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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1

A Stochastic Model for Chain Collisions of VehiclesEquipped With Vehicular Communications

Carolina Garcia-Costa, Esteban Egea-Lopez, Juan Bautista Tomas-Gabarron,Joan Garcia-Haro, Member, IEEE, and Zygmunt J. Haas, Fellow, IEEE

Abstract—Improvement of traffic safety by cooperative vehicu-lar applications is one of the most promising benefits of vehicularad hoc networks (VANETs). However, to properly develop suchapplications, the influence of different driving parameters on theevent of vehicle collision must be assessed at an early design stage.In this paper, we derive a stochastic model for the number ofaccidents in a platoon of vehicles equipped with a warning collisionnotification system, which is able to inform all the vehicles aboutan emergency event. In fact, the assumption of communicationsbeing used is key to simplify the derivation of a stochastic model.The model enables the computation of the average number ofcollisions that occur in the platoon, the probabilities of the dif-ferent ways in which the collisions may take place, as well as otherstatistics of interest. Although an exponential distribution has beenused for the traffic density, it is also valid for different probabilitydistributions for traffic densities, as well as for other significantparameters of the model. Moreover, the actual communicationsystem employed is independent of the model since it is abstractedby a message delay variable, which allows it to be used to evaluatedifferent communication technologies. We validate the proposedmodel with Monte Carlo simulations. With this model, one canquickly evaluate numerically the influence of different model para-meters (vehicle density, velocities, decelerations, and delays) on thecollision process and draw conclusions that shed relevant guide-lines for the design of vehicular communication systems, as wellas chain collision avoidance applications. Illustrative examples ofapplication are provided, although a systematic characterizationand evaluation of different scenarios is left as future work.

Index Terms—Chain collision, collision avoidance, road acci-dents, stochastic model, vehicle platoon, vehicle safety, vehicularcommunications.

I. INTRODUCTION

INTERVEHICLE communications based on wireless tech-nologies pave the way for novel applications in traffic safety,

driver assistance, traffic control, and other advanced services

Manuscript received March 28, 2011; revised July 29, 2011; ac-cepted September 30, 2011. This work was supported in part by theMICINN/FEDER project under Grant TEC2010-21405-C02-02/TCM (CALM)and in part by Fundación Seneca RM under Grants 00002/CS/08 FORMAand 04549/GERM/06. The work of J. Garcia-Haro was supported by GrantPR2009-0337. The work of E. Egea was supported by the Universidad Politéc-nica de Cartagena under Grant PMPDI-UPCT-2011. The work of J. Tomas-Gabarron was supported by Grant AP2008-02244. The work of C. Garcia-Costawas supported by Grant 12347/FPI/09. The Associate Editor for this paperwas Z. Li.

C. Garcia-Costa, E. Egea-Lopez, J. B. Tomas-Gabarron, and J. Garcia-Haro are with the Department of Information and Communication Technolo-gies, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain (e-mail:[email protected]).

Z. J. Haas is with the School of Electrical and Computer Engineering,Wireless Networks Laboratory, Cornell University, Ithaca, NY 14853 USA.

Digital Object Identifier 10.1109/TITS.2011.2171336

that will make future intelligent transportation systems. Theadvances in technology and standardization, particularly withthe allocation of dedicated bandwidth to vehicular communica-tions, from the mid 1990s have increased research and devel-opment efforts on vehicular ad hoc networks (VANETs) fromthe networking and mobile communications community [1],although early research on “automated highways systems” goesback to the 1960s and later [2]. Improvement of traffic safety bycooperative vehicular applications is one of the most promisingtechnical and social benefits of VANET [3], [4]. However, todesign and implement such applications, a deep understandingof the vehicle collision processes is needed. The influence ofdifferent driving parameters on the collision event must beassessed at an early design stage to develop applications thatcan timely adapt vehicle dynamics to avoid or at least mitigatethe danger [5].

Very detailed models of vehicle motion and collision dy-namics can be found [6], [7], but the equations are completelydeterministic, whereas in reality, randomness is always presentas an effect of human behavior or noisy operation introduced bysensors or other reasons. To account for it, the usual methodol-ogy is to evaluate deterministic models by applying a MonteCarlo or stochastic analysis over an extensive range of theirparameters [2], [6], [8]. However, to the authors’ knowledge,little effort has been devoted to develop models that are sto-chastic in nature and, in particular, for rear-end chain collisionsof vehicles. Some reasons behind it are the difficulties ofevaluating all the possible ways in which a collision may occurand the complexity posed by the fact that the motion equationsfor those possibilities involve a dependence on the parametersof the preceding vehicles. That is, the driver reacts to variationsin the driving conditions of the preceding vehicle, as in acar-following approach [9], [10]. However, if vehicles use acommunication system that is able to inform all the vehiclesabout an emergency event, those difficulties can be overcome.The key is that, in that case, it can be assumed that driversreact as soon as they receive a warning message, and they startbraking independent of the preceding vehicle behavior. This isin fact the goal of warning collision systems or electronic brakewarning applications. This assumption removes the dependenceof motion equations on the preceding vehicles and facilitates thedevelopment of a stochastic model.

In this paper, we take this approach. Our goal is to describeand analyze the risk of colliding for a set of moving vehiclesforming a platoon (or chain) and equipped with a warningcollision system when the leading vehicle stops suddenly. Tothis aim, we derive a stochastic model for the number of

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2 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS

accidents that occurs in this scenario. The model allows us tocompute the average number of collisions that occur in theplatoon, among other statistics of interest. The scenario underconsideration is basically a platoon of vehicles moving along aunidimensional road in the same direction in which the leadingvehicle suddenly comes to a complete stop. To consider a worstcase scenario, we add two strong assumptions. First, the leadingvehicle stops instantly (it may also be considered that a fixedobstacle lays on the road). Second, vehicles will not be able tochange their direction of movement to cope with the unexpectedincident. Our model is stochastic because all its parametersmay be described by random variables. We derive the equationsassuming always a random intervehicle spacing, in particularfor an exponentially distributed spacing, although the model isvalid for other distributions. When additional parameters are as-sumed random, the solutions have been computed numerically.Additionally, it should be observed that the model is indepen-dent of the communication technology, since the operation ofthe communication system is abstracted by the use of a messagereception/notification delay variable. Finally, the probabilitiesfor all the ways the collision may take place are also derived,which can be further used to evaluate the severity of accidentsin higher detail, for instance, by assigning different severityweights to different types of collision. A deeper discussion onthis topic, however, is out of the scope of this paper.

The main practical utility of this model lays in its abilityto quickly evaluate numerically the influence of different pa-rameters on the collision process without the need to resortto complex simulations in a first stage. Such an evaluationprovides relevant guidelines for the design of vehicular com-munication systems as well as chain collision avoidance (CCA)applications. As an example, it can quickly reveal for whichrange and distributions of the parameters the communicationdelay has a serious impact on the metric of interest, which canbe the average number of accidents, but also the probability ofcollision of every vehicle in the chain. Since it turns out that insome scenarios a low delay is not relevant for the outcome, acommunication system could trade it off for additional reliabil-ity mechanisms. Moreover, in this paper, we set either constantor purely random parameters, but the model can be used witharbitrary parameters to evaluate more specific applications. Forinstance, to evaluate multihop communications, we can set upa vector of delays with progressively increasing values. Weprovide examples of use in Section V, but in any case, a carefulcharacterization of the model parameters for the scenarios andapplications is a necessary previous step.

Therefore, in summary, in this paper, our goal is to providethe derivation and validation of the model and show its utilitywith a few illustrative examples. A proper characterizationand evaluation of different scenarios and metrics is left asfuture work.

The remainder of this paper is organized as follows. InSection II, we briefly review the related work. The derivationand validation of the model are provided in Sections III and IV.Section V illustrates how to use the model to evaluate the influ-ence of different parameters on the collision process and to ob-tain qualitative conclusions relevant to the design of vehicularcommunication, as well as CCA applications. Conclusions and

future work are remarked in Section VI, whereas the necessaryauxiliary material is relegated to Appendices A and B.

II. RELATED WORK

Our model assumes that there is a communication systembetween vehicles that allows them to receive warning messagesto start braking in the event of a sudden stop of the leadingcar. However, such a system is abstracted in the model andcharacterized by the use of a message reception/notificationdelay variable. Therefore, our model is actually independentof it and can be applied to any communication system whoseoperation can be abstracted by an appropriate delay variable.For instance, current VANET standards specify the use of IEEE802.11p based on contention [carrier sense multiple access(CSMA)] medium access control (MAC) [1]. Such a sharedchannel MAC technique can be abstracted in our model by adelay random variable with an appropriate probability distrib-ution [11]. Further details on current VANET communicationtechnologies can be found in [1].

