# Theoretical predictions for vehicular headways and their clusters

Milan Krbálek  Department of Mathematics, Faculty of Nuclear Sciences and
Physical Engineering
Czech Technical University in Prague, Prague – Czech Republic

###### Abstract

This article presents a derivation of analytical predictions for steady-state
distributions of netto time gaps among clusters of vehicles moving inside a
traffic stream. Using the thermodynamic socio-physical traffic model with
short-ranged repulsion between particles (originally introduced in [Physica A
333 (2004) 370]) we firstly derive the time-clearance distribution in the
model. Consecutively, the statistical distributions for the so-called time
multi-clearances are calculated by means of theory of functional convolutions.
Moreover, all the theoretical surmises used during the above-mentioned
calculations are proven by the statistical analysis of traffic data. The
mathematical predictions acquired in this paper are thoroughly compared with
relevant empirical quantities and discussed in the context of three-phase
traffic theory.

###### keywords:

vehicular traffic , headway distribution , socio-physical traffic model

###### PACS:

05.40.-a , 89.40.-a , 05.45.-a

††journal: Physica A

##  1 Introduction

Explorations of traffic micro-quantities (i.e. quantities belonging to
individual vehicles) and efforts to predict their statistics are as old as
traffic research itself. Virtually, all important reviews on traffic science
(for example [1, 2, 3], or [4]) try to explain at least some basic knowledge
on statistical distributions of time-intervals or distance-gaps among moving
cars. Although the theoretical prediction of time-evolution for headway-
distributions is still extremely vague, some partial achievements have been
reached in the last years (for example [5, 6, 7, 8, 9, 10, 12, 13, 14, 15,
16], or [17]). We try to pick up the threads of those results and get closer
to the heart of the matter.

This contribution is focused predominantly on the time-evolution of netto time
intervals (so-called _time-clearances_) between succeeding cars passing a
given point (a traffic detector, typically) located at an expressway.
Investigations of the traffic clearances are significantly advantageous
(contrary to explorations of distance-gaps) because of their direct
measurability. Indeed, the most of traffic detectors gauges the time of
vehicle’s passage directly, which means that those data do not show any
systematic or mediated errors (provided the detector is not damaged). Except
the time-clearances we concentrate our attention on the so-called _multi-
clearances_ , i.e. cumulated time-clearances among $n$ consecutive cars. By
analogy, such a quantity is (similarly to time-clearance) directly measurable,
which opens a possibility for detailed statistical investigations of multi-
clearance distributions with respect to the location of traffic ensemble in
the fundamental diagram.

In fact, the empirical and/or theoretical investigations of traffic multi-
headways are not sporadic in the physics of traffic. Some of the previous
scientific works, e.g. [18, 19, 20, 21], or [22], briefly analyze the multi-
clearance distributions (predominantly from the empirical point of view) or
their statistical variances. The reasons for investigations of such a type are
obvious, since deeper comprehension of changes in traffic microstructure will
provide a more thorough insight into the convoluted traffic interactions.

Figure 1: Mean modified velocity w (and variance) as a function of traffic
density. In the main part of the figure there is displayed the average value
$\bar{w}$ which is drawn separately for the fast and main lanes. The
fluctuations of velocity quantified by the velocity-variance
$\mathtt{VAR}(W)=n^{-1}\sum_{k=1}^{n}(w_{k}-\bar{w})^{2}$ depending on traffic
density are visualized in the inset.

##  2 Preliminary evaluation of empirical observations

The vehicular-data records analyzed in this article are typically of the
following types. The set

$T^{(\mathrm{in})}_{\ell}=\bigl{\\{}\tau^{(\mathrm{in})}_{k\ell}\in\mathbb{R}^{+}|~{}k=1,2,\ldots,N\bigr{\\}},\quad\quad\bigl{(}\ell\in\\{0,1,2\\}\bigr{)}$
(1)

includes the chronologically-ordered times when the front bumper of $k$th car
has intersected the detector-line (located at $\ell$th lane of an expressway).
The respective times $\tau^{(\mathrm{out})}_{k\ell}$ when the rear bumper of
$k$th car has intersected the detector-line are summarized in the set
$T^{(\mathrm{out})}_{\ell}.$ Analogously, the set of velocities of individual
vehicles recorded by the detector is

$V_{\ell}=\bigl{\\{}v_{k\ell}\in\mathbb{R}^{+}|~{}k=0,1,2,\ldots,N\bigr{\\}},\quad\quad\bigl{(}\ell\in\\{0,1,2\\}\bigr{)}.$
(2)

