IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. X, NO. X, XX XX 1
A stochastic model for chain collisions of vehicles
equipped 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 VANET.
However, in order to develop properly such applications, the
influence of the different driving parameters on the event of a
vehicle collision must be assessed at an early design stage. In this
paper we derive a stochastic model for the number of accidents
in a platoon of vehicles equipped with a warning collision
notification system, which is able to inform all the vehicles about
an emergency event. In fact, the assumption of communications
being used is key to simplify the derivation of a stochastic model.
The model enables the computation of the average number of
collisions that occur in the platoon, the probabilities of the
different ways in which the collisions may take place as well as
other statistics of interest. Although an exponential distribution
has been used for the traffic density, it is also valid for different
probability distributions for the traffic densities as well as for
other significant parameters of the model. Moreover, the actual
communication system employed is independent of the model
since it is abstracted by a message delay variable, which allows
it to be used to evaluate different communication technologies.
We validate the proposed model with Monte-Carlo simulations.
With this model one can quickly evaluate numerically the
influence of the different model parameters (vehicle density,
velocities, decelerations and delays) on the collision process
and draw conclusions that shed relevant guidelines for the
design of vehicular communication systems as well as Chain
Collision Avoidance (CCA) applications. Illustrative examples of
application are provided, though a systematic characterization
and evaluation of different scenarios is left as future work.
Index Terms—Vehicle safety, vehicular communications, chain
collision, vehicle platoon, collision avoidance, stochastic model,
road accidents
I. INTRODUCTION
I
NTER-VEHICLE communications based on wireless tech-
nologies pave the way for novel applications in traffic
Manuscript received March 23, 2011; revised XX XX, 2011. This re-
search has been supported by the MICINN/FEDER project grant TEC2010-
21405-C02-02/TCM (CALM) and Fundación Seneca RM grant 00002/CS/08
FORMA. It is also developed in the framework of “Programa de Ayudas
a Grupos de Excelencia de la Región de Murcia, de la Fundación Séneca,
Agencia de Ciencia y Tecnología de la RM”. J. Garcia-Haro acknowledges
personal grant PR2009-0337. J. B. Tomas-Gabarron thanks the Spanish
MICINN for a FPU (REF AP2008-02244) pre-doctoral fellowship. C. Garcia-
Costa also acknowledges the Fundación Seneca for a FPI (REF 12347/FPI/09)
pre-doctoral fellowship. E. Egea acknowledges UPCT for a PMPDI-UPCT-
2011 grant.
C. Garcia-Costa, E. Egea-Lopez, J. B. Tomas-Gabarron and J. Garcia-
Haro are with the Department of Information and Communication Tech-
nologies, Universidad Politécnica de Cartagena (UPCT), Spain, e-mail: es-
teban.egea@upct.es.
Z. J. Haas is with the School of Electrical and Computer Engineering,
Wireless Networks Lab. Cornell University, Ithaca, NY, USA.
safety, driver-assistance, traffic control and other advanced
services which will make future Intelligent Transportation
Systems (ITS). The advances in technology and standardiza-
tion, especially with the allocation of dedicated bandwidth
to vehicular communications, from the mid 1990s have in-
creased research and development efforts on Vehicular Ad-
Hoc Networks (VANET) from the networking and mobile
communications community [1], though early research on
“Automated Highways Systems” goes back to the 1960s
and later [2]. Improvement of traffic safety by cooperative
vehicular applications is one of the most promising technical
and social benefits of VANET [3], [4]. However, in order to
design and implement such applications, a deep understanding
of the vehicle collision processes is needed. The influence of
the different driving parameters on the collision event must
be assessed at an early design stage in order to develop
applications that can timely adapt vehicle dynamics to avoid
or at least mitigate the danger [5].