Regarding collision models for chains of vehicles, two dif-ferent groups of studies can be found: 1) statistical models ofthe frequency of accident occurrence and their circumstances[12], [13]; and 2) models of the collision process itself based onphysical parameters [2], [6], [8]. This paper falls on the lattercategory and additionally assumes that an automated warningsystem is in place. In most of these studies, deterministicequations for the occurrence of collisions are derived, and toaccount for random variability, stochastic analysis or MonteCarlo simulations over a wide range of model parameters areperformed afterward to obtain an estimate of the collisionprobability or other metrics of interest. Our approach is dif-ferent, and the model shown here is directly stochastic andassumes that at least the intervehicle distance is a randomvariable, which is, in fact, a realistic assumption, as shownin [14]. We also perform Monte Carlo simulations, but unlikethe previously mentioned papers, we use them to validate ourmodel rather than to obtain metrics of interest. Looking intothese works in particular, in an early study, Glimm and Fenton[2] define an accident cost function to evaluate the severity ofvehicle collisions. The collision model used is derived for anautomatically controlled1 platoon of vehicles that advance atconstant speed with a constant intervehicle spacing. A morerecent work [8] provides a similar collision model for a four-car platoon of vehicles, assuming that just one of the vehiclesis equipped with an autonomous intelligent cruise control. Inboth cases, the collision model defines how vehicles decelerateto obtain a deterministic equation for the collision. Afterwards,the evaluation is done by randomizing some parameters ofthe model and running a Monte Carlo simulation. In [10],the authors derive conditions necessary for a chain collision,starting from a car-following model. However, they assumethat all the vehicles are driving with equal initial speeds andintervehicle distances.

1Let us note that early research, which goes back to the 1960s, consideredthe hypothesis of achieving “automated highway systems,” where most of thedriving tasks were automatically controlled.


GARCIA-COSTA et al.: MODEL FOR CHAIN COLLISIONS OF VEHICLES WITH VEHICULAR COMMUNICATIONS 3

Fig. 1. Scenario under consideration.

Interestingly, the vehicle collision model proposed in thispaper is more general; it explicitly accounts for random interve-hicle spacing and can be used to assign arbitrary variables, evenrandom ones, to the kinematic parameters of each vehicle, aswell as the warning message communication delay. Moreover,there are additional applications of our model; for instance, itcan readily be used to evaluate the severity of collisions, as in[2]: since we compute the probability of collisions occurringin several manners, we could assign a severity weight to eachpossibility, that is, we may assign more severity to a collisionwhen both vehicles are in motion than to other cases, forexample, although this topic is not treated in the present paper.On the other hand, some of the results in [2] are similar toours; for instance, the sensitivity shown to the decrease indeceleration capabilities of the subsequent vehicles. In all thecases as well as in this paper, only rear-end collisions areconsidered. Head-on collisions are evaluated in [6] based ona very detailed analytical model of the vehicle.

Finally, in this paper, we provide examples about the kindof results that can be drawn from the proposed model, whichare useful for the design of CCA applications. A review onintelligent collision avoidance algorithms can be found in [5].In particular, the influence of delay notification on differentscenarios is useful to set appropriate time horizons for CCAsystems based on trajectory prediction [3].

III. COLLISION MODEL

We consider a platoon (or chain) of N vehicles followinga leading vehicle (see Fig. 1), where each vehicle Ci, i ∈1, . . . , N , moves at constant velocity Vi. The leading vehicleC0 collides with an obstacle on the road, at time t0 = 0, and im-mediately, it sends a warning message to the following vehicles.The rest of the vehicles start to brake at constant deceleration2

ai when they are aware of the risk of collision, that is, after atime lapse δi. Let us remark here that this time lapse is mainlydetermined by the reception of the warning message generatedby the communication system, and therefore, the reaction of thedriver is independent of the movement state of the precedingvehicle. That is, a warned driver will decelerate, even if thepreceding car has not started to decelerate. In a classical car-following approach, on the contrary, the deceleration would bea consequence of a change in the speed or intervehicle spacingof the preceding vehicle. For the sake of simplicity, we assumethat every vehicle has the same length L, and its position isgiven by the x coordinate of its front bumper. The leadingvehicle stops at coordinate x0 = 0, and the initial intervehiclespacing is si = xi − (xi−1 + L). We assume that at least theintervehicle spacing is a random variable. Its probability distri-bution and the variability of the other variables ai, Vi, and δi

2To simplify the notation, in the remainder of the paper, we consider ai adeceleration and, therefore, assign it a positive sign.

Fig. 2. Probability tree diagram that defines the model. Si,j represents thestate with i collided vehicles and j successfully stopped vehicles.

are discussed in Section IV. Vehicles cannot change lanes orperform evasive maneuvers.

With this model, the final outcome of a vehicle depends onthe outcome of the preceding vehicles. Therefore, the collisionmodel is based on the construction of the probability treedepicted in Fig. 2. We consider an initial state in which novehicle has collided. Once the danger of collision has beendetected, the first vehicle in the chain C1 (immediately afterthe leading one) may collide or stop successfully. From both ofthese states, two possible cases spring as well, that is, eitherthe following vehicle in the chain C2 may collide or stopsuccessfully, and so on, until the last vehicle in the chain,denoted by CN . At the last level of the probability tree, thereare N + 1 possible outcomes (final outcomes) that representthe number of collided vehicles, that is, from 0 to N possiblecollisions. Observe that Si,j represents the state with i collidedvehicles and j successfully stopped vehicles.

The transition probability between the nodes of the tree isthe probability of collision of the corresponding vehicle in thechain pi (or its complementary). These probabilities are crucialto the model and will be calculated recursively, as described inthe next section. Let us note how every path in the tree from theroot to the leaves leads to a possible outcome involving everyvehicle in the chain. The probability of a particular path resultsfrom the product of the transition probabilities that belong to thepath. Since there are multiple paths that may lead to the samefinal outcome (a particular leaf node in the tree), the probabilityof that outcome will be the sum of the resulting probabilities ofevery possible path reaching it.

To compute the probabilities of the final outcomes, wecan construct a Markov chain whose state diagram isbased on the previously discussed probability tree. It isa homogeneous Markov chain with ((N + 1)(N + 2)/2)states, (S0,0, S1,0, S0,1, . . . , SN,0, SN−1,1, . . . , S1,N−1, S0,N ).The transition matrix P of the resulting Markov chain is asquare matrix of dimension ((N + 1)(N + 2)/2), which is asparse matrix, since from each state, it is only possible to moveto two of the other states. For the sake of clarity, a brief examplewith two vehicles is illustrated in Fig. 3.

Then, we need to compute the probabilities of going from theinitial state to each of the N + 1 final states in N steps, which



Fig. 3. Probability tree and transition matrix for a chain with N = 2 vehicles.

Fig. 4. Parameters of the kinematic model used to compute the vehiclecollision probabilities.

are given by PN . Therefore, the final outcome probabilities arethe last N + 1 entries of the first row of the matrix PN .

Let Πi be the probability of reaching the final outcomewith i collided vehicles, that is, state Si,N−i. Then, Πi =PN (1, ((N + 1)(N + 2)/2) − i). We obtain the average of thetotal number of accidents in the chain using the weighted sum

Nacc =N∑

i=0

i · Πi. (1)

IV. COMPUTATION OF THE VEHICLE

COLLISION PROBABILITIES

Computing the collision probabilities is the main problemin our model. In this section, we start from a deterministickinematic model and compute the collision probabilities whendifferent parameters of the kinematic model are consideredvariables. The results are validated by Monte Carlo simulations.Hence, we start from a basic kinematic collision model pro-vided by [15] that can be summarized as follows.

