# Physical models of traffic safety at crossings

Andreas Leich  Ronald Nippold  Andreas Schadschneider  Peter Wagner

###### Abstract

Traffic safety at intersections is studied quantitatively using methods from
Statistical Mechanics on the basis of simple microscopic traffic flow models.
In order to determine a relationship between traffic flow and the number of
crashes, the modelling focus is on the building block of any road network,
namely the crossing of two streams. In this paper, it is shown that the number
of crossing conflicts is proportional to the product of the two traffic flows
from which a simple model is developed. This model substantiates known
empirical findings. Since real crash data are obtained by an involved process
from such building blocks, there is a difference between the theoretical and
empirical results. This process is modelled here as well and narrows the gap
between theory and observation.

###### keywords:

Traffic safety , Conflict rates , Traffic models

††journal: Physica A

[dlr]organization=Institute of Transport Systems, German Aerospace
Centre,city=Berlin, postcode=12489, country=Germany

[uc]organization=Institute of Theoretical Physics, University of
Cologne,city=Cologne, postcode=50937, country=Germany

[tub]organization=Institute of Land and Sea Traffic, Technical University of
Berlin,city=Berlin, postcode=10553, country=Germany

By using physical models for the conflicts at two crossing streams of
vehicles, the relationship between the traffic flows of two crossing streams
and the number of conflicts can be mapped.

For small flows, a simple analytical model is in line with the simulation
results, while it seems different from empirical results.

While this might be due to human factors, this work instead models the step
from the simulation to reality as well. This leads to a narrowing of the gap
between simulation results and empirical results.

##  1 Introduction

The relation between the number of crashes and the number of vehicles at some
location of a road network is fundamental for road safety research. It has
been studied extensively in the traffic engineering literature [1, 2, 3, 4,
5], mostly by analyzing empirical data with regression models that lack a
physical or mechanistic model behind.

In addition, some theoretical studies, often by physicists, have investigated
the occurrence of accidents or dangerous situations in specific scenarios.
However, almost none of these investigations has done a comparison with
empirical data. They are often theoretical approaches that focus on rear end
collisions due to violations of the safety distance.

One of the first studies [6] considered the effect of a block, caused by a
crash, in a network. Later, a crash model was introduced [7, 8] that is based
on simple criteria derived from the motion rules of the model (often variants
of the Nagel-Schreckenberg model [9]). However, the ensuing crashes are never
executed explicitly. Instead, the simulation is continued as without an
accident. Therefore, the terminology “accident” has been replaced by
“dangerous situations” in later works. Such a procedure will lead to
correlations as it becomes rather likely that the criteria for a potential
crash are again fulfilled immediately after such a “dangerous situation”. This
leads to an overestimation of the number of crashes in such a setting,
especially since most works considered periodic boundary conditions.

Here we attempt to bridge the gap between theoretical and empirical studies
using concepts from Statistical Mechanics and simple microscopic traffic flow
models to obtain quantitative results that can be compared with empirical
data. This delivers a deeper insight into the emergence of road crashes and
the process of measuring crash rates.

The approach here is restricted to conflicts between two vehicles, which is
the overwhelming majority of all crashes (more than 90 % in urban areas), and
to the crossing of two streams of vehicles. In urban areas, more than 50 % of
all crashes are of this type.

###  1.1 Modelling Crashes using Cellular Automata

Cellular automata (CA) have been used to simulate traffic flow since the early
1990s [9], from simple set-ups where all vehicles run in a circle to fairly
complex network geometries, and to the simulation of different modes of
traffic (cars, bicycles, pedestrians). It is not easy to give a concise review
here, but see [10] (updated in [11]) for some more or less complete overview
of early work. Road traffic safety simulation has received less attention in
both the physics and transportation safety research communities. For a few
examples see [7, 12, 13, 14] and [15] for the traffic safety community. In
physics, especially the TASEP (Totally Asymmetric Simple Exclusion Process)
and related models have been used to study crossings, however usually focusing
on the traffic flow aspect and not on safety [16]. The reasons for this are
that those models are constructed crash-free and so it is not clear whether
any concept of traffic safety might apply here. Furthermore, crashes are rare
events. Therefore, it is difficult to pin down detailed reasons for each crash
that then can be modelled in a physically appealing manner.

