to, to 3, until it aives at the end of the oad segment), whee infomation tansmission has the highest oppotunity to success accoding to the successful tansmission condition intoduced in Pat C of this section. (ii) Infomation popagating fom one end of the oad segment to the othe along the diection that the oad extends is studied. The cuvatue of the oad is ignoed. (iii) Physical dimensions of smat vehicle ae ignoed. Smat vehicle is epesented by a small ectangle without consideing its physical dimensions. Infomation may flow in eithe the same o the opposite diection to the taffic flow. Thus, fou possible cases only with smat vehicles illustated in Fig. ae in consideation. Moe exactly, Case (a) o Case (b) epesents the situation whee infomation flows in same o opposite diection as the taffic flow on a one-way (possibly multiple-lane) oad. Case (c) and Case (d) ae essentially the same, epesenting a situation whee the infomation flows opposite to one of the taffic flows on a two-way oad. The poposed study will cove all these fou cases. Fig.. Taffic flow and infomation flow on a segment Infomation Flow j Taffic Flow 3

C. Successful Condition This study assumes that the infomation tansmission between smat vehicles applies a dedicated shot-ange communications (DSRC) adio tuned to the 5.9 GHz fequency, allocated by the Fedeal Communications Commission fo tanspotation safety and mobility applications on vehicle and infastuctue. The successful commination between two consecutive vehicles is identified by SINR condition, whose standad fomulation is shown in Eq., in which vehicle is the tansmitte and vehicle is the eceive. It indicates that vehicle will successfully tansmit infomation to vehicle, if SINR value at the eceive is geate than a theshold value. Pw xwi SINR () N I Whee, epesents tansmission powe of node ; epesents the distance between tansmitte vehicle and eceive vehicle ; is the signal powe decay, typically ; epesents the backgound noise on the fequent channel utilized by netwok; is the theshold which depends on the designing modulation and code ate (values which indicate the data tansmission ate duing a wieless connection) of wieless netwok. = 0.5 is ecommended fo VV communication [6]; epesents the sum of intefeence powe fom othe vehicles except vehicle to eceive vehicle i., if vehicle is in tansmission status, othewise,. SINR is a physical model to detemine successful eception of a tansmission ove one hop in wieless netwok. It consides many envionment factos: the distance between two nodes, path loss of signal and wieless intefeence. Thus, using SINR will make ou fomulations captue moe communication eality. The standad SINR fomulation can be futhe simplified by consideing the communication featues in VV wieless communication netwok. Fist, smat vehicles in VV usually apply boadcast potocol, thus, we have. Second, existing liteatue [7] shows that the backgound noise in VV communication usually follows nomal distibution with zeo mean. Accodingly, this study applies and coesponding to fee space infomation popagation [8]. Last, assuming all smat vehicles adapt the same tansmission powe (this assumptions have been widely used in liteatue such as [9]), we have. With the above fou featues holding in VV communication netwok, the standad SINR fomulation in Eq. is tansfomed to Eq., and futhe pocessed to obtain the elationships in Eq. 3. SINR n x j, jw wi x ji j, jw xwi n xwi x ji xwi i () (3) Whee, ( ). Eq. 3 epesents the condition of a successful tansmission between two smat vehicles. It not only factos the distance between the tansmitte and eceive, but also the distibution of all othe vehicles aound them (epesented by ), eflecting the instantaneous taffic condition. Howeve, vaies with the location of the tansmitte and eceive. Thus, Eq. 3 is the fomulation of individuality, which implies mico-level vehicle distibution infomation is needed. It is not a pope fomulation to be used. The poposed study then exploes a unifom fomulation (a pseudo tansmission ange deived fom SINR condition), which can be used to identify successful infomation tansmission by known aggegated taffic infomation. We pesent ou method below. D. Pseudo Tansmission Range To develop the unifom successful tansmission condition fom SINR, this study labels smat vehicles fom the left to ight by numbe, as shown in Fig. 3. Next, the spacing is appoximated by, whee h is the expected spacing between two adjacent smat vehicles. Accodingly, can be appoximated by Eq. 4, whee is the label of the eceive. i ni i (4) h m m m m Taffic flow 0 i Intefeence to vehicle i Fig. 3. Intefeence at vehicle i Note that implies the infomation is tansmitted fom vehicle 0 to vehicle ; the case is not consideed since infomation stats fom vehicle 0 and it will only be a tansmitte athe than a eceive in this study; case indicates that infomation aives at the last vehicle so thee is no intefeence coming fom its ight side. This study futhe woks on Eq. 4 and obtains the appoximation fo in Eq. 5 accoding to [30], Moe, we know that the two lowe bounds shown in Eq. 6 and Eq. 7 exist, accoding to [3]. i+ i+ n (5) (6) (7) 4