Let li represent the total distance traveled by vehicle Ci sincethe emergency event occurs at time instant t0 = 0 until thevehicle completely stops or collides with Ci−1. Let δi be thetime lapse that goes between the detection of the emergencyevent until vehicle Ci actually begins to slow down. We call δi

the notification delay, which models the delay between the timeinstant t0 = 0 and the instant the driver of vehicle Ci is aware ofit and starts to brake. These parameters are depicted in Fig. 4.The notification delay plays an important role if we considera communication system in operation between the vehicles. Inthis case, we can assume that the driver starts to brake when itreceives a warning message; therefore, if the emergency event

Fig. 5. Distance li traveled by a vehicle when there is a collision (b) that isshorter than the distance needed by it to stop successfully (a) ds,i. (a) VehicleCi is able to stop successfully, and then, li = ds,i. (b) Vehicle Ci collideswith Ci−1. In this case, the actual distance covered by Ci up to the collisionis shorter than ds,i, as given by (2). Now, it is li = si + li−1 and depends onthe distance covered by Ci−1.

occurs at t0 = 0, the warning message is received at t = δi bythe vehicle Ci. However, we assume a more realistic case inwhich there is also a reaction time before the driver actuallystarts to brake. Therefore, δi = Tm,i + Tr,i, where Tm,i is themessage reception delay, and Tr,i is the driver reaction time.

Considering a constant deceleration ai, the distance neededby vehicle Ci to completely stop if it does not collide isgiven by

ds,i =V 2

i

2ai+ Viδi. (2)

However, when a collision occurs, the actual distance traveledby the car dc,i is no longer given by (2), but one has to considerthe way the collision has occurred. For example, if a vehiclecrashes, its actual distance to stop is obviously shorter than ds,i,as illustrated in Fig. 5, as well as different when both vehiclesare still in motion when the crash occurs.

Let us remark at this point that (2) implies that a communi-cation system is in place and all vehicles start to brake whenthey receive the message, independently of the behavior of thepreceding vehicles. Otherwise, drivers would start to brake onlywhen they sensed the braking of its nearest forward neighboras in a car-following approach [9], [10]; therefore, (2) wouldbecome a function of the parameters of the preceding vehicle,that is, ds,i = f(Vi, Vi−1, ai, ai−1, δi, δi−1), and the problemwould become more complex.

In all the cases, the probability of collision of vehicle Ci

depends on the relationship between its distance to stop ds,i,the total distance traveled by the preceding vehicle li−1, and theinitial intervehicle space si. That is, when ds,i < li−1 + si, thevehicle is able to stop without colliding.

At this point, we also assume another simplification: if twovehicles collide, we consider that they instantly stop at thepoint of collision. This way, we keep on assuming a worst caseevaluation. There are more realistic approaches, for instance,to take into account the conservation of the linear moments tocompute the displacement due to the crash [7].

As can be seen from the previous equation, the number ofcollisions depends on the vector of velocities Vi, decelerationsai, notification delays δi, and intervehicle distances si, whichwe refer to as kinematic parameters. When all the parameters



are given, the model is completely deterministic. However, weare interested in a more realistic case involving the randomvariability of the parameters. To study the influence of thedifferent parameters on collisions, we introduce variability ondifferent model parameters as follows: For all the cases, weconsider that si is an exponentially distributed random variablewith parameter λ. This parameter represents the density ofvehicles on the road, which is defined as the average number ofvehicles per meter. Let us remark that si can adopt a differentdistribution, and the following model is still valid. The reasonfor this is that since si is the intervehicle spacing when theemergency event occurs, we can consider it independent of therest of parameters of the model, which means that the followingequations would be essentially the same, but substituting theexponential probability density function by the correspondingnew one. We have selected an exponential distribution be-cause it simplifies the computations, and it has been shownto describe well the intervehicle spacing when traffic densitiesare small [14], whereas high traffic densities show log-normaldistributions [14].

Once we have described our collision model, we next derivea basic model for the vehicle collision probabilities, in which allthe parameters are constant except for the intervehicle distance.Then, we extend the model by considering variable the restof the kinematic parameters. This way, we can evaluate theeffects of the different parameters on the vehicle collisionmodel.

A. Basic Model

Our first step is to evaluate the basic model, consideringall the parameters constant, except for si, which is assumedexponentially distributed. If a vehicle is able to stop withoutcolliding, and the kinematic parameters are constant, it alwaystravels at the same distance ds. However, if there is a collision,a vehicle only travels the initial intervehicle distance plus thedistance traveled by the preceding vehicle until it collides.Therefore, we have to compute the collision probability con-ditioned on the distance traveled by the previous vehicle. In thefollowing sections, we first compute this probability exactly,and then we provide an approximation that allows us to sim-plify the computations when additional variable parameters areconsidered in the model.

1) Case 1. Exact Computation of Collision ProbabilitiesWith Constant Kinematic Parameters: In this case, we computethe collision probability exactly. For the sake of clarity, ourassumptions are summarized as follows.

1) All vehicles move at the same constant velocity V .2) All vehicles begin to slow down at the same constant

deceleration a.3) The delay δ is the same for all drivers. It implies that

all the drivers receive the warning message at the sameinstant.

Since Vi, δi, and ai are constants, from (2), we obtain

ds =V 2

2a+ V δ. (3)

For 1 ≤ i ≤ N , the collision probability will be computed asfollows:

pi =P (ds ≥ li−1 + si)=P (li−1 + si ≤ ds | li−1 ≤ ds)P (li−1 ≤ ds)

+ P (li−1 + si ≤ ds | li−1 > ds)P (li−1 > ds) (4)

where li−1 is a random variable that represents the distancetraveled by the preceding vehicle (assuming that l0 = 0, sincevehicle C0 stops instantly at x0 = 0).

In this simple case, if vehicle Ci−1 does not collide, thenneither does vehicle Ci, because the velocity, deceleration,and reaction time are the same for both of them. Moreover,if vehicle Ci−1 collides, it means that all of the precedingvehicles have collided. From these observations, we can con-clude that li−1 = s1 + s2 + · · · + si−1 ∼ Erlang(i − 1, λ),and P (li−1 + si ≤ ds | li−1 > ds) = 0.

Now, we need to compute pi = P (li−1 + si ≤ ds | li−1 ≤ds)P (li−1 ≤ ds).

The joint probability density function of X = li−1 + si andY = li−1 is

g(x, y) =λ2(λy)i−2e−λx

(i − 2)!, for 0 ≤ y ≤ x. (5)

Therefore, the joint cumulative distribution function is

G(x, y) =

y∫0

t∫0

λ2(λs)i−2e−λt

(i − 2)!ds dt

+

x∫y

y∫0

λ2(λs)i−2e−λt

(i − 2)!ds dt

=γ(i, λy)(i − 1)!

+(λy)i−1

(i − 1)!(e−λy − e−λx), for 0 ≤ y ≤ x

(6)

where γ is the incomplete gamma function, which is defined asγ(a, x) =

∫ x

0 ta−1e−tdt.Finally, for 1 < i ≤ N

pi =P (li−1+si ≤ ds | li−1≤ds)P (li−1 ≤ ds)

=G(ds, ds)Fy(ds)

·Fy(ds)=G(ds, ds)

=γ(i, λds)(i − 1)!

+(λds)i−1

(i − 1)!(e−λds−e−λds)=

γ(i, λds)(i − 1)!

(7)

holds, where F is the cumulative distribution function of theexponential distribution exp(λ), and λ is the vehicle density (invehicles per meter).

At this point, if the metric of interest is the average numberof accidents, then the procedure to obtain it is as follows. Oncewe have computed the collision probability for each vehicle,we have to construct the matrix P described in Section III.The next step is to calculate the final outcome probabilities Πi,and finally, the average number of accidents can be obtainedthrough (1).



As can be seen, in this case, it is relatively easy to computethe collision probability conditioned on the distance traveledby the preceding vehicle li−1. However, in the following cases,it becomes increasingly difficult. In addition, it can be seenthat the collision probability basically depends on the differ-ence ds,i − li−1 of any two cars being greater than the initialintervehicle distance si. From this observation, and to simplifythe following computations, in the next section, we compute thecollision probability using the average distance traveled by thepreceding vehicle and compare it with the results of this section.

2) Case 2. Approximate Computation of Collision Proba-bilities With Constant Kinematic Parameters: As previouslydiscussed, in this section, we compute an approximation tothe collision probability for the basic model, where we use theaverage distance traveled by the preceding vehicle and compareit with the exact computation of Case 1. For the sake of clarity,our assumptions are summarized as follows.

1) All vehicles move at the same constant velocity V .2) All vehicles begin to slow down at the same constant

deceleration a at the same time (the delay δ is the samefor all drivers).

3) We use the average distance traveled by the precedingvehicle to calculate the collision probabilities.

As in Case 1, the distance traveled by a vehicle until itcompletely stops if it does not collide is given by (3).