Very detailed models of vehicle motion and collision dy-
namics can be found [6], [7], but the equations are com-
pletely deterministic, whereas, in reality, randomness is always
present as an effect of human behaviour or noisy operation
introduced by sensors or other reasons. To account for it,
the usual methodology is to evaluate deterministic models by
applying a Monte-Carlo or stochastic analysis over an exten-
sive range of their parameters [2], [6], [8]. However, to the
authors’ knowledge, little effort has been devoted to develop
models which are stochastic in nature, and in particular for
rear-end chain collisions of vehicles. Some reasons behind
it are the difficulties of evaluating all the possible ways in
which a collision may occur and the complexity posed by the
fact that the motion equations for those possibilities involve a
dependence on the parameters of preceding vehicles. That is,
the driver reacts to variations in the driving conditions of the
preceding vehicle, as in a car-following approach [9], [10].
However, if vehicles use a communication system which is
able to inform all the vehicles about an emergency event, those
difficulties can be overcome. The key is that, in that case,
it can be assumed that drivers react as soon as they receive
a warning message and they start braking independently of
the preceding vehicles behavior. This is in fact the goal of
warning collision systems or Electronic Brake Warning (EBW)
applications. This assumption removes the dependence of the
motion equations on the preceding vehicles and facilitates the
development of a stochastic model.
In this paper we take this approach. Our goal is to describe
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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. X, NO. X, XX XX 2
and analyze the risk of colliding for a set of moving vehicles
forming a platoon (or chain) and equipped with a warning
collision system when there is a sudden stop of the leading ve-
hicle. To this aim, we derive a stochastic model for the number
of accidents that occurs in this scenario. The model allows us
to compute the average number of collisions that occur in the
platoon, among other statistics of interest. The scenario under
consideration is basically a platoon of vehicles moving along a
unidimensional road in the same direction in which the leading
vehicle suddenly comes to a complete stop. To consider a
worst case scenario we add two strong assumptions: first,
the leading vehicle stops instantly (it may also be considered
that a fixed obstacle lays on the road). Second, vehicles will
not be able to change their direction of movement to cope
with the unexpected incident. Our model is stochastic because
all its parameters may be described by random variables.
We derive the equations assuming always a random inter-
vehicle spacing, in particular for an exponentially distributed
spacing, though the model is valid for other distributions.
When additional parameters are assumed random, the solutions
have been computed numerically. Additionally, it should be
observed that the model is independent of the communication
technology, since the operation of the communication system
is abstracted by the use of a message reception/notification
delay variable. Finally, the probabilities for all the ways the
collision may take place are also derived, which can be further
used to evaluate the severity of accidents in higher detail, for
instance, by assigning different severity weights to different
types of collision. A deeper discussion on this topic, however,
is out of the scope of this paper.
The main practical utility of this model lays in its ability
to quickly evaluate numerically the influence of the differ-
ent parameters on the collision process, without the need
to resort to complex simulations in a first stage. Such an
evaluation provides relevant guidelines for the design of ve-
hicular communication systems as well as Chain Collision
Avoidance (CCA) applications. As an example, it can quickly
reveal for which range and distributions of the parameters the
communication delay has a serious impact on the metric of
interest, which can be the average number of accidents but
also the probability of collision of every vehicle in the chain.
Since it turns out that in some scenarios a low delay is not
relevant for the outcome, a communication system could trade
it off for additional reliability mechanisms. Moreover, in this
paper we set either constant or purely random parameters, but
the model can be used with arbitrary parameters to evaluate
more specific applications. For instance, to evaluate multi-
hop communications we can set up a vector of delays with
progressively increasing values. We provide examples of use
in Section V, but in any case, a careful characterization of
the model parameters for the scenarios and applications is a
necessary previous step.
So, in summary, in this paper our goal is to provide the
derivation and validation of the model and show its utility
with a few illustrative examples. A proper characterization and
evaluation of different scenarios and metrics is left as future
work.
The remainder of this paper is organized as follows. In
Section II we briefly review the related work. The derivation
and validation of the model is provided in Sections III and
IV. Section V illustrates how to use the model to evaluate
the influence of different parameters on the collision process
and to obtain qualitative conclusions relevant to the design
of vehicular communication as well as CCA applications.
Conclusions and future work are remarked in Section VI, while
the necessary auxiliary material is relegated to the Appendices
A and B.
II. RELATED WORK
Our model assumes that there is a communication system
between vehicles that allows them to receive warning messages
to start braking in the event of a sudden stop of the leading
car. However, such a system is abstracted in the model and
characterized by the use of a message reception/notification
delay variable. Therefore, our model is actually independent
of it and can be applied to any communication system whose
operation can be abstracted by an appropriate delay variable.