Plugging Eq. 6 and Eq. 7 into Eq. 4, the lowe bound fo is given in Eq. 8. ( ) ( ) (8.a) ( ) (8.b) Accoding to Appendix A and A, it is obseved that the lowe bound in Eq. 8.a is a tight bound to. Fo example, Equation 8.a is with the maximum elative eo equal to 4% as and 3% as, which happens at bounday points; as, the elative eo of Eq. 8.a is significantly educed (less than %). Equation 8.b ( is with a vey small elative eo (0.3% as and 0.08% as ). In addition, it is obseved that a lage value leads to a smalle elative eo and tighte lowe bound. Substituting in Eq. 3 by the lowe bounds given above, the SINR condition is tansfomed to Eq. 9. i h, i n n 3 i n i xwi (9) i n h,i n 6 i Obseving, we know in Eq. 9. In addition, it is ecognized that eaches to the minimum value at. By applying the tightest bound fo, SINR condition is led to the fomat in Eq. 0. n x ji min i h i 4 (0) 3 n Whee, is consideed as a pseudo tansmission ange which limits the successful tansmission. Note that hee is deived fom SINR condition. It eflects the taffic flow influence by factoing space headway between vehicles as well as the vehicle distibution aound the eceive on the oad. It is diffeent to the fixed tansmission ange specified by tansmission powe and fequency. III. METHODOLOGY This section pesents ou methodology to captue the time delay of infomation speading on a oad segment in Case (a). Namely, the infomation delay on a one-way oad is fist studied, consideing infomation flows in the same diection as taffic flow. Futhemoe, we demonstate that the poposed methodology is applicable to Cases (b), (c) and (d). A. Time Delay of Intemittent Tansmission Due to the effect of taffic flow dynamics on wieless communication connection, it has been ecognized that intemittent communication epesents a geneal infomation tansmission fashion, in which wieless connection is intemittently connected (leading to instantaneous tansmission) and boken (leading to fey tansmission) due to elative movement between vehicles. Intemittent tansmission usually happens in a taffic flow with mild congestion. Pue fey and instantaneous communication ae two extemes of intemittent tansmission. They usually happen in vey spase taffic and highly congested taffic flow, whee the wieless connection between two vehicles happens aely o constantly. Thus, the methodology poposed below will focus on intemittent communication. Fig. 4 povides an illustation about the infomation popagation in Case (a) duing a time inteval. An intemittent tansmission is consideed to follow a patten in which instantaneous tansmission and fey tansmission altenatively happen until the infomation aives at the end of the oad segment. As an instantaneous tansmission occus, seveal vehicles ae well connected and infomation is smoothly tansmitted, such as fom node to node in Fig. 4. The coesponding time delay is calculated by:, whee epesents the tansmission time and epesents the numbe of the hops. Note that the communication time delay is taken into account to make the poposed model applicable to captue the time delay of infomation dissemination ove a lage scale netwok, whee the accumulated effects of a huge numbe of instantaneous communication time delay shows impact, even though it is ignoable on a shot oad segment. As taffic is vey spase, fey tansmission happens. Namely, the infomation will be feied by a vehicle until it meets anothe vehicle, such as the infomation popagation at node is boken, and then feied by node until it eaches to node at anothe time in Fig. 4. The coesponding time delay is calculated by, whee epesents the caying-on distance and epesents the aveage speed of the vehicle caying infomation. Ignoing the bounday case in which instantaneous tansmission o fey tansmission happens one moe times than the othe, the expected time delay of a piece of infomation tavesing on a oad segment is estimated by the totally time delay fo one intemittent