Of special interest is the relationship between the number of crashes $N$ (per
time-interval) and the traffic flow $Q$, since this is the most important
factor that determines the number of crashes. In the following, capital
letters are used to name the variables as measured in traffic safety studies,
which is in most cases the long-term sum of the number of crashes $N$
(typically aggregated over one or more years and across different crash-
types), and the so called Average Daily Traffic $Q$ (often named ADT). An ADT-
flow is the average of the daily traffic of many days; in the following, to
avoid confusion, $Q$ is often named ADT-flow.

The simulation models below work on a shorter time-scale and in a fully
controllable environment. This produces similar variables $n$ (the number of
conflicts) and traffic flows $q_{1},q_{2}$ for a crossing. It is assumed that
the relationship found between $n$ and $q_{1},q_{2}$ can be transferred to the
macroscopic variables $N$ and $Q_{1}$, $Q_{2}$, or sometimes even to the total
flow $Q:=Q_{1}+Q_{2}$. This transformation is non-trivial, the section 3.2
will deal with it.

###  1.2 Empirical Examination of Crashes vs. Traffic Flow

The traffic safety community has come up with a lot of empirical work on this
topic, without reaching a clear conclusion. From a recent meta-analysis [5],
which assumed $N\propto Q^{\beta}$, at least an idea for the dependency of the
number of crashes $N$ on the ADT-flow $Q$ may be found. Here, $Q$ is the total
ADT-flow observed at a road or an intersection. In this case, the average of
the studies included in the meta-study leads to exponents $\beta=0.522$ for
single vehicle crashes, and $\beta=1.210$ for multi-vehicle crashes. Other
work dealing with intersections claim that $N\propto
Q_{1}^{\beta_{1}}Q_{2}^{\beta_{2}}$ (here $Q_{1},Q_{2}$ are the ADT-flows of
the crossing roads, one is the major flow, one is the minor flow), where
$\beta_{i}<1$ [17]. Note, however, that the power-law is at best an
approximation to small demand. For a larger demand, a kind of saturation might
occur [4, 18].

In general, traffic safety is modelled by so called safety performance
functions where it is assumed that the number of crashes $N$ depends on the
ADT-flows $Q_{i}$ and on other factors $x_{i}$ as follows [2, 3, 19, 20, 21]:

$N=\beta_{0}Q_{1}^{\beta_{1}}Q_{2}^{\beta_{2}}\exp\left(\sum_{i>2}\beta_{i}x_{i}+\xi\right)\\!\\!.$
(1)

Here, the $x_{i}$ are factors believed to influence $N$ as for instance the
speed-limit, intersection organization and the like, and the $\beta_{i}$ are
the coefficients that determine how strongly each $x_{i}$ influences the final
outcome. In the case of the flows, they are the exponents with which the
crash-rate grows as function of the $Q_{i}$. As mentioned already, the traffic
flows in this equation are the average daily traffic (ADT); these $Q_{i}$ are
the result of a sum over complex traffic flow time-series $q_{i}(t)$ which are
often not known exactly. In addition, in empirical work it is often difficult
to come up with reasonable and trustworthy values of $Q_{i}$ at all the places
where crashes occur. This is because, especially for small crash numbers,
there are issues with under-sampling, and even with under-reporting. And
finally, Eq. (1) is just a regression analysis where a detailed model in the
physical or mechanical sense is missing.

The noise-term $\xi$ is constructed so that the distribution of $N$ follows a
negative binomial distribution (NBD). The NBD is a generalization to the
Poisson distribution (PD). Its variance is larger than that of the PD with a
parameter $\gamma$ that describes how the variance $\sigma^{2}$ depends on the
mean value $\mu$ of the distribution. This is also named over-dispersion,
since the width of the crash-number distribution is larger than what could be
expected from a Poisson distribution. While $\sigma^{2}=\mu$ holds for PD, the
NBD has

$\sigma^{2}=\mu+\gamma\mu^{2},$ (2)

i. e. the variance increases quadratically with the mean-value. For
$\gamma=0$, the NBD is a PD. Eq. (2) is an exact relation that holds for any
NBD distribution.

The particular form in Eq. (1) is used since it is possible to fit real data
to the mean of $\log N$ by a generalized linear model (GLM), and rely on the
strong mathematical and statistical foundation of GLM’s. The downside is that
it is often difficult to check if this form is really consistent with the
data. Admittedly, doing so is not trivial, as Fig. 1 demonstrates (included
here only as an illustration). Fig. 1 displays the number of crashes versus
the ADT-flow $Q$ for a subset of roundabouts in Germany, together with a
fitted GLM. It can be seen that parameter values of the model fits are
strongly influenced by a few data-points for large $Q$, and that the bulk of
the data at smaller ADT-flow could be better described by a model where
$N\propto Q$.