For 1 ≤ i ≤ N , vehicle Ci will collide with Ci−1 if and onlyif the distance needed by Ci to stop is greater than the distancebetween them plus the average distance traveled by Ci−1, li−1,so the collision probability of Ci is

pi = P (ds ≥ li−1 + si) = F (ds − li−1). (8)

The average distance traveled by a vehicle li must be com-puted recursively, starting from l0 = 0. For 1 ≤ i ≤ N , theaverage distance traveled by vehicle Ci is li = ds(1 − pi) +dc,ipi, where dc,i is the average distance traveled by the vehiclein case of collision

dc,i =1pi

ds−li−1∫0

(li−1 + x)λe−λxdx

=1pi

(li−1 +

1λ−

(ds +

1λ

)e−λ(ds−li−1)

). (9)

Then, the equation for li is

li ={

ds(1 − pi) + dc,ipi, pi > 0ds, pi = 0.

(10)

Now, like in Case 1, we have to construct the matrix P andcalculate the average number of accidents through (1).

3) Validation and Discussion: Fig. 6 shows the results ofcomputing the basic model described in the previous sections.The number of vehicles in the chain is N = 20, and the restof the parameters have been fixed at a = 8 m/s2, which isthe maximum deceleration of what is considered a normalvehicle [16], V = 33 m/s, and δ = Tm,i + Tr,i = 0.1 + 0.9 s.In this case, Tm,i = 0.1 s is the maximum delay for warningmessages that vehicular communication standards specify [17],whereas Tr,i = 0.9 s is an average driver reaction time [18].

Fig. 6. Average percentage of accidents versus average intervehicle distances = (1/λ − L) m for basic model, with exact solution (Case 1), approximatesolution (Case 2), and Monte Carlo simulation with 99.5% confidence intervals.

Fig. 6 illustrates the curves for the exact and approximate basicmodels. In addition, a Monte Carlo simulation of the systemhas been also conducted to validate our model. All the MonteCarlo simulations in this paper have been performed with tenreplications per simulation point, and the results are shown with99.5% confidence intervals. As can be seen using the averagedistance traveled by the preceding vehicle, li−1, computedin Case 2, provides an excellent approximation to the exactcollision probability, since the mean square error between theresults of both cases is less than 0.5%. Moreover, the simulationresults confirm that the model is correct enough, since the meansquare error between the results of Case 2 and the Monte Carlosimulation does not exceed 2%.

B. Influence of Variability on Deceleration, Velocity, andNotification Delay

In this section, the basic model is extended by consideringnotification delays δi, velocities Vi, and decelerations ai as vari-ables. In most of the cases, they should be considered randomvariables with their appropriate probability density functions tomodel some particular effect. At this point, we do not assumeany particular probability distribution for them. A discussion onthis matter is provided later in Section IV-B1.

When deriving a model where all the involved parametersvary simultaneously, several problems arise. Our approach hasbeen to derive a first model considering constant decelerationsai = a, and then another one considering constant notificationdelays δi = δ. Later in this section, this approach will bediscussed and justified in detail.

Therefore, let us first consider constant decelerations andvariable velocities and notification delays. In this case, thedistance needed to stop is not constant and equal for eachvehicle anymore but given by

ds,i =V 2

i

2a+ Viδi. (11)

As in Case 1, for 1 ≤ i ≤ N , the collision probability isgiven by

pi = F (ds,i − li−1). (12)



Again, the average distance traveled and the collision prob-abilities must be recursively computed. However, in this case,vehicle collisions may occur in four different ways: 1) VehiclesCi and Ci−1 have not started to brake; 2) only one of them isbraking; 3) both of them are braking; or 4) vehicle Ci−1 hasstopped. Each one of these possibilities results in a differentdistance to stop, i.e., dc1,i, dc2,i, dc3,i, and dc4,i, which must beweighted by its probability of occurrence and added to get theaverage distance traveled li as

li =ds,i(1−pi)+ dc1,iqc1,i+ dc2,iqc2,i+ dc3,iqc3,i+ dc4,iqc4,i

(13)

where

dc1,i =1

qc1,i

sup1∫inf1

Vitc1,i(x)λe−λxdx, Vi >Vi−1 (14)

dc2,i =

1qc2,i

sup2∫inf2

(Vi−1tc2,i(x)+x) λe−λxdx, δi <δi−1

1qc2,i

sup2∫inf2

Vitc2,i(x)λe−λxdx, δi >δi−1

(15)

dc3,i =1

qc3,i

sup3∫inf3

(Vitc3,i(x) − a

2(tc3,i−δi)2

)λe−λxdx

(Vi − Vi−1)+a(δi−δi−1) �=0 (16)

dc4,i =1

qc4,i

sup4∫inf4

(li−1+x)λe−λxdx (17)

qcj ,i =P (infj ≤si≤supj)=F (supj)−F (infj) (18)

for j = 1, . . . , 4.The functions tc1,i(x), tc2,i(x), and tc3,i(x) represent the

time instants at which the collisions (1)–(3) have occurred,where x is the distance between Ci and Ci−1. The derivationof these time functions as well as equations (14)–(17) and theappropriate values for the integration limits infj and supj forj = 1, . . . , 4 are provided in Appendix A. A discussion aboutthe circumstances that cause the different ways of colliding isoffered there as well.

At this point, we can justify our previously discussed ap-proach: if all the parameters are assumed variable, then thenumber of possible ways in which collisions may occur in-creases remarkably, and all of them have to be taken intoaccount for the computation of the average distance traveledby a vehicle, as dc1,i, dc2,i, dc3,i, and dc4,i in (13). This fact

makes the resulting equations cumbersome and hard to solveand makes it also difficult to describe the reasons why thoseevents happen and to distinguish the influence of the differentparameters on them. On the contrary, with our approach, wecan still obtain solutions for most of the cases by computing thesolutions of the model for a range of the constant parameter.That is, we can plot a family of curves for the foregoing model,varying the deceleration, and another family of curves for thenext model, varying the notification delay, as shown later inSection V. Moreover, the first case with variable notificationdelay models a scenario where communications are in use, butdrivers have control over brake; therefore, a driver reaction timehas meaning and must be taken into consideration. The secondcase with constant notification delay exemplifies a scenariowhere communications are in use, and the car is automaticallybraked as soon as a warning message is received.

Therefore, in the second step, we consider δi constant. Thedistance to stop without collision of vehicle Ci now includes avariable ai and becomes

ds,i =V 2

i

2ai+ Viδ. (19)

Using the same arguments as above, the collision probabilitiesare given by (12).

In this case, collisions may occur only according to threedifferent ways: 1) Vehicles Ci and Ci−1 have not started tobrake; 2) both of them are already braking; or 3) vehicle Ci−1

has stopped, with their respective actual distances traveled,dc1,i, dc2,i, and dc3,i. Therefore, the average distance traveledby vehicle Ci is given by

li = ds,i(1 − pi) + dc1,iqc1,i + dc2,iqc2,i + dc3,iqc3,i (20)

where qc1,i, qc2,i, and qc3,i are given by (18), dc1,i and dc3,i

have the same form of (14) and (17), respectively, with slightlydifferent integration limits and time functions, which are de-rived in Appendix B, and dc2,i is expressed by (21), shown atthe bottom of the page.

Note that in all the cases the above equations provide ad-ditional meaningful information since we have derived theprobability of the different ways in which vehicle collisionsmay occur (qcj ,i) as well as the average distance traveled bythe vehicles (dcj ,i). This information can be used, for instance,to design a measure about the severity of collisions that assignsdifferent weights to every particular type of collision. Finally,one may wonder whether it is possible to adopt a simplerapproximation instead of the average distance li to compute the

dc2,i =

1qc2,i

sup2∫inf2

(Vitc2,i(x) − ai

2 (tc2,i(x) − δ)2)λe−λxdx, Vi > Vi−1, ai−1 − ai = 0,

1qc2a,i

sup2a∫inf2a

(Vit1c2,i(x) − ai

2 (t1c2,i(x) − δ)2)λe−λxdx

+ 1qc2b,i

sup2b∫inf2b

(Vit2c2,i(x) − ai

2 (t2c2,i(x) − δ)2)λe−λxdx, ai−1 − ai �= 0

(21)



probabilities, or whether there is significant difference betweenli. Indeed, for vehicles located in the chain far enough fromthe leading vehicle, li is closer to (2), that is, the influence ofthe outcome of the previous vehicle is weak, and the outcomedepends mainly on the particular values of the parameters.However, for vehicles close to the leading vehicle, there is astrong dependence on the outcome of the previous vehicle, and(13) and (20) cannot be neglected.