For instance, current VANET standards specify the use of
IEEE 802.11p which is based on contention (CSMA) Medium
Access Control [1]. Such a shared channel MAC technique can
be abstracted in our model by a delay random variable with
an appropriate probability distribution [11]. Further details on
current VANET communication technologies can be found in
[1].
Regarding collision models for chains of vehicles two dif-
ferent groups of studies can be found: (i) statistical models of
the frequency of accidents occurrence and their circumstances
[12], [13], and (ii) models of the collision process itself based
on physical parameters [2], [6], [8]. This paper falls on the
latter category and additionally assumes that an automated
warning system is in place. In most of these studies determin-
istic equations for the occurrence of collisions are derived and,
to account for random variability, stochastic analysis or Monte-
Carlo simulations over a wide range of model parameters are
performed afterwards to obtain an estimate of the collision
probability or other metrics of interest. Our approach is
different and the model shown here is directly stochastic and
assumes that at least the inter-vehicle distance is a random
variable, which is in fact a realistic assumption as shown in
[14]. We also perform Monte-Carlo simulations but, unlike
the previously mentioned papers, we use them to validate
our model rather than to obtain metrics of interest. Looking
into these works in particular, in an early study Fenton [2]
defines an accident cost function to evaluate the severity of
vehicle collisions. The collision model used is derived for an
automatically controlled
1
platoon of vehicles which advance
at constant speed with a constant inter-vehicle spacing. A
more recent work [8] provides a similar collision model for
a four-car platoon of vehicles assuming that just one of the
vehicles is equipped with an autonomous intelligent cruise
control. In both cases, the collision model defines how vehicles
decelerate in order to obtain a deterministic equation for the
1
Let us note that early research, which goes back to the 1960s, considered
the hypothesis of achieving “automated highway systems”, where most of the
driving tasks were automatically controlled.
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. X, NO. X, XX XX 3
collision. Afterwards, the evaluation is done by randomizing
some parameters of the model and running a Monte-Carlo
simulation. In [10], authors derive conditions necessary for a
chain collision, starting from a car-following model. However,
they assume that all the vehicles are driving with equal initial
speeds and inter-vehicle distances.
Interestingly, the vehicle collision model proposed in this
paper is more general, it explicitly accounts for random inter-
vehicle spacing, and can be used to assign arbitrary variables,
even random ones, to the kinematic parameters of each ve-
hicle as well as the warning message communication delay.
Moreover, there are additional applications of our model, for
instance, it can be readily used to evaluate the severity of
collisions, as in [2]: since we compute the probability of
collisions occurring in several manners, we could assign a
severity weight to each possibility, that is, we may assign more
severity to a collision when both vehicles are in motion than
to other cases, for example, though this topic is not treated
in the present paper. On the other hand, some of the results
in [2] are similar to ours, for instance the sensitivity shown
to the decrease in deceleration capabilities of the subsequent
vehicles. In all the cases as well as in this paper, only rear-end
collisions are considered. Head-on collisions are evaluated in
[6], based on a very detailed analytical model of the vehicle.
Finally, in this paper we provide examples about the kind
of results that can be drawn from the proposed model which
are useful for the design of CCA applications. A review on
intelligent collision avoidance algorithms can be found in [5].
In particular, the influence of delay notification on different
scenarios is useful to set appropriate time horizons for CCA
systems based on trajectory prediction [3].