tansmission (i.e. an instantaneous tansmission followed by a fey tansmission and vice vesa) multiple by expected times of the intemittent tansmission happens. Mathematically, this idea is pesented by Eq. below. ( ) ( ) () Whee, and epesent the infomation popagation distance following instantaneous and fey tansmission espectively; and epesent thei expected values; epesents the expected numbe of hops in instantaneous tansmission; is the length of the oad segment. Eq. coves vaious taffic flow conditions. dominates the time delay in congested taffic condition, but mainly accounts fo the time delay in spase taffic condition; and togethe captues the featue in the intemediate congested taffic condition. The follows of this pape futhe pesent ou appoaches to develop the fomulations fo and incopoating taffic flow featues and communication limits. 5

i Taffic flow i+ i+ i+k x Infomation flow y j i+k Fig. 4. Infomation popagation on one-way oad B. Expected Infomation Popagation Distance in an Instantaneous Tansmission As the space headway between a tansmitte and eceive satisfy, infomation popagates by instantaneous tansmission. Consideing infomation always popagates to its neaest neighbo, the conditional andom vaiable epesents the space headway in instantaneous tansmission. Given an instantaneous tansmission includes hops on aveage, we obtain the fomulation fo below. () Whee, is the expected space headway in instantaneous tansmission; can be calculated by Eq. 3, which is deived by the mathematical pocess given in Eq. 4 & Eq. 5, given the spacing distibution is known. E( S S ) 0 0 sf s ds f u du j (3) obsevation, whee a piece of infomation popagates on a oad segment with same vehicle distibution, but the instantaneous tansmission Fig. 5 (a) follows scenaio I, i.e. a tansmitte only each to the neaest neighbo; the instantaneous tansmission in Fig. 5 (b) o (c) follows scenaio II, i.e. a tansmitte eaches to the neaest two o thee neighbos espectively. Then, if vehicle 5 fails to each its neaest neighbo, vehicle 6, then we know that vehicle 5 cannot each othe vehicle futhe in Fig. 5 (b) and (c) accoding to ou successful tansmission condition given in Eq. 0. In this context, vehicle 4 will fail to each vehicle 6 in both Fig. 5 (b) and (c) since is moe difficult to building up connection between vehicle 4 and 6 than vehicle 5 and 6. Following the same thought, vehicle 3 cannot connect to vehicle 6 eithe in Fig. 5 (c). As a esult, the infomation popagates the same distances fom vehicle to vehicle 5 unde the thee examples in Fig. 5, even though one tansmitte only connects the neaest neighbo in scenaio I, two o thee eceives in scenaio II. This is a good quality fo the poposed methodology in Eq.. (iii) The instantaneous tansmission time is vey small (mico seconds), so the time delay diffeence between Scenaio I and Scenaio II is small and negligible. Hence, this study focus on Scenaio I to study the infomation popagation time delay on oad segment. 3 4 5 6 7 (a) Scenaio I: connected to the neaest eceive 3 4 5 6 7 (b) Scenaio II: connected to the neaest two eceives df S S f s f S S ds f u du 0 (4) (5) This study noticed that in eality, one tansmitte may successfully tansmit one piece of infomation to multiple eceives (efeed to as scenaio II) athe than only to the neaest neighbo (efeed to as scenaio I). Howeve, we focus on scenaio I in this study fo the thee easons. (i) It is had to decide how many vehicles one tansmission will cove. It depends on the vehicle distibution aound a tansmitte. As the taffic distibution is not unifom, this numbe is uncetain and becomes vey difficult to decide. (ii) This study obseved that the infomation popagates the same distance along the oad segment unde these two scenaios given we ignoe the distance between vehicles vetical to diection that the oad extends. Fig. 5 povides examples to illustate this 3 4 5 6 7 (c) Scenaio II: connected to the neaest thee eceives Fig. 5. One tansmitte has one, two o thee eceives C. Expected Infomation Popagation Distance in a Fey Tansmission As the space headway between a tansmitte and a eceive (two consecutive vehicles) satisfy, infomation will be spead by a fey tansmission. A fey tansmission will stop as the spacing between this tansmitte and a eceive satisfies. Theefoe, the expected infomation popagation distance by a fey tansmission,, can be calculated by Eq. 6. E S S E y vi (6) v ij Whee, epesents the expected spacing given a fey tansmission happens; is the aveage elative speed between two vehicles i and j. Consideing as a 6