1) Validation and Discussion: The next stage would be toassign to the kinematic parameters and notification delays ap-propriate values that model realistic scenarios. As an example,to take into account an underlying communication model, thenotification delay should be assumed to be a random variablewith an appropriate probability density function. This way, in-formation packet collisions in a heavily loaded shared commu-nications channel can be modeled with an appropriate randomvariable for access delay and characterized by Tm,i [11]. Fur-thermore, since vehicles move at different speeds, the velocityshould be assumed to be a random variable too. Let us notethat, in most of the practical cases, intervehicle distances andvelocities represent the state of the system when the incidentoccurs; therefore, they should be considered random variables,although determining their distributions and ranges requires aproper characterization of the scenario of interest. Accelera-tions and delays can be controlled by different means after theincident, and therefore, depending on the application evaluated,they can be considered constant or assigned particular values.

However, with regard to the analysis, introducing additionalrandom variables makes it hard to obtain a closed-form solutionfor the collision probabilities, even for the simple case whenparameters are assigned uniform distributed random variables,and the benefits are not clear. Therefore, the solutions whena parameter is a random variable have been computed nu-merically. The parameters are supposed to be uniform randomvariables, and (1) has been computed 100 times and averaged.In all the cases, we assume a chain of N = 20 vehicles.

A solution for the model with constant deceleration has beencomputed for three different scenarios. In the first one, δi isassumed to be a uniform random variable ranging between 0.5and 1.5 s, whereas the velocity has been fixed at V = 33 m/s.In the second scenario, Vi is assumed to be a uniform randomvariable between 30 and 36 m/s, and the notification delay hasbeen fixed at δ = 1 s. In the last simulation, both the velocityand the notification delay are assumed to be uniform randomvariables ranging between 0.5 and 1.5 s and between 30 and36 m/s, respectively. In all the simulations, the deceleration iskept constant at 8 m/s2.

The validation of the model with constant notification delayhas also been done for three different scenarios. In the firstscenario, deceleration ai is assumed to be a uniform randomvariable between 4 and 8 m/s2, whereas the velocity has beenfixed at V = 33 m/s. In the second scenario, Vi is assumedto be a uniform random variable between 30 and 36 m/s,and the deceleration has been fixed at a = 8 m/s2. In the lastsimulation, both the deceleration and the velocity are assumedto be uniform random variables between 4 and 8 m/s2 andbetween 30 and 36 m/s, respectively. In all the simulations, thenotification delay is kept constant at 1 s.

Fig. 7. Validation of the model with constant deceleration through the eval-uation of three different scenarios. (a) δi ∼ U(0.5, 1.5) s, Vi = 33 m/s, andai = 8 m/s2. (b) δi = 1 s, Vi ∼ U(30, 36) m/s, and ai = 8 m/s2. (c) δi ∼U(0.5, 1.5) s, Vi ∼ U(30, 36) m/s, and ai = 8 m/s2.

Finally, to validate the results for our solutions, the cor-responding Monte Carlo simulations have been conductedas well.

Figs. 7 and 8 show the results of this section. Let us remarkthat these pictures are provided to validate that our modelcorrectly shows the dynamics of the system. A discussion on theinfluence of the parameters on the collision process is deferredto the next section. The average number of accidents computed



Fig. 8. Validation of the model with constant notification delay throughevaluation of three different scenarios. (a) δi = 1 s, Vi = 33 m/s, andai ∼ U(4, 8) m/s2. (b) δi = 1 s, Vi ∼ U(30, 36) m/s, and ai = 8 m/s2.(c) δi = 1 s, Vi ∼ U(30, 36) m/s, and ai ∼ U(4, 8) m/s2.

with our model for each of the six cases is compared withthe aforementioned Monte Carlo simulations. The standarddeviation has been computed and shown as error bars. Dashedlines show the 95% confidence interval of the correspondingsimulation. In all the cases, the results reasonably confirm thevalidity of our model, even using li−1 as approximation, sincethe mean square error between the results of the analysis andthe simulation remains between 3.5% and 6% for all the cases.

Fig. 9. Performance of the model with different (a) constant decelerationsand (b) message reception delay. (a) Average percentage of accidents fordifferent decelerations when Vi ∼ U(30, 36) m/s and δi ∼ U(0.5, 1.8) s.(b) Average percentage of accidents for different notification delays whenVi ∼ U(30, 36) m/s and ai ∼ U(4, 8) m/s2.

V. APPLICATIONS AND DISCUSSION OF THE MODEL

Once our model has been validated in the previous sections,we use it to evaluate the influence of the different parameters onthe vehicle collision process. In this section, we present someresults as an example of the utility of our model. The metricused here is the average percentage of accidents in the chain,but the model could also provide information about the prob-ability of collision or the average distance traveled for thedifferent manners a collision can occur. A systematic evaluationof the different scenarios as well as the different metrics is leftas future work. First, we focus on qualitative aspects of theinfluence of the parameters on the collision process that ourmodel can quickly reveal. Then, we discuss quantitative aspectsof the results provided in this section.

As for the qualitative evaluation, we first provide a set offigures that show the influence of the different parameters.Fig. 9 shows a family of curves for both instances of the modelover a range of their constant parameter a or δ. As can beseen in Fig. 9(a), the number of accidents is clearly sensitiveto the deceleration capabilities of the vehicles, which agreeswith the results obtained in [2]. However, it does not seem tobe a statistical difference for different notification delays when



Fig. 10. Evaluation of the impact of the parameters’ variability on the numberof vehicle collisions. (a) Variability of deceleration with different notificationdelays. (b) Variability of velocity and notification delay. (c) Performance of themodel with fixed parameters.

the deceleration and velocities are variable. This result is also inaccordance with [2], where it is shown that moderate changes inthe notification delay cause small variations in accident severity.Later, in this section, we discuss when the delay actually has animportant influence on the number of accidents.

Fig. 10(a) shows the results when the velocities are randomlydistributed. In this case, if either deceleration or notificationdelay are kept constant, it causes a reduction of the number ofaccidents. In fact, in this case, it is noticeable that the positive

effect of a communication system is able to deliver warningmessages with short maximum delays and automatic vehicleresponse. Fig. 10(b) shows similar results when deceleration iskept constant at a = 6 m/s2. The results, however, reveal that ingeneral the variability of the kinetic parameters has a negativeimpact on the number of accidents. If the system is able to keepconstant some of the parameters during the emergency event,an improvement can be achieved. The benefits of a warningcollision system are even clearer in Fig. 10(c). When all theparameters remain constant, a shorter notification delay alwaysresults in fewer vehicle accidents.

Overall, these results suggest that a cooperative warningcollision notification system combined with a vehicle controlsystem able to smooth out the variations of speed anddeceleration of the platoon of vehicles may improve the driverand passenger safety. In fact, more detailed conclusions can beextracted. As we said in Section I, our model is useful to providegeneral guidelines about the design and operation of a CCAapplication. For instance, a CCA application may be based on awarning message delivered by an appropriate human–machineinterface (HMI) [19], a specific sound for instance as in [20],or by a fully automated braking response [21], [22]. The latteris expected to provide better performance, but one may wonderif the former benefits from a communication system and forwhich range of parameters. Now, how can our model be appliedto obtain relevant conclusions about these questions? First, itmust be taken into account that for a reactive CCA application,the only parameters in our model that can be controlled aredelays, with the communication system, and decelerations,with some automated control response to the warning message.Let us recall that either velocities of the vehicles and theintervehicle distance model the state of the traffic when theincident occurs; therefore, both of them should be considered arandom variable. Therefore, in the best case, the CCA is able toprovide a constant and short delay and enforce an appropriateconstant deceleration. Therefore, the curve in Fig. 10(b)provides the results for this case. If there is a warning messagebut the driver still keeps the control of braking, then a randomreaction time should be added before the brake. This case isprovided by the curve with uniform delay again in Fig. 10(b),assuming that the deceleration can be kept constant, which isnot quite realistic. It is more reasonable that every driver alsoapplies a different deceleration, which is the case shown inFig. 9(b). However, as shown, in this case, the actual delayis of little relevance. This has important implications in thedesign of the CCA. The usual approach is to consider that theemergency messages must be sent as fast as possible [15], [23],but according to these results, a higher delay could be tradedoff for other features, such as reliability of warning messagereception. For instance, adding a request-to-send/clear-to-sendmechanism to avoid packet collisions due to hidden nodes[1], or more importantly, the CCA application should providean acceleration control mechanism, so the margin in delaycan be used to collect all the necessary information fromneighbor vehicles to perform such control properly. This kindof insight on delay requirements is also important for designingCCA applications based on predicting trajectory conflicts todetermine the time horizon for trajectory estimation [3].