The lengths of vehicles are denoted as $d_{k\ell}$ and summarized in the set
$D_{\ell}.$ We remark that the indices $\ell=0,1,2$ correspond to the slow,
main, and overtaking lanes of a freeway, respectively. The slow lane is
intended predominantly for long vehicles (trucks or lorries) and therefore the
respective data will not be considered for this research. On contrary, the
drivers use the third lane $(\ell=2)$ particularly if they are overtaking (and
after overtaking-manoeuvre they turn back in the main lane), or if they are
significantly faster that other cars. All of the above-mentioned quantities
are (for purposes of this article) considered to be primary, which means that
they are directly detecable by the traffic detectors.

Except the primary traffic quantities we now introduce some important
secondary quantities, whose values are obtained vicariously, i.e. they are not
included in the traffic-detector’s records. The cardinal secondary traffic
micro-quantities are _the time-headways_

$z_{k\ell}:=\tau^{(\mathrm{in})}_{k\ell}-\tau^{(\mathrm{in})}_{(k-1),\ell}$
(3)

and _time-clearances_

$t_{k\ell}:=\tau^{(\mathrm{in})}_{k\ell}-\tau^{(\mathrm{out})}_{(k-1),\ell}.$
(4)

Although both of them are not explicitly included in traffic-data files, their
values are not burdened with any additional error. Above that, brutto space-
gaps between successive vehicles (usually called as _the distance-headways_)
are traditionally approximated by the relation
$s_{k\ell}:=v_{k\ell}~{}z_{k\ell}$ that presupposes the constant velocity
$v(\tau)=v_{k\ell}$ during a time period when
$\tau\in[\tau^{(\mathrm{in})}_{(k-1),\ell},\tau^{(\mathrm{in})}_{k\ell}].$ As
well known, such a precondition is questionable, especially in the region of
small traffic densities where the time-headways are too large. However, the
influence of a possible error is of marginal importance, as apparent from the
fact that the headway distributions analyzed in small-density regions do not
show any noticeable deviation from exponential distribution expected for
infrequent events (see [11, 2], or [6]). Similarly, _the distance-clearance_
is calculated via $r_{k\ell}:=v_{k\ell}~{}t_{k\ell}$ and represents the
estimated netto distance between $k$th car and its predecessor. Contrary to
the time-headways (clearances) the distance-headways (clearances) are burdened
by the systematic error that is discussed above. Such a error devalues the
knowledge on distance-headway distributions and their evolution.

Denoting the sampling size by $m$ (in this study there is consistently
considered $m=50$) and number of data-sets by $M_{\ell}$ (which therefore
implies that $M_{\ell}=\lfloor N_{\ell}/m\rfloor$, where
$N_{\ell}=\bigl{|}\bigl{\\{}\tau^{(\mathrm{in})}_{k\ell}|~{}\ell\text{ is
fixed}\bigr{\\}}\bigr{|}),$ one acquires the main data-samples

$S^{(\mathrm{main})}_{j}=\Bigl{\\{}\bigl{(}\tau^{(\mathrm{in})}_{k\ell},\tau^{(\mathrm{out})}_{k\ell},v_{k\ell},d_{k\ell}\bigr{)}\in
T^{(\mathrm{in})}_{1}\times T^{(\mathrm{out})}_{1}\times V_{1}\times
D_{1}|~{}\\\ k=(j-1)m+1,(j-1)m+2,\ldots,jm~{}\wedge~{}\ell=1\Bigr{\\}},$ (5)

and the secondary data-samples

$S^{(\mathrm{fast})}_{i}=\Bigl{\\{}\bigl{(}\tau^{(\mathrm{in})}_{k\ell},\tau^{(\mathrm{out})}_{k\ell},v_{k\ell},d_{k\ell}\bigr{)}\in
T^{(\mathrm{in})}_{2}\times T^{(\mathrm{out})}_{2}\times V_{2}\times
D_{2}|~{}\\\ k=(i-1)m+1,(i-1)m+2,\ldots,im~{}\wedge~{}\ell=2\Bigr{\\}},$ (6)

where $j=1,2,\ldots,M_{1}$ and $i=1,2,\ldots,M_{2}.$ For each data-sample
$S^{(\mathrm{main})}_{j}$ (or $S^{(\mathrm{fast})}_{i}$ alternatively) we
calculate the local flux