Figure 1: Fit of a GL-model to crash data from German roundabouts. Plotted is
the number of crashes versus the total flow $Q$ (sum over the flows of all
approaches), both for roundabouts in urban areas and in rural areas (for more
details, see [22]).

###  1.3 Outline

This work is restricted to one of the building blocks of a road traffic
system: the crossing of two traffic streams. Each real-life intersection is
constructed of many conflict areas where two streams cross, see Figure 2 for a
visualization.

Figure 2: Scheme of the conflict areas of a 3-leg or T-intersection. The
crosses show the three crossing conflicts, the three rectangles rear-end
(sometimes named diverging conflicts), and the triangles could be both
(merging conflicts). Such a T-intersection has six possible flows.

The simplest 3-leg intersection has three areas where streams cross, three
areas where vehicles merge, and three areas where rear-end conflicts could
occur, each with its own pair of demands (this counts only the conflicts
between vehicles and ignores the ones between vehicles and pedestrians and is
true for one-lane roads only). This work concentrates on just one conflict
area (always named X in the following). With regard to traffic safety, this
simplifies the matter greatly to essentially two types of conflicts: the ones
that stem from the crossing itself, and the ones that are there due to the
fact that a crossing is a capacity bottleneck which causes more braking
compared to free flow, thus leading to rear-end conflicts in the approaching
links. And it makes it treatable by simple simulation models, so that some
complexity of the real-world can be dropped.

Note that empirical data are always a mixture of these and many more
conflicts-types. However, since crashes are rare, it is in most cases not
possible to discern the different conflict types and their dependency on
traffic flow and other external circumstances, so they are lumped together in
one crash number.

To connect models with empirical data, we concentrate on the dependency of the
number of crashes $N$ on flow only. We focus on the pure mechanics, without
considering the many human factors that may influence the crash probability.
To do so, two carefully crafted CA models of a conflict area X are used as
mechanistic models for the conflicts and their number, and the related number
of crashes.

##  2 A short theory and two simulation models

###  2.1 Crossing and rear-end conflicts

From the set-up above, two simple considerations might be put forward. Given
are two streams of traffic objects with volumes (traffic flows) $q_{1},q_{2}$
wanting to cross a conflict area of size $L_{1}\times L_{2}$. The streams are
assumed to be statistically independent, which means that each flow $q_{i}$
emits with a certain probability $p_{i}=q_{i}\Delta t$ a vehicle at random,
where $\Delta t$ is the time-step size. Each vehicle has a length $\ell_{i}$
(they can be made different, but here it is assumed that they are equal for
all of them), and that means that it needs the time
$\tau_{i}=(L_{i}+\ell_{i})/v_{i}$ to cross X, where $v_{i}$ is the speed right
at X. The average time between two vehicles is the gross time headway
$T_{i}=1/q_{i}$. From the assumption of independence the following deduction
holds for the probability $\Pi$ that these two end up being in conflict
($L_{i}=\ell_{i}=\ell$ has been set to simplify the resulting expressions):

$\Pi_{\text{X}}=\frac{\tau_{1}}{T_{1}}\frac{\tau_{2}}{T_{2}}=4\ell^{2}\frac{q_{1}\,q_{2}}{v_{1}\,v_{2}}$
(3)

Here, the subscript ’X’ has been used to indicate crossing conflicts. The same
can be done for the conflicts caused by rear-end interactions (subscript
’RE’):

$\Pi_{\text{RE}}=\frac{\tau_{1}}{T_{1}}\frac{\tau_{1}}{T_{1}}+\frac{\tau_{2}}{T_{2}}\frac{\tau_{2}}{T_{2}}=4\ell^{2}\left(\left(\frac{q_{1}}{v_{1}}\right)^{2}+\left(\frac{q_{2}}{v_{2}}\right)^{2}\right)$
(4)

For completeness, the total number of conflicts may then be written as:

$\Pi=4\ell^{2}\left(\frac{q_{1}\,q_{2}}{v_{1}\,v_{2}}+\left(\frac{q_{1}}{v_{1}}\right)^{2}+\left(\frac{q_{2}}{v_{2}}\right)^{2}\right)$
(5)

The notable property of these equations is that they connect the probability
for conflict with two traffic flow parameters – the speed and the flow.
Clearly, an alternative formulation would be to use the densities
$\rho_{i}=q_{i}/v_{i}$; this is not done here, since in traffic safety speeds
are typically not available, except for freeway data.