Fig. 11. Average percentage of accidents in a low-speed scenario withVi ∼ U(10, 16) m/s.

However, if we consider a low-speed and high-density sce-nario, then the delay has a remarkable influence. Fig. 11 showsthe average percentage of accidents when velocities are uni-formly distributed within 10 and 16 m/s. This scenario wouldmodel an urban road, where speed is relatively low, but thevehicle density is high.3 In addition, in this case, particularlyat short intervehicle distances corresponding to urban roads,the influence of delay is more noticeable, higher than that ofdeceleration. Therefore, we can conclude that the use of an HMImessage might not be sufficient to ensure safety, and a specialemphasis should be placed on providing automatic decelerationcontrol. Moreover, in this scenario, it is particularly difficult fora communication system based on contention channel access(CSMA) to provide low delays, since the number of neighborsin range is high, unless additional congestion control mecha-nisms such as transmit power control are applied.

In fact, some of these conclusions can be drawn by directlyexamining (12), that is, for high speeds, it is more importantto have good deceleration capabilities rather than to press thebrake quickly, and conversely for low speeds. The previousdiscussion is provided to show that the model has potentialto provide interesting qualitative and quantitative conclusionsabout the colliding process and the mechanisms that CCAapplications should include, but we have to remark that aprecise definition of the scenarios of interest is still necessaryto set appropriate distributions and ranges for the parameters,which is left as future work.

Finally, as for the quantitative aspects of the results, thepercentage of accidents might seem higher than expected,above 10% in many cases, as well as the slow decay of itfor high intervehicular distances. This is first a consequenceof the extreme case we are evaluating here, that is, the lead-ing vehicle stops completely and immediately. It makes thecollision of the first car of the platoon almost unavoidable inmost of the cases. As a worst-case approach, better outcomesare expected in reality. However, also, these results have to be

3Just for the sake of example, let us remark that a log-normal distribution forinter-vehicle distances describes more accurately high-density scenarios.

interpreted with care, since using average intervehicle distancesmay lead to misleading conclusions. As an example, with theparameters used in Fig. 10(c), i.e., V = 36 m/s, a = 6 m/s2,and δ = 0.1, the distance needed to stop is 111.6 m. For anexponentially distributed intervehicle distance with mean s =60 m, the probability of si being less than 100 m is 0.81,and even with a mean s = 150 m, this probability is still 0.48.Therefore, the probability of collision is higher than one mayintuitively think, particularly for the first vehicles in the chain.Therefore, even at relatively high intervehicular distances, thecollisions are mainly suffered by the first and second vehicles,which accounts for 10% of accidents for our example withN = 20 vehicles.

VI. CONCLUSION

In this paper, we have proposed and derived a stochasticmodel for the probability of collisions in a chain of vehicleswhere a warning collision system is in operation. The fact thata warning notification system is used allows us to overcome thedifficulties for obtaining stochastic models for such vehicularscenarios, since we can assume that all the drivers/vehiclesreact to the warning message independently, and therefore, themotion equations can be simplified. We also propose a goodmatching approximation to the exact model to further reducethe required computations to calculate the vehicle collisionprobabilities. In both cases, its validity has been confirmed byMonte Carlo simulations.

The model is independent of the particular communicationsystem employed as long as its operation can be abstractedand characterized by an appropriate message notification delay,including communication latency and driver reaction times.Therefore, it also enables the performance evaluation of dif-ferent technologies. Indeed, a future line of this work is toassess the performance of current VANET technology based oncontention (CSMA) MAC protocols for those cases where delayis actually relevant for the collision process outcome. Similarly,different probability distributions for the intervehicular spacingcan seamlessly be incorporated into the model due to thefact that the distribution of the initial intervehicle spacing isindependent of the actions that drivers make after receivingthe warning messages. Here, we have used an exponentialdistribution, which is considered appropriate for low vehicletraffic densities. As a future work, we plan to employ a log-normal distribution that describes well high vehicle trafficdensities. Finally, we compute the probability that collisionsoccur in different forms (both vehicles in motion, one stoppedand one in motion, etc.), which opens a promising way todefine detailed accident severity functions, that is, by assign-ing different grades of severity to each collision possibility.This is an interesting approach that we leave as future workas well.

Although we have shown some examples of the applicationof the model, a quantitative evaluation requires a careful defin-ition of the scenarios of interest. Therefore, we leave as futurebut imminent direction to pursue a systematic characterizationand evaluation of the different scenarios for a wider and moreaccurate extent of the model parameters.



APPENDIX ACOMPUTATION OF THE DISTANCE TRAVELED

BY A VEHICLE FOR VARIABLE VELOCITY

AND NOTIFICATION DELAY

Let us recall that when velocity and notification delay arenot constant, the collisions may occur in four different ways:1) Vehicles Ci and Ci−1 have not started to brake; 2) only one ofthem is braking; 3) both of them are braking; or 4) vehicle Ci−1

has stopped. Each one of these possibilities results in a differentdistance to stop, i.e., dc1,i, dc2,i, dc3,i, and dc4,i, respectively,which must be weighted by the probability of the event andadded to get the average li, as in (13).

The computation of these distances is as follows.1) Collision When the Vehicles Have Not Started to Brake

(dc1,i): This event may happen if the difference of initial ve-locities makes the vehicles crash before receiving the warningmessage.4

A time instant tc1,i(si) should exist so that

Vitc1,i(si) = Vi−1tc1,i(si) + si (22)0 ≤ tc1,i(si) ≤ min{δi, δi−1}. (23)

Solving (22), we obtain

tc1,i(si) =si

Vi − Vi−1. (24)

Therefore, the distance to stop in this case is d = Vitc1,i(si),which is a function of si (exponentially distributed). Then, theaverage distance is computed as follows:

dc1,i =1

F (sup1) − F (inf1)

sup1∫inf1

Vix

Vi − Vi−1λe−λx dx. (25)

Now, it remains to compute the range of si where a collisioncan happen, that is, the integration limits denoted as inf1

and sup1.1) If Vi − Vi−1 ≤ 0, there is no solution (no collision can

occur). Let us define appropriate limits

inf1 = 0 (26)sup1 = 0. (27)

2) If Vi − Vi−1 > 0, then (23) holds if and only if 0 ≤ si ≤(Vi − Vi−1) · min{δi, δi−1}. Let us define

inf1 = 0 (28)sup1 = (Vi − Vi−1) · min{δi, δi−1}. (29)

The same procedure is applied for the following cases: thecomputation of the actual distance traveled, in this case, d =Vitc1,i(si), where tc1,i(si) = f(si, a, Vi, Vi−1, δi, δi−1), whichis multiplied by the exponential pdf and integrated within theappropriate limits, which are also derived.

4Let us note that this case implies that the vehicles would collide even ifthere is no obstacle ahead on the road or the rest of vehicles in the chain are notbraking. For instance, a driver notices that his actual speed is higher than that ofthe preceding vehicle but does not reduce it and lets his car collide. One mightthink of a situation where bad weather conditions, like a very thick fog, preventthe driver from noticing the risk. In a normal driving situation, this case shouldbe highly unlikely, but it has to be considered to get a consistent result.

2) Collision When Only One Vehicle Is Braking (dc2,i): Inthis case, the collision event depends on the relative reactiontimes of the drivers. That is, due to a high reaction time, one ofthe drivers starts to brake too late.

1) If δi = δi−1, then we have to skip to case 3, later in thetext; therefore, let us define

inf2 = sup1 (30)sup2 = sup1. (31)

2) If δi < δi−1, then vehicle Ci starts to brake before Ci−1

does.A time instant tc2,i(si) should exist so that

Vitc2,i(si) −a

2(tc2,i(si) − δi)

2 = Vi−1tc2,i(si) + si, (32)

δi ≤ tc2,i(si) ≤ δi−1. (33)

Solving this equation, we obtain the followingsolutions:

t1c2,i(si) =Vi − Vi−1

a+ δi

−

√(Vi − Vi−1

a+ δi

)2

− δ2i − 2si

a(34)


a+ δi

+

√(Vi − Vi−1

a+ δi

)2

− δ2i − 2si

a. (35)

The term in the square root is positive if and only ifsi ≤ δi(Vi − Vi−1) + ((Vi − Vi−1)2/2a).