$J_{j}=\frac{m}{\tau^{(\mathrm{out})}_{jm,\ell}-\tau^{(\mathrm{in})}_{(j-1)m+1,\ell}}$
(7)

and local average velocity
$\bar{v}_{j}=m^{-1}\sum_{k=(j-1)m+1}^{jm}v_{k\ell}.$ The local density
$\varrho_{j}$ is then estimated (as suggested in [2]) via the fluid-dynamic
equation

$\varrho_{j}=\frac{J_{j}}{\bar{v}_{j}}.$ (8)

Such a expression is understood as one of the approximations suitable for
estimation of vehicular density. We add that the incorrectness of the
definition (8) is caused by the mixing of time and spatial averaging. Besides
the macroscopic description of each sample $S_{j}$ we now introduce the mean
time-clearance (or distance-clearance) by means of definitions

$\bar{t}_{j}=\frac{1}{m}\sum_{k=(j-1)m+1}^{jm}t_{k\ell},\quad\quad\bar{r}_{j}=\frac{1}{m}\sum_{k=(j-1)m+1}^{jm}r_{k\ell},$
(9)

respectively.

Figure 2: Correlation coefficient $\omega_{wx}$ as a function of traffic
density. The value $\omega_{wx}$ quantifies a statistical correlation between
the modified vehicular velocities $w_{k\ell}$ and scaled gaps $x_{k\ell}$ to
the preceding car (in the regions of fixed traffic densities). The squares
represent cars moving in the main lane, whereas circles correspond to the
fast-lane cars. Note that the vehicles in the fast lane show the stronger
correlations than others. Such a tendency is accented in the regions where the
mental strain of drivers is stronger (i.e. in the regions of over-
saturations). Vertical abscissae demonstrate a degree of fluctuations
(quantified by the standard deviation) in $\omega_{wx}$ for main-lane data
analyzed from different data sources.

Since aiming to investigate the essential properties of micro-distributions
(i.e. statistical distributions of microscopic traffic quantities) we
eliminate (in the next part of this text) the global trends in those
distributions. Predominantly, we eliminate the changes of the average headways
caused by the varying traffic density. Such an approach corresponds to the
technique examined in the articles ([8, 11, 9, 21], or [17]), and allows more
sophisticated comparison for different traffic regimes or for traffic data
originated from different countries. The procedure of the headway re-scaling
(see the text below) represents in fact the trivial variant of the so-called
_Savitzky–Golay smoothing filter_ (for details please see the Ref. [23])
applied to the matrix spectra in the Random Matrix Theory (see [24]), for
example. Thus, we define the _scaled distance-clearances_ (for the sample
$S_{j}$) as

$x_{k\ell}=\frac{r_{k\ell}}{\bar{r}_{j}}\quad\text{for all}\quad
k\in\bigl{\\{}(j-1)m+1,(j-1)m+2,\ldots,jm\bigr{\\}}$ (10)

and the _modified velocities_ (representing a local traffic flow, in fact) as

$w_{k\ell}=\frac{v_{k\ell}}{\bar{r}_{j}}\quad\text{for all}\quad
k\in\bigl{\\{}(j-1)m+1,(j-1)m+2,\ldots,jm\bigr{\\}}.$ (11)

It implies that the mean clearance in each sample is re-scaled to the unit. In
the Fig. 1 we demonstrate the changes of averages of the modified velocities
$w_{k\ell}.$

For intentions of analytical calculations executed in the section 3 it is now
necessary to verify a statistical independence (or negligible dependency)
between individual spatial clearances and car velocities. The standard way how
to inspect this property is to investigate the associated correlation
coefficient. Denoting $X=\\{x_{1},x_{2},\ldots,x_{n}\\}$ and
$W=\\{w_{1},w_{2},\ldots,w_{n}\\}$ the sets of scaled distance-clearances or
modified velocities respectively, one can calculated the Pearson’s correlation
coefficient using the relation

$\omega_{wx}=\frac{\mathtt{COV}(X,W)}{\mathtt{VAR}(X)\cdot\mathtt{VAR}(W)}=\frac{\sum_{k=1}^{n}(x_{k}-1)(w_{k}-\bar{w})}{\sqrt{\sum_{k=1}^{n}(x_{k}-1)^{2}}~{}\sqrt{\sum_{k=1}^{n}(w_{k}-\bar{w})^{2}}},$
(12)

where $\bar{w}=\sum_{k=1}^{n}w_{k}/n.$ Such a coefficient
$\omega_{wx}\in[-1,1]$ represents a measure for stochastic dependency between
$X$ a $W,$ so that the values being close to zero indicate a weak correlation.