conditional andom vaiable, can be calculated by Eq. 7, which is deived by the cumulative distibution and pobability density fomulations fo given by Eq. 8 and Eq. 9 espectively. E( S S ) B i i- i i+ i+ i+3 A A 0 sf s ds sf s ds f u du f u du df S S f s f S S db f u du D. Expected Hops in an Instantaneous Tansmission (7) (8) (9) A piece of infomation may popagate multiple hops in an instantaneous tansmission along the well-connected vehicle netwok on a oad segment, until the communication link is boken and the infomation popagation tuns to fey tansmission. To develop the fomulation fo (the expected numbe of hops in an instantaneous tansmission), we conside thee ae numbe of vehicles unning on the oad segment, and label the vehicles fom left to ight with the numbe fom 0 to n as we did befoe. Fig. 6 shows an example. Based on that, we conside as a andom vaiable and exploe its expectation by the events defined below. Fig. 6. Instantaneous tansmission on a segment () Event epesents an instantaneous tansmission stating at the vehicle,. Consideing an instantaneous communication may stat at any individual vehicles evenly, we have. () Event, epesents the c th hop in an instantaneous communication. Fo instance: if the instantaneous tansmission stats at nd vehicle, A epesents tansmission between nd and 3 d vehicle, A epesents tansmission between 3 d and 4 th vehicle, etc. With the pseudo tansmission ange (deived fom section II.D), we calculate, the pobability of a successful infomation tansmission fom any vehicles to its neaest neighbo by Eq. 0. (0) (3) Event epesents hops of successive tansmission, Using the same notation fo the event and its pobability, we have. (4) Event epesents an instantaneous tansmission stats at vehicle and only successively popagates hops. Then, we have. It is noticed that fo a given,. Namely, if an instantaneous tansmission stats fom the i th vehicle, its maximum numbe of successive hops is. Table povides the calculations fo all possible. (5) Event epesents an instantaneous tansmission with only hops.. Table below demonstates the calculations of and as and. Fo example, when and. It indicates that the instantaneous tansmission stats fom vehicle, popagates two hops, and then the connection is boken. Each ow in Table povides the pobabilities that an instantaneous tansmission speads hops given this instantaneous tansmission stats at any vehicle. By summing in each ow, we obtain the geneal fomulation: ( ) (note that the same notation is used fo event and its pobability). With the solution given in Table, can be calculated by Eq. below: n k g g ngn k kg... () n kpk P Pk g k n Clealy, is the key component to calculate. The following study investigates the fomulation fo as well as unde diffeent taffic conditions. ) Fee flow taffic condition Unde fee flow condition, the lage spacing between vehicles guaantees vehicle s movement feedom without concening safety. This implies the independence of the spacing and successful infomation tansmission between any two consecutive vehicles so that, and then can be calculated by Eq.. k k n kp P P g k () n Based on that, it can be poved that the sum of the last column in Table equals, thus the coectness of the pobability distibution in Table is veified. In addition, we obtain the closed-fom fomulation to pedict the expected hops of an instantaneous tansmission in Eq. 3. (3) ) Congested flow Unde congested taffic condition, the spacing between two vehicles is elatively small. The movement of a following vehicle needs to conside the movement of the leading vehicle in font to keep safety, theefoe, the spacing between any two consecutive vehicles is dependent. Accodingly, it bings the difficulty to accuately calculate and the associated in Eq.. As a compomise, this study develops the lowe and uppe bounds of, which futhe lead to lowe and uppe bounds of. Below pesents ou methods. Accoding to Bonfeoni bound [3] and Caen bound [33], the 7