For an intersection, there is of course a complicated relationship between
these four variables, which in most cases can only be mapped by simulations,
especially for larger flows. In addition, it depends on the organization of
the intersection, in this paper the symmetric “first come first serve” (FCFS)
will be used, but others can be considered as well.

###  2.2 From conflicts to crashes

This approach might also be a good starting point for more detailed crash
models. The simplest one, which is used here, states that the actual crash
probability $\pi$ is simply a constant factor $\alpha$ to be multiplied with
$\Pi$, where $\alpha$ is a very small number (e. g. $\alpha=10^{-3}\ldots
10^{-6}$), and it might be different for crossing as well as for rear-end
conflicts:

$\pi=\alpha\Pi$ (6)

Note, too, that the traffic safety research community’s predominant models for
a whole intersection [17, 23] are not exactly the same as Eq. (5):

$N=\pi T_{\text{obs}}\propto Q_{1}^{\beta_{1}}Q_{2}^{\beta_{2}}$

Here, $T_{\text{obs}}$ is an observation period. To our knowledge, speeds have
not been used in such models apart from a general speed-limit that may or may
not be in place at an intersection. With the exception of freeway data, speeds
are almost never available for intersections.

More complicated dependencies of $\alpha$ on the traffic state might be
considered, but this means to enter the realm of human factors. This will not
be done here. Instead, we concentrate on a better understanding of the
relationship between $\Pi,\pi$ and the traffic state described by
$q_{i},v_{i}$. If $\alpha$ is in fact a constant, then the number of crashes
is directly proportional to the number of conflicts, and therefore it suffices
to regard the latter ones instead of real crashes. From Eq. (5) it might be
expected that $n\propto q_{1}q_{2}+q_{1}^{2}+q_{2}^{2}$ is the right
relationship between $n$ and $q$. As will be seen in the results section, this
is only true for small flows, as long as the speed $v$ is mostly independent
of the traffic flow $q$. For larger flows, the interaction between the objects
gains influence and changes this simple picture.

###  2.3 Models

Two models have been chosen here, the CA model with maximum speed
$v_{\text{max}}=2$ (measured in sites/time-step, where the length of a site is
the generalized vehicle length of $\lambda=7.5\,m$, and the TASEP (totally
asymmetric simple exclusion process) with parallel update. The time-step
$\Delta t$ has been chosen to be 1 second for the CA, and $\Delta t=0.5\,s$
for the TASEP, respectively, which leads to the same speed of 15 m/s when
expressed in real-world units. Each stream is a set of vehicles that are
described by their position as well as their speed,
$\\{(x_{i}(t),v_{i}(t))\\}_{i=1,\ldots,K(t)}$. The number of objects $K(t)$ in
each stream is time-dependent. Open boundary conditions are used: Objects try
to enter with a certain rate $p_{i}\Delta t$ and can enter only if there is a
free space at the beginning of the link. Therefore, the flow that is finally
achieved and that determines the number of crashes is always smaller or equal
to $q_{i}\leq p_{i}\Delta t$. The downstream boundary is completely open
without any restriction on leaving. This outflow is what is named flow $q_{i}$
in the following, and this is the variable that determines the conflict-rates
in Eq. (5). The crossing site is located $v_{\text{max}}$ sites upstream of
the last site, this helps to catch the interaction between vehicles
approaching X and the ones leaving X.

Choosing these two models has an advantage over more complex modelling
approaches (like continuous models as those implemented in micro-simulation
software): A conflict can be determined unambiguously and therefore can be
counted. The speed and position update rules, which are repeated here for the
sake of completeness ($g$: number of empty cells in front,
$v,v^{\prime},\hat{v}$: current, updated, and intermediate speeds,
respectively, rand() a random-number in $(0,1]$, and $p_{\text{brake}}=0.25$ a
randomization probability) are:

$\displaystyle\hat{v}=\min\\{v+1,v_{\text{max}},g\\},$ (7) $\displaystyle
v^{\prime}=\left\\{\begin{array}[]{ll}\max\\{0,\hat{v}-1\\}&\text{if}\,\,\text{rand()}<p_{\text{max}}\\\
\hat{v}&\text{else}\end{array}\right.,$ (10) $\displaystyle
x^{\prime}=x+v^{\prime}.$ (11)

They have to be supplemented with rules that are applied to X, which will be
stated below.