It can be proved that (33) does not hold for t2c2,i(si);therefore, the only possible solution is t1c2,i(si).1) If Vi − Vi−1 > a(δi−1 − δi), then (33) holds for

t1c2,i(si) if and only if

δi(Vi−Vi−1) ≤ si ≤ δi−1(Vi−Vi−1)−a

2(δi−δi−1)2. (36)

Let us define

inf2 = δi(Vi − Vi−1) (37)

sup2 = min{

δi(Vi − Vi−1) +(Vi − Vi−1)2

2a

δi−1(Vi − Vi−1) −a

2(δi − δi−1)2

}. (38)

2) If 0 < Vi − Vi−1 ≤ a(δi−1 − δi), then (33) holds fort1c2,i(si) if and only if

δi(Vi − Vi−1) ≤ si ≤ δi(Vi − Vi−1) +(Vi − Vi−1)2

2a. (39)

Let us define

inf2 = δi(Vi − Vi−1) (40)

sup2 = δi(Vi − Vi−1) +(Vi − Vi−1)2

2a. (41)

3) Otherwise, there is no solution; therefore, let us define

inf2 = sup1 (42)sup2 = sup1. (43)



3) If δi > δi−1, then vehicle Ci−1 starts to brake beforeCi does.


Vitc2,i(si) = Vi−1tc2,i(si) −a

2(tc2,i(si) − δi−1)

2 + si (44)

δi−1 ≤ tc2,i(si) ≤ δi. (45)

Solving (44), we obtain the following solutions:

t1c2,i(si) =Vi−1 − Vi

a+ δi−1

−

√(Vi−1 − Vi

a+ δi−1

)2

− δ2i−1 +

2si

a(46)

t2c2,i(si) =Vi−1 − Vi

a+ δi−1

+

√(Vi−1 − Vi

a+ δi−1

)2

− δ2i−1 +

2si

a. (47)

The term in the square root is positive if and only ifsi ≥ δi−1(Vi − Vi−1) − ((Vi−1 − Vi)2/2a).

It can be proved that (45) does not hold for t1c2,i(si),so the only possible solution is t2c2,i(si).1) If Vi−1 − Vi < 0, then (45) holds for t2c2,i(si) if and

only if

δi−1(Vi − Vi−1) ≤ si ≤ δi(Vi − Vi−1) +a

2(δi − δi−1)2.

(48)Let us define

inf2 = δi−1(Vi − Vi−1) (49)

sup2 = δi(Vi − Vi−1) +a

2(δi − δi−1)2. (50)

2) If 0 ≤ Vi−1 − Vi ≤ a(δi − δi−1), then (45) holds fort2c2,i(si) if and only if

δi−1(Vi − Vi−1) −(Vi−1 − Vi)2

2a≤ si

≤ δi(Vi − Vi−1) +a

2(δi − δi−1)2. (51)

Let us define

inf2 = δi−1(Vi − Vi−1) −(Vi−1 − Vi)2

2a(52)

sup2 = δi(Vi − Vi−1) +a

2(δi − δi−1)2. (53)


inf2 = sup1 (54)

sup2 = sup1. (55)

3) Collision When Both Vehicles Are Braking (dc3,i): In thiscase, both vehicles are aware of the danger and have started tobrake, but they are not able to avoid the collision, due to theirinitial speeds and reaction times, and they collide in motion.

A time instant should exist so that

Vitc3,i(si) −a

2(tc3,i(si) − δi)2

= Vi−1tc3,i(si) −a

2(tc3,i(si) − δi−1)2 + si (56)

δi ≤ tc3,i(si) ≤Vi

a+ δi (57)

δi−1 ≤ tc3,i(si) ≤ Ti−1(li−1) (58)

where Ti−1(li−1) is the time needed by vehicle Ci−1 to travelthe distance li−1, and it is calculated by the function

Ti(x) =

{ xVi

, if x ≤ Viδi,Vi

a + δi −√

2a (ds,i − x), if x > Viδi.

(59)

Solving (56), we obtain

tc3,i(si) =si + a

2

(δ2i − δ2

i−1

)(Vi − Vi−1) + a(δi − δi−1)

. (60)

To simplify the notation, we call num = (a/2)(δ2i − δ2

i−1)and den = (Vi − Vi−1) + a(δi − δi−1).

1) If den = 0, then there is no solution. Let us define

inf3 = sup2 (61)

sup3 = sup2. (62)

2) If den > 0, then (57) and (58) hold if and only if

den · max{δi, δi−1} − num ≤ si

≤ den · min{

Vi

a+ δi, Ti−1(li−1)

}− num. (63)

Therefore, let us define

inf3 = den · max{δi, δi−1} − num (64)

sup3 = den · min{

Vi


}− num. (65)

3) If den < 0, then (57) and (58) hold if and only if

den · min{

Vi


}− num ≤ si

≤ den · max{δi, δi−1} − num. (66)

Then, let us define

inf3 = den · min{

Vi


}− num (67)

sup3 = den · max{δi, δi−1} − num. (68)

4) Collision When Vehicle Ci−1 Has Stopped (dc4,i): Thepreceding vehicle has been able to stop safely, but a rearcollision still occurs.

In this case, si should directly satisfy si ≤ ds,i − li−1.We set

inf4 = sup3 (69)

sup4 = ds,i − li−1. (70)



APPENDIX BCOMPUTATION OF THE DISTANCE TRAVELED BY A

VEHICLE FOR VARIABLE VELOCITY AND DECELERATION

Let us recall that when velocity and deceleration are notconstant, the collisions may occur in three different ways:1) Vehicles Ci and Ci−1 have not started to brake; 2) bothof them are already braking; or 3) vehicle Ci−1 has stopped,with their respective actual distances to stop dc1,i, dc2,i, anddc3,i. These distances are computed using the same procedurein Appendix A.

1) Collision When the Vehicles Have Not Started To Brake(dc1,i): This event may happen if the difference of initial ve-locities makes the vehicles crash before receiving the warningmessage.


Vitc1,i(si) =Vi−1tc1,i(si) + si (71)0 ≤ tc1,i(si) ≤ δ. (72)

Solving (71), we have

tc1,i(si) =si

Vi − Vi−1. (73)

Now, it remains to compute the integration limits inf1

and sup1.1) If Vi − Vi−1 ≤ 0, then there is no solution (no collision

can occur). Let us define appropriate limits

inf1 = 0 (74)sup1 = 0. (75)

2) If Vi − Vi−1 > 0, then (72) holds if and only if 0 ≤ si ≤δ(Vi − Vi−1). Let us define

inf1 = 0 (76)sup1 = δ(Vi − Vi−1). (77)

2) Collision When Both Vehicles Are Braking (dc2,i): A timeinstant tc2,i(si) should exist so that

Vitc2,i(si) −ai

2(tc2,i(si) − δ)2

= Vi−1tc2,i(si) −ai−1

2(tc2,i(si) − δ)2 + si (78)

δ ≤ tc2,i(si) ≤ min{

Vi

ai+ δ, Ti−1(li−1)

}. (79)

To simplify the notation, we call min = min{(Vi/ai) +δ, Ti−1(li−1)} − δ = min{(Vi/ai), Ti−1(li−1) − δ}.

1) If ai−1 − ai = 0 in solving (78), we obtain

tc2,i(si) =si

Vi − Vi−1. (80)

1) If Vi − Vi−1 ≤ 0, then there is no solution. Let usdefine

inf2 = sup1 (81)sup2 = sup1. (82)

2) If Vi − Vi−1 > 0, then (79) holds if and only if

δ(Vi − Vi−1) ≤ si ≤ (Vi − Vi−1) · min . (83)

Therefore, let us define

inf2 = δ(Vi − Vi−1) (84)sup2 = (Vi − Vi−1) · min . (85)

2) If ai−1 − ai �= 0 in solving (78), then we obtain thefollowing solutions:


ai − ai−1+ δ

−

√(Vi − Vi−1

ai − ai−1+ δ

)2

− δ2 +2s(i)

ai−1 − ai, (86)


ai − ai−1+ δ

+

√(Vi − Vi−1

ai − ai−1+ δ

)2

− δ2 +2s(i)

ai−1 − ai. (87)

The term in the square root is positive if and onlyif (si/(ai−1 − ai)) ≥ (δ(Vi − Vi−1)/(ai−1 − ai)) −((Vi − Vi−1)2/2(ai−1 − ai)2).