lowe and uppe bounds fo ae given in Eq. 4 and Eq. 5, espectively. k Pk P(A A... A k ) P(A ) P (k ) c c k (4) ( ) (5) By obseving, Eq. 5 povides a new uppe bound fo in Eq. 6 below. Accodingly, we obtain the lowe and uppe bound fo in Eq. 7 and Eq. 8 espectively. ( ) (7) (8) At this point, by combining Eq., Eq., Eq. 6, Eq. 3, Eq. 7, and Eq. 8, we ae eady to calculate the expected time delay of infomation popagation along a oad segment in Case (a). (6) TABLE I. THE PROBABILITY THAT AN INSTANTANEOUS TRANSMISSION PROPAGATES K HOPS GIVEN IT STARTS AT I TH VEHICLE Hops 0 0 0 0 k 0 0 0 0 n- 0 0 0 0 0 n- 0 0 0 0 0 n 0 0 0 0 0 0 E. Extension to Othe Cases This study next demonstates the applicability of the poposed appoach to case (b), case (c) and case (d) in Fig.. ) Case (b): One-way oad segment with taffic and infomation flowing in the opposite diection Case (b) epesents a situation that infomation flow speads in an opposite diection to taffic flow. It is obseved that infomation popagation in case (b) also follows the same patten, altenatively pesenting instantaneous tansmission and fey tansmission. Moe specifically, as infomation speads by instantaneous tansmission, we may ignoe the movement of smat vehicles since it is much slowe than wieless infomation spead. Accodingly, the time delay esulting fom instantaneous tansmission in case (b) can be measued by. Howeve, as a fey tansmission happens in case (b), the vehicles conducting fey tansmission may cay infomation backwad to the infomation popagation diection fo time. Hence, the distance that a piece of infomation being spead fowad in one cycle of the tansmission patten (an instantaneous tansmission followed by a fey tansmission o vice vesa) equals to. Accodingly, the time delay of the infomation popagation along a oad segment unde case (b) will be calculated by Eq. 9. ( ) (9) Note that (i) as, the infomation will neve each to the end of the oad segment since it is always caied back by fey tansmission; (ii) Eq. 9 is a vaiant of Eq. ; all the elements such as and can be measued by the fomulations poposed in pevious sections. Theefoe, ou appoach also woks fo case (b). ) Case (c) and (d): Two-way oad Case (c) and Case (d) ae essentially the same. They both illustate a situation that infomation speads in a same diection to one of the taffic flows (such as in East Bound (EB)) but opposite to the othe way (such as in West Bound (WB)). The time delay fomulation developed fo instantaneous tansmission in Case (a) still woks fo these two cases. This study next povides moe discussions about fey tansmission based on the examples in Fig. 7 and Fig. 8, whee epesents the spacing distance between vehicles on EB diection and vehicles on WB diection; and epesent the spacing distance between vehicles on EB and WB diection espectively. It is obseved that thee ae two possible scenaios fo a fey tansmission. Scenaio (). The example shown in Fig. 7 indicates that a pevious multi-hops instantaneous tansmission stops at 8

vehicle, with the last hop fom vehicle in WB diection occuing. Thus a fey tansmission stats on a fey vehicle (vehicle ) caies infomation fowad (i.e. given S EB >, and S >, vehicle caies infomation in EB diection). This fey tansmission will be stopped by an instantaneous tansmission between the fey vehicle and the othe vehicle in the same way (such as vehicle 4) and implies, o in the othe way (such as vehicle 3) and implies. These two possible fey tansmissions ae denoted as and with pobability and ; they lead to the expected fowad fey tansmission distance and espectively. Scenaio (). The example shown in Fig. 8 indicates that a pevious multi-hops instantaneous tansmission stops at vehicle with the last hop fom vehicle in EB diection. Thus, the fey tansmission stats on a fey vehicle (such as vehicle ) caying infomation backwad (i.e. given S WB >, and S >, vehicle caies infomation in WB diection). This fey may stop at a vehicle behind itself on the same way (such as vehicle 3), but not on the othe way (such as vehicle 4) due to the opposite moving diection. Howeve, as this backwad caying happens, infomation is still possible to move fowad since it is noticed that vehicle will cay infomation and move fowad. This study next pefoms moe elaboate discussions fo this scenaio, which may include two othe situations. (a) If vehicle 3 meets vehicle ( 3 ) befoe it meets vehicle (i.e. 3 ), then infomation is caied fowad by vehicle. The fey tansmission conducted by vehicle is simila to the fowad fey we discussed in Scenaio (). The only diffeence is that the fowad fey hee will cancel the popagation of infomation esulting fom pevious instantaneous infomation tansmission fom vehicle to vehicle. Consideing it is had to measue the infomation popagation distance fom vehicle to vehicle unde this situation, this study ignoes this details and consides it as the fowad fey. Note that the backwad fey conducted by vehicle unde this situation does not have an effect on infomation popagating fowad. (b) If vehicle 3 meets vehicle befoe it meets vehicle, then infomation is caied