The TASEP update rule is just the one of the CA with $v_{\text{max}}=1$. Some
simulation experiments have also been performed with a modified randomization
rule for the TASEP (where the probability to move after standstill increases
with standing time), but they yield very similar results and so are only
mentioned here to confirm the robustness of the results below.

The conflicts are handled as follows: The crossing area X has an own discrete
variable, which can be either free, or a block to one of the two links. For
the TASEP (starting from an unblocked state), in each time-step it is checked
whether there is an object on the two sites upstream of X. If just one of
these two sites is occupied, the other TASEP’s site X is blocked. This
blocking state is retained until the first object has cleared X, which needs
at least two time-steps. If both sites upstream of X are occupied (no matter
the speed of the two objects), then this is recorded as a conflict, and it is
resolved by deciding randomly which object is allowed to run first. This
conflict is counted, and the respective exit times are recorded as well – they
will be used in the subsequent analysis below.

For the CA, the handling of conflict area X is more complicated: Here, all
sites that can reach X in the next time-step are checked for the presence of
objects and counted if at least one is present. If on one link the count is
larger than zero and the other is zero, then the other link gets blocked. If
there are objects on both links that can reach X in the next time-step, then
it is decided randomly which one moves first towards X. And, as in the case of
the TASEP, it is counted as a conflict. Again, the block is removed after the
object has cleared X completely; this may happen in this case in one time-
step.

Figure 3: Space-time diagram of one link of CA objects approaching conflict
site X, which is located at site 60. Here, X corresponds to one of the
crossing points in Figure 2. Note that $v_{\text{max}}$ additional sites
downstream of X have been included to catch the interaction between a vehicle
approaching X and the one leaving X. Each point is colored according to the
speed.

A typical space-time diagram for one link is displayed in Figure 3.

All simulations have been run for at least $10^{5}$ time-steps per traffic
flow value. For small demand, typically more time-steps are needed to reach a
reasonable number of events: In this case, the simulation is run until at
least 100 conflicts had been recorded.

The rear-end conflicts are counted differently. For the TASEP, any interaction
where the following vehicle is blocked by a leading vehicle is counted as a
conflict. For the CA model, all interactions where the speed was adjusted to
the distance ahead, i. e. where $\hat{v}=g$ had been applied, are counted as a
conflict. This includes two standing objects, since in principle the follower
could move and crash.

Clearly, at least for the CA-model a kind of severity might be assigned to
each conflict, depending on the speed-difference between them. Again, for the
sake of simplicity, this work refrains from such additional complications.

##  3 Results

###  3.1 Conflicts versus flows

The result of the simulations are several functions $f(q_{1},q_{2})$ that
characterize the system’s response. Here, $f$ are chosen to be the average
speed, the speed difference $\Delta v=v_{1}-v_{2}$ between the two links, the
number of crossing conflicts $n_{\text{X}}(q_{1},q_{2})$ and the number of
rear-end conflicts $n_{\text{RE}}(q_{1},q_{2})$. Plots of those quantities are
displayed in Figure 4 for the CA model. In the following, a normalized
conflict rate will be used: All numbers of conflicts are divided by the length
of the time-interval within which they occurred, while the rear-end conflicts,
in addition, are divided by the number of cells used in the simulation,
therefore giving a rate per cell. The normalized conflict rates are named as
$r_{\text{X}}(q_{1},q_{2})$ and $r_{\text{RE}}(q_{1},q_{2})$, respectively,
and the sum of both is the conflict rate $r(q_{1},q_{2})$.

Figure 4: The average speed $v$ (top left), the speed difference $\Delta
v=v_{1}-v_{2}$ (top right), and the two crash-rates $r_{\text{X}}$ (lower
left) and $r_{\text{RE}}$ (lower right) as function of the two flows
$q_{1},q_{2}$ for the CA model.

In Eq. (5), conflict rates are simple functions of the traffic state $z$,
where $z_{\text{X}}=q_{1}q_{2}/(v_{1}v_{2})$ for the crossing conflicts, and
$z_{\text{RE}}=(q_{1}/v_{1})^{2}+(q_{2}/v_{2})^{2}$ for the rear-end
conflicts. Therefore, Figure 5 displays the rates versus the two $z_{x}$
variables for the two simulation models, together with a linear fit of $r_{x}$
versus $z_{x}$ (the fit is done with weights of $1/r_{x}^{2}$ to favor small
values of $z$). This shows that the considerations above are very well in line
with the simulation results.