III. COLLISION MODEL
We consider a platoon (or chain) of N vehicles following a
leading one (see Fig. 1), where each vehicle C
i
, i ∈ 1 . . . N,
moves at constant velocity V
i
. The leading vehicle C
0
collides
with an obstacle on the road, at time t
0
= 0, and immediately
it sends a warning message to the following vehicles. The
rest of the vehicles start to brake at constant deceleration
2
a
i
when they are aware of the risk of collision, that is, after
a time lapse δ
i
. Let us remark here that this time lapse is
mainly determined by the reception of the warning message,
generated by the communication system, so the reaction of the
driver is independent of the movement state of the preceding
vehicle. That is, a warned driver will decelerate even if the
preceding car has not started to decelerate. In a classical car-
following approach, on the contrary, the deceleration would
be a consequence of a change in the speed or inter-vehicle
spacing of the preceding vehicle. For the sake of simplicity,
we assume that every vehicle has the same length L and its
position is given by the x coordinate of its front bumper. The
leading vehicle stops at coordinate x
0
= 0 and the initial inter-
vehicle spacing is s
i
= x
i
−(x
i−1
+L). We assume that at least
the inter-vehicle spacing is a random variable. Its probability
distribution as well as the variability of the other variables a
i
,
2
To simplify the notation, in the remaining of the paper we consider a
i
a
deceleration, and so assign it a positive sign.
Fig. 1. The scenario under consideration.
Fig. 2. Probability tree diagram that defines the model. S
i,j
represents the
state with i collided vehicles and j successfully stopped vehicles.
V
i
, δ
i
is discussed in Sect. IV. Vehicles cannot change lane or
perform evasive maneuvers.
With this model the final outcome of a vehicle depends on
the outcome of the preceding vehicles. Therefore, the collision
model is based on the construction of the probability tree
depicted in Fig. 2. We consider an initial state in which no
vehicle has collided. Once the danger of collision has been
detected, the first vehicle in the chain C
1
(immediately after
the leading one) may collide or stop successfully. From both
of these states two possible cases spring as well, that is either
the following vehicle in the chain C
2
may collide or stop
successfully. And so on until the last vehicle in the chain
denoted by C
N
. At the last level of the probability tree there
are N + 1 possible outcomes (final outcomes) which represent
the number of collided vehicles, that is, from 0 to N possible
collisions. Observe that S
i,j
represents the state with i collided
vehicles and j successfully stopped vehicles.
The transition probability between the nodes of the tree is
the probability of collision of the corresponding vehicle in the
chain p
i
(or its complementary). These probabilities are crucial
to the model and will be calculated recursively, as described in
the next section. Let us note how every path in the tree from
the root to the leaves leads to a possible outcome involving
every vehicle in the chain. The probability of a particular path
results from the product of the transition probabilities that
belong to the path. Since there are multiple paths that may
lead to the same final outcome (a particular leaf node in the
tree), the probability of that outcome will be the sum of the
resulting probabilities of every possible path reaching it.
In order to compute the probabilities of the final out-
comes, we can construct a Markov chain whose state di-
agram is based on the previously discussed probability
tree. It is a homogeneous Markov chain with
(N+1)(N+2)
2
states, (S
0,0
, S
1,0
, S
0,1
, . . . , S
N,0
, S
N−1,1
, . . . , S
1,N −1
, S
0,N
).
The transition matrix P of the resulting Markov chain is a
square matrix of dimension
(N+1)(N+2)
2
, which is a sparse
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. X, NO. X, XX XX 4
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 vehicle
collision probabilities.
matrix, since from each state it is only possible to move to
two of the other states. For the sake of clarity, a brief example
with 2 vehicles is illustrated in Fig. 3.
Then, we need to compute the probabilities of going from
the initial state to each of the N + 1 final states in N
steps, which are given by P
N
. Therefore, the final outcome
probabilities are the last N + 1 entries of the first row of the
matrix P
N
.
Let Π
i
be the probability of reaching the final outcome
with i collided vehicles, that is, state S
i,N−i
. Then, Π
i
=
P
N
(1,
(N+1)(N+2)
2
− i). We obtain the average of the total
number of accidents in the chain using the weighted sum:
N
acc
=
N
X
i=0
i · Π
i
. (1)
IV. COMPUTATION OF THE VEHICLE COLLISION
PROBABILITIES
Computing the collision probabilities is the main problem
in our model. In this section we start from a deterministic
kinematic model and compute the collision probabilities when
different parameters of the kinematic model are considered
variables. The results are validated by Monte-Carlo simula-
tions. Hence, we start from a basic kinematic collision model
provided by [15], that can be summarized as follows.