backwad by vehicle befoe the instantaneous tansmission happens between vehicle and vehicle 3. We denote this backwad fey as happening with pobability. Accodingly, it esults in the expected backwad fey tansmission distance. Clealy, to identify this backwad fey, we need to ecognize the distance between vehicle and vehicle 3 and it is vey difficult to get. To addess this issue, we obseved that if thee ae many othe vehicles between vehicle and vehicle, then vehicle is vey likely elatively fa away, the chance that vehicle meets vehicle 3 ealie than vehicle is low; then we ae sue about the backwad fey when S SB > ; on the othe hand if vehicle is almost adjacent to vehicle, then the aveage distance between vehicle and vehicle 3 can be appoximated by. In this context, we think that the backwad fey happens as and, and. Clealy, ( ), ( ) and can be calculated by the fomulations given in Eq. 6 and Eq. 7. Note that the spacing and elative speed used to calculate ( ) should be measued fo vehicles moving in opposite diections. Infomation EB WB WB Infomation EB S EB > S WB S EB S > Fig. 7. Fey tansmission on vehicle in EB diection on two-way segment S > S WB > Fig. 8. Fey tansmission on vehicle in WB diection on a two-way segment This study next develops the fomulations fo, and based on the examples in Fig. 7 and Fig. 8. It is obseved that the pobability that a fey tansmission stats (the last hop of the pevious instantaneous tansmission stops) on the EB diection o the WB diection depends on the spacing of vehicles on these two diections. Namely, if the vehicles on the EB diections ae spase than on the WB diection, then the chance that a fey tansmission stats fom EB diection is highe than fom the WB diection. With this obsevation, this study estimates the pobability that a fey tansmission happens on the WB o the EB diection by and. Futhemoe, we develop the fomulations fo, and based on the above discussions fo, and. P f p P f f P f ' S EB ' SEB S. PS SEB SWB SEB veb vew P f (30) P f p P f f P f ' S EB ' SEB S. PS SEB SWB SEB veb vew P f (3) P fb pb P fb f P f ' S ' WB SWB S. PS SWB SWB SEB vwb vew P f (3) Whee,,, and epesent the coesponding pobabilities of fowad fey tansmission, and backwad fey tansmission ; epesents the 4 4 3 3 9

pobability of fey tansmission; epesents the elative speed on the lanes in EB diection. epesents the elative speed between the lane on EB diection and the lane on WB diection. Oveall, a piece of infomation may spead on a two-way oad segment though instantaneous and fey tansmissions. By following the ideas to develop the time delay fomulation fo Case (a) and Case (b) (i.e., Eq. and Eq. 9), we develop Eq. 33 to estimate the expected infomation popagation time delay time delay fo Case (c) and Case (d). E y f E y f E yb E T k p p p b v f v f v b (33) L E x p E y p E y p E y f f b b Whee, and ae the aveage vehicle speed in a taffic flow with the same and opposite diection to infomation flow espectively. Note that Eq. 33 implies that the distance that a piece of infomation moving fowad in one cycle including an instantaneous tansmission and a fey tansmission is calculated by the diffeence between fowad tansmission (though instantaneous tansmission and fowad fey) and backwad tansmission (though backwad fey). The expected fey tansmission distance is calculated by the weighted aveage of fowad fey tansmission and backwad fey tansmission. Mathematically, this diffeence is calculated by ( ) ( ). Clealy, Eq. 33 pesents the same undeline logic as Eq. and Eq. 9). So fa, we claim that the poposed appoaches cove all the fou cases in Fig.. To validate the poposed fomulations, this study conducts the numeical expeiments in next section. IV. NUMERICAL EXPERIMENTS This section pesents the numeical expeiments to validate the poposed mathematical fomulations. A. Test-bed and Input Data The field data collected by Next-Geneation Simulation (NGSIM) is used to validate the poposed methodology and fomulations. The data set povides vehicle tajectoy data including the attibutes: vehicle ID, fame ID, total fames, global time, local X, local Y, global X, global Y, vehicle length, vehicle width, vehicle class, vehicle velocity, vehicle acceleation, lane ID, peceding vehicle ID, following vehicle ID, space headway (in same lane) and time headway (also in same lane). Thee test-beds ae selected so that the expeiments cove one-way o two-way oad segment as well as fee flow and congested taffic conditions. Below povides details fo the expeiments which ae setup on a