Figure 5: The two rates $r_{\text{X}}$ (top cloud of points) and
$r_{\text{RE}}$ (bottom cloud of points) are plotted versus the variables
$z=z_{\text{X}}$ or $z_{\text{RE}}$, respectively. They follow the theoretical
approach quite nicely. The color of the points indicates the speed, the red
(for $z_{\text{RE}}$) and violet (for $z_{\text{X}}$) lines are linear fits.
The figure on top is from the TASEP, the one on the bottom from the CA-model.

For medium and large values of $z$, deviations can be seen: The number of
crossing conflicts saturates, while the number of rear-end conflicts first
grows stronger than in theory, but seems to saturate as well when approaching
the capacity of the link.

###  3.2 Empirical analyses and the Average Daily Traffic

The results so far are not completely in line with the empirical results. The
main differences are: The exponents of the $N(Q)$ relationship in empirical
data are smaller than in the simulation, and the distribution of crash numbers
is different from the Poisson distributions obtained in the simulation. This
could be the already mentioned negative binomial (NBD), or any other
distribution with over-dispersion. Therefore, a closer look is needed how the
empirical results are obtained, given the rarity of real crashes.

One data-point $(Q_{k},N_{k})$ in an empirical analysis is obtained e. g. by
the number of crashes $N_{k}$ over a certain time-period (typically of the
order of months and years) at a certain place $k=1,\ldots,K$, while the
corresponding ADT-value $Q_{k}$ in ideal cases is counted over the same
period. More often, it is obtained by short-term counts extrapolated, or it is
based on a travel demand model. Even in good conditions, each $Q_{k}$ is a sum
over all the different traffic flows (enumerated by $\nu$) in this place
(think of the six streams at the three-leg intersection above), and it
averages over time $t$:

$Q_{k}=\frac{1}{T}\sum_{t,\mu}q_{k}^{\nu}(t)$ (12)

The daily curves $q_{k}^{\nu}(t)$ display a wide range of shapes, one example
could be seen in Figure 6 with data from one detector in Berlin, the data-set
has been obtained from [24].

Figure 6: Scaled flow values (gray values) of all flow values, and ten daily
courses $\hat{Q}_{i}(t)$ (lines) for one typical loop detector in Berlin.

Similar data-sources can be found elsewhere [25]. It is assumed that there
will be no principal difference between the data from Berlin and data from
other places.

This sampling process can be modelled as well, with a few assumptions about
the form of the various distributions needed:

  * 1.

The distribution of $q_{k}(t)$-values that are used to compute $Q_{k}$ is
simply a uniform distribution in $[0,2\,Q_{k}]$.

  * 2.

The distribution of the $Q_{k}$-values itself is again a uniform distribution,
this time in the interval $10^{3},5\cdot 10^{4}$ vehicles/day.

  * 3.

From this, artificial crashes are generated by setting a crash probability
$p=\alpha q^{\beta}$, resulting in a reasonable number of collisions.

Doing this for a simulated for 1000 days and $K=1000$ places, the resulting
$N(Q)$-curve still reproduces the exponent and the crash-number distribution
is $P$ (see Fig. 7, left plot).

Figure 7: Simulated measuring process. Left, without error, the red line is a
fit to a model with $Q^{\beta}$ and a Poisson distribution. The plot on the
right is the same crash data, in this case all $Q$-values have been shuffled
as described in the text. Now, the exponent has changed. The red line is again
a fit with $Q^{\beta}$, as is the green line. The red line is from a Poisson
distribution, while the green is for a negative binomial one.

Several approaches have been tried, the by far the simplest one is to assume
that the $Q_{k}$-data are strongly disturbed. By multiplying each $Q$-value
with a number drawn from a uniform distribution in $[0.25,1.75]$ and then
repeating the fitting process, the exponent changes, and the distribution
switches to an over-dispersed one.

The result of this procedure is shown in Fig. 7. It is very likely that in the
real world more complicated processes are in place that modify the Poisson
distribution at the heart of this process, but for this paper the simplest
possible explanation is used.