Let l
i
represent the total distance traveled by vehicle C
i
since the emergency event occurs at time instant t
0
= 0
until the vehicle completely stops or collides with C
i−1
. Let
δ
i
be the time lapse that goes between the detection of the
emergency event until vehicle C
i
actually begins to slow
down. We call δ
i
the notification delay which models the
delay between the time instant t
0
= 0 and the instant the
driver of vehicle C
i
is aware of it and starts to brake. These
parameters are depicted in Fig. 4. The notification delay plays
an important role if we consider a communication system in
operation between the vehicles. In this case, we can assume
that the driver starts to brake when it receives a warning
message, so if the emergency event occurs at t
0
= 0 the
warning message is received at t = δ
i
by the vehicle C
i
.
However, we assume a more realistic case in which there
is also a reaction time before the driver actually starts to
brake. Therefore δ
i
= T
m,i
+ T
r,i
, where T
m,i
is the message
reception delay and T
r,i
is the driver reaction time.
Considering a constant deceleration a
i
, the distance needed
by vehicle C
i
to completely stop if it does not collide is given
by:
d
s,i
=
V
2
i
2a
i
+ V
i
δ
i
. (2)
However, when a collision occurs, the actual distance traveled
by the car, d
c,i
, is not given by (2) anymore, but one has to
consider the way the collision has occurred. For example, if a
vehicle crashes, its actual distance to stop is obviously shorter
than d
s,i
, as illustrated in Fig. 5, and also different when both
vehicles are still in motion when the crash occurs.
Let us remark at this point that (2) implies that a com-
munication system is in place and all vehicles start to brake
when they receive the message, independently of the behavior
of the preceding vehicles. Otherwise, drivers would start to
brake only when they sensed the braking of its nearest forward
neighbor as in a car-following approach [9], [10], so (2) would
become a function of the parameters of the preceding vehicle,
that is, d
s,i
= f (V
i
, V
i−1
, a
i
, a
i−1
, δ
i
, δ
i−1
) and the problem
would become more complex.
In all the cases the probability of collision of vehicle C
i
depends on the relationship between its distance to stop, d
s,i
,
the total distance traveled by the preceding vehicle, l
i−1
, and
the initial inter-vehicle space, s
i
. That is, when d
s,i
< l
i−1
+s
i
the vehicle is able to stop without colliding.
At this point we also assume another simplification: if two
vehicles collide we consider that they instantly stop at the
point of collision. This way we keep on assuming a worst case
evaluation. There are more realistic approaches, for instance,
to take into account the conservation of the linear moments to
compute the displacement due to the crash [7].
As can be seen from the previous equation, the number of
collisions depends on the vector of velocities V
i
, decelerations
a
i
, notification delays δ
i
, and inter-vehicle distances s
i
, which
we refer to as kinematic parameters. When all the parameters
are given, the model is completely deterministic. However,
we are interested in a more realistic case involving random
variability of the parameters. To study the influence of the
different parameters on collisions we introduce variability on
different model parameters as follows: for all the cases we
consider that s
i
is an exponentially distributed random variable
with parameter λ. This parameter represents the density of
vehicles on the road, defined as the average number of
vehicles per meter. Let us remark that s
i
can adopt a different
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. X, NO. X, XX XX 5
(a) Vehicle C
i
is able to stop successfully, then l
i
= d
s,i
.
(b) Vehicle C
i
collides with C
i−1
. In this case, the actual distance
covered by C
i
up to the collision is shorter than d
s,i
as given by (2).
Now it is l
i
= s
i
+ l
i−1
and depends on the distance covered by C
i−1
.
Fig. 5. The distance l
i
traveled by a vehicle when there is a collision (b) is
shorter than the distance needed by it to stop successfully (a), d
s,i
.
distribution and the following model is still valid. The reason
for this is that since s
i
is the inter-vehicle spacing when
the emergency event occurs, we can consider it independent
of the rest of parameters of the model, which means that
the following equations would be essentially the same, but
substituting the exponential probability density function by
the corresponding new one. We have selected an exponential
distribution because it simplifies the computations and it has
been shown that describes well inter-vehicle spacing when
traffic densities are small [14], whereas high traffic densities
show log-normal distributions [14].