Dividing the complete data-sample into the sub-samples of almost-constant
traffic density we demonstrate in the Fig. 2 that coefficient $\omega_{wx}$ is
localized in the range $[0.05;0.30]$ and the corresponding values depend on a
traffic phase (in the three-phase-traffic interpretation) and freeway lane.
However, in all density-intervals the Pearson’s correlation is restrained,
which means that the hypothesis on a weak dependence between individual
clearances and velocities seems to be legitimate.

##  3 Mathematical derivation of time-clearance distribution

As introduced in [8], justified in [14], and elaborated in [11], one of the
possible ways how to acquire theoretical predictions for distributions of
empirical distance-clearances is to use the socio-physical traffic model whose
elements are repulsed by the short-ranged power-law forces and randomized by
the socio-physical noise, which is supposed to be of thermal-like nature. The
influence of such a thermodynamical component to the model can be
increased/reduced by the socio-physical coefficient $\beta\in[0,\infty)$
reflecting a mental strain under which the drivers are during a given traffic
situation. Specifically, in the free traffic regimes (where the psychological
pressure caused by the traffic situation is weak) the parameter $\beta$ is
almost zero. On contrary, for the congested phase (where interactions among
drivers are reinforced) the mental strain coefficient $\beta$ is large. For
the detailed changes of $\beta$ one can inspect the articles [8, 11], or [21].

To be specific, we consider dimensionless particles moving along a ring whose
velocities are $w_{1},w_{2},\ldots,w_{N}$ and mutual distances (between
subsequent particles) are $x_{1},x_{2},\ldots,x_{N}.$ Introducing the short-
ranged repulsive potential

$U(x_{1},x_{2},\ldots,x_{N})=\sum_{k=1}^{N}\frac{1}{x_{k}}$ (13)

and socio-physical hamiltonian (see [11])

$\mathcal{H}(w_{1},w_{2},\ldots,w_{N},x_{1},x_{2},\ldots,x_{N})=\frac{1}{2}\sum_{k=1}^{N}\bigl{(}w_{k}-w_{\mathrm{d}}\bigr{)}^{2}+U(x_{1},x_{2},\ldots,x_{N})$
(14)

one can derive (for details please see [11]) that velocities of such a
ensemble (analyzed in the steady state) are gaussian-distributed, i.e. the
associated probability density reads

$q(w)=\frac{1}{\sqrt{2\pi}\sigma}\mathtt{e}^{-\frac{(w-w_{\mathrm{d}})^{2}}{2\sigma^{2}}},$
(15)

where $\sigma^{2}$ is the second statistical moment (variance) and
$w_{\mathrm{d}}$ is the optimal velocity of drivers. By analogy, in the
articles [11, 21] there has been deduced that the scaled distance-clearance
distribution reads as

$\wp(r)=A~{}\Theta(r)~{}\mathtt{e}^{-\beta/r}\mathtt{e}^{-Dr},$ (16)

where $\Theta(x)$ is the Heaviside’s step-function and the normalization
factors are

$D\approx\beta+\frac{3-\mathtt{e}^{-\sqrt{\beta}}}{2},$ (17)
$A^{-1}=2\sqrt{\frac{\beta}{D}}\mathcal{K}_{1}\bigl{(}2\sqrt{D\beta}\bigr{)}.$
(18)

As verified in [8, 11, 19] the one-parametric distribution-family (16) is in
an satisfactory agreement with the clearance distribution observed in the
real-road data. We add that the noise-parameter $\beta$ is related to the
corresponding traffic density $\varrho.$

With the help of formulas (15) and (16) one can mathematically derive the
analytical form of probability density $\eta(t)$ for clear time-intervals
between two consecutive vehicles (particles of a model). Since the joint
probability density for distance and velocity can be (under the condition on
weak distance-vs-speed dependency – see the Fig. 2) predicted as
$g(w,x)=q(w)\wp(x),$ the joint probability density for time gaps and
velocities reads therefore as $h(w,t)=wq(w)\wp(wt).$ Then the time-clearance
distribution represents a marginal density

$\eta(t)=\int_{\mathbb{R}}h(w,t)\,\mathtt{d}w=\int_{\mathbb{R}}wq(w)\wp(wt)\,\mathtt{d}w.$
(19)

After expanding a function $f(w)=w\wp(wt)$ into the Taylor’s series about the
optimal velocity $w_{\mathrm{d}}$ we acquire