##  4 Conclusions

This research has tried to find a relationship between the traffic flow and
the number of crashes at the crossing of two traffic streams. This is achieved
by investigating the behavior of two simulation models and therefore yields a
mechanism that can explain some of the empirical observations. It turned out
to be important to make a distinction between the microscopic picture drawn by
the simulation, which leads to a relationship between the traffic flows
$q_{i}$ and the number of conflicts $n$, and the macroscopic picture from
empirical results, which yields a similar relationship between the macroscopic
ADT-flows $Q_{i}$ and the number of crashes $N$.

The simulation results confirm a theoretical approach that shows that the
number of conflicts behaves as simple functions of the traffic state, see Eq.
(5). The main result is that especially for small flows, the number of
conflicts $n$, and with it the number of crashes, should in fact be $n\propto
q_{1}q_{2}$ for crossing conflicts, and $n\propto q_{1}^{2}+q_{2}^{2}$ for
rear-end conflicts. For large flows, more complicated behavior becomes
visible: For intermediate flows, the crash-rate seems to grow stronger than
$n\propto q_{1}q_{2}$, while at very large flows close to capacity, saturation
becomes visible. The increase is due to an increased number of interactions
caused by the crossing, while the final saturation might be due to a decrease
in speeds when approaching capacity. This saturation effect has so far only
been found in a small number of empirical studies, so this study lends further
support to it. Furthermore, it provides a complete picture of the number of
conflicts in relationship to the traffic state in general.

There are differences between the simulation results and the empirical
observations. While $n\propto q_{1}q_{2}$ for the simulation results, and the
number of crashes derived from the number of conflicts must follow a Poisson
distribution, real data have $N\propto Q_{1}^{\beta_{1}}Q_{2}^{\beta_{2}}$
with exponents $\beta_{i}<1$ and they follow a negative binomial distribution.
Therefore, an observation model has been investigated to try to understand how
to do the transition from the microscopic variables $(n,q_{i})$ to the
macroscopic ones $(N,Q_{i})$. It turns out that the simplest explanation we
have found so far is to assume that the ADT-flows $Q_{i}$ that enter into the
empirical analyses are strongly disturbed by a number of processes. E. g., as
the example of one loop detector in Fig. 6 shows, even what is associated with
one $Q_{i}$-value carries considerable variance. Furthermore, in reality the
process how the $Q_{i}$ are measured (or estimated) is prone to errors. A
second problem is with the modelling in typical traffic safety analyses; they
are too strongly bound to equations where a GLM can be used. Eq. (5) is not of
this type, and the saturation effects seen in the simulation results are also
not very well described by the power-law approach of Eq. (1). The approach
here shows that at least for small traffic flows a mechanistic explanation is
possible that we deem better than the simple regression used in most empirical
studies.

Note that there is an additional twist in the transformation from the
simulation variables to the empirical ones: Since the $Q_{i}$ are average
values, they do a poor job in sampling maximum values – the $Q_{i}$ will never
reach the areas where there are strong deviations from the low-flow behavior.
However, the $N$ do sample from the whole area of flow values that make up
$Q_{i}$, further muddling the picture. The analyses where saturation have been
seen work in fact with short-term flows (e. g. hourly values), and not with
the ADT-values of all the other analyses. Which is a nice explanation why some
studies do in fact see saturation, while others do not.

Finally, as pointed out already, the crash probability (named $\alpha$ in the
description above) itself could be a function of the traffic state and
transform the relationships found here by simulation: One effect is when
speeds of the vehicles decrease, then at least the severity of the crashes
decreases, and one may argue that slower speeds give drivers a longer time-
span to react, so the number of crashes might go down with speed as well. This
paper has not investigated this avenue, which might be better left to human
factors experts, and it will come on top of the simple model explored here.

Clearly, it might be interesting to see whether the approach here holds also
for different intersection organizations and for more complicated models,
which will be the topic of future work. In addition, the approach here makes a
prediction about the ratio between rear-end and crossing conflicts as a
function of the traffic flow, which is in principle testable with real data.

## CRediT author statement

Andreas Leich: Conceptualization, Investigation, Methodology, Writing-
Reviewing and Editing Ronald Nippold: Methodology, Writing- Reviewing and
Editing Andreas Schadschneider: Conceptualization, Investigation , Writing-
Reviewing and Editing Peter Wagner: Conceptualization, Methodology, Software,
Writing- Original draft preparation, Writing- Reviewing and Editing

## Declaration of interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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