Once we have described our collision model, we next derive
a basic model for the vehicle collision probabilities in which
all the parameters are constant except for the inter-vehicle
distance. Then, we extend the model by considering variable
the rest of the kinematic parameters. This way we can evaluate
the effects of the different parameters on the vehicle collision
model.
A. Basic model
Our first step is to evaluate the basic model, considering
all the parameters constant, except for s
i
, which is assumed
exponentially distributed. If a vehicle is able to stop without
colliding and the kinematic parameters are constant it always
travels the same distance d
s
. But if there is a collision, a
vehicle only travels the initial inter-vehicle distance plus the
distance 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
the following subsections we first compute this probability
exactly and then we provide an approximation that allows us to
simplify the computations when additional variable parameters
are considered in the model.
1) Case 1. Exact computation of collision probabilities
with constant kinematic parameters: In this case we compute
the collision probability exactly. 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.
3) The delay δ is the same for all drivers. It implies that
all the drivers receive the warning message at the same
instant.
Since V
i
, δ
i
and a
i
are constants, from (2) we obtain:
d
s
=
V
2
2a
+ V δ. (3)
For 1 ≤ i ≤ N , the collision probability will be computed
as follows:
p
i
= P (d
s
≥ l
i−1
+ s
i
) =
= P (l
i−1
+ s
i
≤ d
s
| l
i−1
≤ d
s
)P (l
i−1
≤ d
s
) +
+ P (l
i−1
+ s
i
≤ d
s
| l
i−1
> d
s
)P (l
i−1
> d
s
), (4)
where l
i−1
is a random variable that represents the distance
traveled by the preceding vehicle (assuming that l
0
= 0, since
vehicle C
0
stops instantly at x
0
= 0), and F is the cumulative
distribution function of the exponential distribution, exp(λ),
with λ the vehicle density (in ve h/m).
In this simple case, if vehicle C
i−1
does not collide then
neither does vehicle C
i
, because the velocity, the deceleration
and the reaction time are the same for both of them. Moreover,
if vehicle C
i−1
collides, it means that all of the preceding ve-
hicles have collided. From these observations we can conclude
that l
i−1
= s
1
+ s
2
+ . . . + s
i−1
∼ Erlang(i − 1, λ), and
P (l
i−1
+ s
i
≤ d
s
| l
i−1
> d
s
) = 0.
Now, we need to compute p
i
= P (l
i−1
+ s
i
≤ d
s
| l
i−1
≤
d
s
)P (l
i−1
≤ d
s
).
The joint probability density function of X = l
i−1
+ s
i
and
Y = l
i−1
is:
g(x, y) =
λ
2
(λy)
i−2
e
−λx
(i − 2)!
for 0 ≤ y ≤ x. (5)
So, the joint cumulative distribution function is:
G(x, y) =
Z
y
0
Z
t
0
λ
2
(λs)
i−2
e
−λt
(i − 2)!
ds dt+ (6)
+
Z
x
y
Z
y
0
λ
2
(λs)
i−2
e
−λt
(i − 2)!
ds dt =
=
γ(i, λy)
(i − 1)!
+
(λy)
i−1
(i − 1)!
(e
−λy
− e
−λx
), for 0 ≤ y ≤ x.
where γ is the incomplete gamma function, defined as
γ(a, x) =
R
x
0
t
a−1
e
−t
dt.
Finally, for 1 < i ≤ N it holds:
p
i
= P (l
i−1
+ s
i
≤ d
s
| l
i−1
≤ d
s
)P (l
i−1
≤ d
s
) =
=
G(d
s
, d
s
)
F
y
(d
s
)
· F
y
(d
s
) = G(d
s
, d
s
) =
=
γ(i, λd
s
)
(i − 1)!
+
(λd
s
)
i−1
(i − 1)!
(e
−λd
s
− e
−λd
s
) =
γ(i, λd
s
)
(i − 1)!
. (7)
At this point, if the metric of interest is the average number
of accidents, the procedure to obtain it is: once we have
computed the collision probability for each vehicle, we have to
construct the matrix P described on Section III. The next step
is to calculate the final outcome probabilities, Π
i
, and finally
the average number of accidents can be obtained through (1).