one-way and two-way oad segment. ) One-way taffic flow The validation expeiments fo one-way taffic flow was conducted on a oad segment of US Highway 0 in Los Angeles, CA (see Fig. 9 (a)). The study aea was a one-way segment with 00-feet long, and five lanes on which vehicle moving fom Noth to South thoughout the section. Taffic data was collected duing two 5-minutes peiods on June 5th 005. Given the speed limit of the oad segment is 55 mph, the data set collected duing (7:50 a.m. 8:05 a.m.) epesents fee flow taffic condition with a flow ate equal to 9500 vph and aveage speed equal to 48 mph; the data set collected duing (8:0 a.m. - 8:35 a.m.) epesents an intemediate congested taffic flow with aveage flow ate equal to 7800 vph and aveage speed equal to 5 mph. ) Two-way taffic flow The validation expeiments fo two-way taffic flow wee conducted on two oad segments: Peachtee Steet, Atlanta (GA) (see Fig. 9 (b)) fo fee flow and Lankeshim Blvd, Los Angeles (CA) (see Fig. 9 (c)) fo congested flow. The segment of Peachtee Steet is 650 feet long with five lanes. Taffic data is collected duing (:50 p.m. :00 p.m.) on Novembe 8th 006. Given the speed limit of 55 mph, the data indicates fee flow taffic condition with the aveage speed equal to 50 mph. The segment of Lankeshim Blvd is 600 feet long with six lanes. Taffic data is collected duing (8:50 a.m. 9:00 a.m.) on June 6th 005. Given speed limit of 55 mph, the collected data indicates a congested flow with aveage speed equal to 3 mph. (a) US Highway 0 (b) Peachtee St (c) Lankeshim Blvd B. Expeiment Design Fig. 9. Test-beds This section designs the expeiments to evaluate the pefomance of the poposed mathematical estimation fo the time delay that a piece of infomation popagates though a oad segment. Consideing the expected numbe of hops in an instantaneous tansmission ( ), is one of the key components in those mathematical estimation fomulations, we also check the accuacy of its fomulation. The oveall ideas of the expeiment ae pesented fist. Based on the field data collected fom the selected test-beds, this study fist measues field infomation popagation time delay as well as the expected numbe of hops in an instantaneous tansmission, and then we calculate the coesponding mathematical estimations fo the time delay and the expected numbe of hops, espectively. Afte that, we compae to and ( to, and demonstate the accuacy by oot-mean-squae eo (RMSE) and elative eo ( ). The field infomation popagation time delay, is defined as the time inteval that a piece of infomation popagates though a oad segment, given a successful tansmission between any evey two vehicles is identified by SINR condition in Eq.. is consideed as the gound tuth 0

in this study. Mathematical estimation, is calculated by the poposed fomulations, given the needed distibution and paametes ae obtained fom the field data. These expeiments select Log-nomal distibution to epesent the space headway distibution vehicles on one-way o two-way oad segment afte it was calibated by the field data. But, the applicability of the poposed appoaches does not depend on the distibution selection. Along the pocess to calculated and, and ae also calculated though field counts and poposed mathematical estimation fomulations, espectively. epesents the gound tuth. Next, we povide the expeimental pocedue. Fo evey ( 6 o 5) seconds, the expeiment stats to tack a piece of infomation just launching on the stat of the oad segment until it eaches to the end of the oad segment. Accodingly, ( and ) and ( and ) ae checked vey seconds. Accoding to the field dataset, 5 (o ) pieces of infomation in total ae tacked fo one-way (o two-way) testbeds. The detailed expeiments steps to tack infomation popagation ae given below. Step : at time, a piece of infomation launches on the stat of the oad segment. Step : tack instantaneous (o fey) infomation popagation until it is boken; ecod. Step 3: check if the infomation eaches to the end of the oad segment.. Yes, ecod cuent time ; T g = -, = aveage ( ), calculated and ; go to Step 4. No, change tansmission scenaio to fey (instantaneous), go back to step Step 4: if all data examined, stop, othewise, = +, go to Step. The accuacy of is evaluated by Root-Mean-Squae- Eo (RMSE) and elative eo (e) given in Eq. 34 and Eq. 35 below. RMSE demonstates the aveage diffeence between and ove all tacked infomation. Relative eo (e) measues the pecentage of the eo between and to, thus gives us the idea how significant the eo is. ( ) (34) (35) Whee, N epesents