First we compute the limits for t1.If ai−1 − ai > 0, then (79) does not hold for t1c2,i(si);

therefore, let us define

inf2a = sup1 (88)sup2a = sup1. (89)

If ai−1 − ai < 0, then we have the following:1) If (Vi − Vi−1/ai − ai−1) ≤ min, then (79) holds for

t1c2,i(si) if and only if

δ(Vi − Vi−1) ≤ si ≤ δ(Vi − Vi−1) −(Vi − Vi−1)2

2(ai−1 − ai). (90)

Let us define

inf2a = δ(Vi − Vi−1) (91)

sup2a = δ(Vi − Vi−1) −(Vi − Vi−1)2

2(ai−1 − ai). (92)

2) If (Vi − Vi−1/ai − ai−1) > min, then (79) holds fort1c2,i(si) if and only if

δ(Vi − Vi−1) ≤ si ≤ai−1 − ai

2· min2

+ (Vi − Vi−1) · min +δ(Vi − Vi−1). (93)

Let us define

inf2a = δ(Vi − Vi−1) (94)

sup2a = min{

δ(Vi − Vi−1) −(Vi − Vi−1)2

2(ai−1 − ai),ai−1 − ai

2min2

+ (Vi − Vi−1) · min +δ(Vi − Vi−1)}

. (95)

We still need to compute the limits for t2c2,i(si).If (Vi − Vi−1/ai − ai−1) > min, (79) does not

hold for t2c2,i(si); therefore, let us define

inf2b = sup2a (96)sup2b = sup2a. (97)



If (Vi − Vi−1/ai − ai−1) ≤ min, then we have thefollowing.

3) If ai−1 − ai > 0 and Vi − Vi−1 > 0, then (79) holdsfor t2c2,i(si) if and only if

δ(Vi − Vi−1) ≤ si ≤ai−1 − ai

2· min2

+(Vi − Vi−1) · min +δ(Vi − Vi−1). (98)

Let us define

inf2b = δ(Vi − Vi−1) (99)

sup2b =ai−1 − ai

2· min2

+ (Vi − Vi−1)min +δ(Vi − Vi−1). (100)

4) If ai−1 − ai > 0 and Vi − Vi−1 < 0, then (79) holdsfor t2c2,i(si) if and only if

δ(Vi − Vi−1) −(Vi − Vi−1)2

2(ai−1 − ai)≤ si ≤

ai−1 − ai

2· min2

+ (Vi − Vi−1) · min +δ(Vi − Vi−1). (101)

Let us define

inf2b = δ(Vi − Vi−1) −(Vi − Vi−1)2

2(ai−1 − ai)(102)

sup2b =ai−1 − ai

2· min2

+ (Vi − Vi−1)min +δ(Vi − Vi−1). (103)

5) If ai−1 − ai < 0 and Vi − Vi−1 > 0, then (79) holdsfor t2c2,i(si) if and only if

ai−1 − ai

2·

2min +(Vi − Vi−1) · min +δ(Vi − Vi−1)

≤ si ≤ δ(Vi − Vi−1) −(Vi − Vi−1)2

2(ai−1 − ai). (104)

Let us define

inf2b =ai−1 − ai

2· min2

+ (Vi − Vi−1)min +δ(Vi − Vi−1) (105)

sup2b = δ(Vi − Vi−1) −(Vi − Vi−1)2

2(ai−1 − ai). (106)


inf2b = sup2a (107)

sup2b = sup2a. (108)

3) Collision When Vehicle Ci−1 Has Stopped (dc3,i): Thepreceding vehicle has been able to stop safely but a rear-endcollision occurs.

In this case, si should directly satisfy that si ≤ ds,i − li−1.We set

inf3 = sup2 (109)

sup3 = ds,i − li−1. (110)

ACKNOWLEDGMENT

J. B. Tomas-Gabarron would like to thank the SpanishMICINN for a FPU (REF AP2008-02244) predoctoral fellow-ship. C. Garcia-Costa would also like to thank the FundaciónSeneca for a FPI (REF 12347/FPI/09) predoctoral fellowship.This work was also developed in the framework of “Programade Ayudas a Grupos de Excelencia de la Región de Murcia, de laFundación Séneca, Agencia de Ciencia y Tecnología de la RM.”

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Carolina Garcia-Costa received the Mathematicsand Master degrees in advanced mathematics fromthe University of Murcia, Murcia, Spain, in 2008 and2009, respectively. She is currently working towardthe Ph.D. degree with the Department of Informa-tion Technologies and Communications, Universi-dad Politécnica de Cartagena, Cartagena, Spain.

Her current research interests are focused onvehicular ad hoc networks, particularly on crashprobability estimation, active safety systems, andperformance analysis of medium access control

protocols.

Esteban Egea-Lopez received the Telecommunica-tions Engineering degree from Universidad Politéc-nica de Valencia, Valencia, Spain, in 2000, the Mas-ter’s degree in electronics from the University ofGavle, Gavle, Sweden, in 2001, and the Ph.D. de-gree in telecommunications from the UniversidadPolitécnica de Cartagena (UPCT), Cartagena, Spain,in 2006.

Since 2001, he has been an Assistant Professorwith the Department of Information Technologiesand Communications, UPCT. His research interest

is focused on radio-frequency identification, vehicular, ad hoc, and wirelesssensor networks.

Juan Bautista Tomas-Gabarron received the B.S.degree in telecommunications engineering in 2008from the Universidad Politécnica de Cartagena,Murcia, Spain (final work project developed in con-junction with the Technical University of Munich,Munich, Germany), and the Master of Researchin communications in 2009 from the UniversidaPolitécnica de Cartagena, Cartagena, Spain, wherehe is currently working toward the Ph.D. degree.

His research interests are vehicular ad hocnetworks, in particular, the design, evaluation, and

testing of cooperative collision avoidance applications to develop commu-nication protocols that can minimize the probability of accident in criticalemergency brake situations.

Joan Garcia-Haro (M’11) received the M.Scand Ph.D degrees in telecommunication engineer-ing from the Universitat Politecnica de Catalunya,Barcelona, Spain, in 1989 and 1995, respectively.

He is currently a Professor with the UniversidadPolitécnica de Cartagena, Cartagena, Spain. He isauthor or coauthor of more than 60 journal papers,mainly in the fields of switching, wireless network-ing, and performance evaluation.

Prof. Garcia-Haro served as the Editor-in-Chiefof the IEEE Global Communications Newsletter,

included in the IEEE Communications Magazine, from April 2002 to December2004. He has been the Technical Editor of the same magazine since March2001. He also received an Honorable Mention for the IEEE CommunicationsSociety Best Tutorial Paper Award (1995).

Zygmunt J. Haas (F’07) received the B.Sc. andM.Sc. degrees in electrical engineering in 1979 and1985 and the Ph.D. degree from Stanford University,Stanford, CA, in 1988.

He subsequently joined the Network ResearchDepartment, AT&T Bell Laboratories, where he pur-sued research on wireless communications, mobil-ity management, fast protocols, optical networks,and optical switching. From September 1994 toJuly 1995, he was with the AT&T Wireless Centerof Excellence, where he investigated various aspects

of wireless and mobile networking, concentrating on Transmission ControlProtocol/Internet Protocol networks. Since August 1995, he has been with theSchool of Electrical and Computer Engineering, Cornell University, Ithaca,NY, where he is currently a Professor. He is an author of numerous technicalpapers and holds 18 patents in the fields of high-speed networking, wirelessnetworks, and optical switching. He has organized several workshops anddelivered numerous tutorials at major IEEE and Association for ComputingMachinery (ACM) conferences. His interests include mobile and wirelesscommunication and networks, biologically inspired networks, and modeling ofcomplex systems.

Dr. Haas is a voting member of the ACM. He has served as Editorof several journals and magazines, including the IEEE TRANSACTIONS

ON NETWORKING, the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS, the IEEE COMMUNICATIONS MAGAZINE, the SpringerWireless Networks journal, the Elsevier Ad Hoc Networks journal, the Journalof High Speed Networks, and the Wiley Wireless Communications andMobile Computing journal. He has been a guest editor of IEEE JOURNAL

ON SELECTED AREAS OF COMMUNICATIONS issues on “gigabit networks,”“mobile computing networks,” and “ad-hoc networks.” He has served in thepast as a Chair of the IEEE Technical Committee on Personal Communications.

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