numbe of scenaios, epesents the field time delay in a scenaio, epesents coesponding mathematical estimation in fee flow (the aveage of the uppe and lowe bounds in congested flow). A negative value indicates an undeestimation ove all expeimental scenaios and a positive value means the othe way aound. The same evaluation wok will be conducted fo C. Expeiment Results and Insights ) One-way segment This section pesents ou numeical expeiment esults and the insights we obtained fo one-way oad segment. The esults given in Fig. 0 indicate that is well bounded by ou mathematical lowe and uppe bounds (calculated by Eq. 7 and Eq. 8) in congested flow, and accuately estimated by the mathematical model (Eq. 3) in fee flow. Moe exactly, RMSE values fo unde congested and fee flow ae.64 and.37 espectively, and the elative eos ae 3.7% and 4.% espectively. In addition, we see that on aveage 8.68 and 9.04 in the tested congested flow, and 6.36 and 6.58 in the tested fee flow; they ae vey close. Thus, ou mathematical fomulations povide eliable estimations fo unde both congested and fee flow on oneway oad segments. k g 8 8 & k M 9 4 RMSE =.64 & e = 3.7% (a) Congested flow k g & k M 58 RMSE =.37 & e = 4.% (b) Fee flow Fig. 0. Compaison of numbe of hops between field and mathematical estimation on one-way segment The esults fo evaluating the mathematical fomulation (Eq.) to estimate the infomation popagation time delay on one-way oad segment ae given in Fig.. Fig. (a) shows that field time delay in congested flow is well bounded by ou mathematical bounds. The elative eo -4.79% and RMSE equal to 5 seconds; and Fig. (b) also indicates that the mathematical fomulation can estimate the field time delay vey well in the fee flow case; the elative eo is -4.34% and RMSE equals to 5.65 seconds. The negative sign of indicates that on the aveage, the poposed mathematical fomulation undeestimates the time delay. In addition, the esults show that 4.07 seconds and 39.84 seconds in the tested congested flow; 46.40 seconds and 44.03 seconds. Clealy, the aveage of the field time delay is vey close to the aveage of the estimated time delay. Moeove, in both cases, the elative eo is aound 4%, so ou mathematical fomulations wok well.

T g 4 7sec & T M 9 84sec RMSE = 5sec & e = -4.79% k g 8 & k M 49 RMSE =.4 & e = 4.5% (a) Congested flow T g 4 4 sec & T M 44 sec RMSE = 5.65sec & e = -4.34% (a) Congested flow k g 5 84 & k M RMSE =.6 & e = 4.00% (b) Fee flow Fig.. Compaison of time delay between field and mathematical estimation on one-way segment ) Two-way segment The pefomances of the poposed appoaches (Eq.33 and all the elated equations) on two-way oad ae also evaluated by the same way that we did fo the one-way segment. The esults in Fig. and Fig. 3 show that the field vales (time delay o the expected numbe of hops) unde congested flow ae well bounded by the mathematical bounds, and they ae accuately estimated by the mathematical fomulations fo fee flow. Moe exactly, the elative eo fo unde fee flow (o congested flow) is 4% (o 4.5%) with RMSE equal to.6 (o.4), implying the aveage diffeence between ou estimation and the field value fo is about. In addition, the esults show that on aveage 6.8 and 6.49 in the tested congested flow, and 5.84 and 6.03 in fee flow. Clealy, they ae vey close. The elative eo fo unde fee flow (o congested flow) is 4.36% (o 4.5%) with RMSE equal to 3. seconds (o 3.4 seconds). Moe, the esults show that on aveage 5 4 seconds and 6.0 seconds in the tested congested flow, and 7.8 seconds and 8.48 seconds in the tested fee flow. Again, they ae vey close. The expeimental esults indicate small estimation eos. Thus, we claim that ou mathematical estimation fomulations fo two-way oad also pefom well. Oveall, the numeical expeiments indicate that the poposed appoaches pefom petty well unde both fee flow and congested taffic flow on a one-way segment and two-way segment. In moe details, ou mathematical estimations pefom a bit bette fo fee flow taffic condition than fo congested taffic flow condition on a one-way oad segment and two-way oad segment, due to pobabilistic bounds athe than closed fom fomulations ae developed to estimate in congested flow. (b) Fee flow Fig.. Compaison of numbe of hops between field and mathematical estimation on two-way segment T g 5 4sec & T M sec RMSE = 3.4sec & e = 4.5% (a) Congested flow T g 7 8sec & T M 8 48sec RMSE = 3.sec & e = 4.36% (b) Fee flow Fig. 3. Compaison of time delay between field and mathematical estimation on two-way segment