$f(w)=f(w_{\mathrm{d}})+\sum_{\ell=1}^{\infty}\frac{1}{\ell!}\frac{\mathtt{d}^{\ell}f}{\mathtt{d}w^{\ell}}(w_{\mathrm{d}})(w-w_{\mathrm{d}})^{\ell},$
(20)

where

$\frac{\mathtt{d}^{\ell}f}{\mathtt{d}w^{\ell}}=\frac{\partial^{\ell}\wp}{\partial(wt)^{\ell}}t^{\ell}w+\ell\frac{\partial^{\ell-1}\wp}{\partial(wt)^{\ell-1}}t^{\ell-1}\quad(\ell\in\mathbb{N}).$
(21)

Hence

$\eta(t)=w_{\mathrm{d}}\wp(w_{\mathrm{d}}t)+\sum_{\ell=1}^{\infty}\frac{1}{\ell!}\frac{\partial^{\ell}\wp}{\partial(wt)^{\ell}}(w_{\mathrm{d}}t)t^{\ell}w_{\mathrm{d}}\int_{\mathbb{R}}(w-w_{\mathrm{d}})^{\ell}q(v)\,\mathtt{d}w~{}+\\\
+\sum_{\ell=1}^{\infty}\frac{1}{\ell!}\frac{\partial^{\ell-1}\wp}{\partial(wt)^{\ell-1}}(w_{\mathrm{d}}t)t^{\ell-1}\int_{\mathbb{R}}(w-w_{\mathrm{d}})^{\ell}q(w)\,\mathtt{d}w~{}=\\\
=w_{\mathrm{d}}\wp(w_{\mathrm{d}}t)+\sum_{\ell=1}^{\infty}\frac{\mu_{\ell}}{\ell!}\left(\frac{\partial^{\ell}\wp}{\partial(wt)^{\ell}}(w_{\mathrm{d}}t)t^{\ell}w_{\mathrm{d}}+\frac{\partial^{\ell-1}\wp}{\partial(wt)^{\ell-1}}(w_{\mathrm{d}}t)t^{\ell-1}\right),$
(22)

where $\mu_{\ell}=\int_{\mathbb{R}}(w-w_{\mathrm{d}})^{\ell}q(w)\,\mathtt{d}w$
is $\ell-$th central statistical moment with three prerogatived cases
$\mu_{0}=1,$ $\mu_{1}=0,$ and $\mu_{2}=\sigma^{2}.$ The latter represents a
statistical variance (see also (15)). As well known the odd central
statistical moments (associated to the Gauss distribution) are zero and even
moments comply with equalities $\mu_{2\ell}=\sigma^{2\ell}(2\ell-1)!!,$ which
leads to the general formula

$\eta(t)=w_{\mathrm{d}}\wp(w_{\mathrm{d}}t)+\sum_{\ell=1}^{\infty}\frac{\sigma^{2\ell}}{(2\ell)!!}\left(\frac{\partial^{2\ell}\wp}{\partial(wt)^{2\ell}}(w_{\mathrm{d}}t)t^{2\ell}w_{\mathrm{d}}+\frac{\partial^{2\ell-1}\wp}{\partial(wt)^{2\ell-1}}(w_{\mathrm{d}}t)t^{2\ell-1}\right).$
(23)

Owing to the facts
$\int_{\mathbb{R}}w_{\mathrm{d}}\wp(w_{\mathrm{d}}t)\,\mathtt{d}t=1$ and

$\int_{\mathbb{R}}\left(\frac{\partial^{2\ell}\wp}{\partial(wt)^{2\ell}}(w_{\mathrm{d}}t)t^{2\ell}w_{\mathrm{d}}+\frac{\partial^{2\ell-1}\wp}{\partial(wt)^{2\ell-1}}(w_{\mathrm{d}}t)t^{2\ell-1}\right)\mathtt{d}t=0,$
(24)

the TC-distribution (23) is normalized correctly. In a certain sense (as
probably understandable from the two previous equations) the first factor in
(23) forms a leading term of TC-distribution, whereas the sum in (23)
represents a perturbation term only. Above that, the weight of the $\ell$th
perturbation summands decreases with a factor $\sigma^{2\ell}/(2\ell)!!.$ For
practical applications (especially for traffic applications) it seems
therefore to be coherent using an approximate expansion

$\eta(t)\approx
w_{\mathrm{d}}\wp(w_{\mathrm{d}}t)+\frac{\sigma^{2}}{2}\left(\frac{\partial^{2}\wp}{\partial(wt)^{2}}(w_{\mathrm{d}}t)t^{2}w_{\mathrm{d}}+2\frac{\partial\wp}{\partial(wt)}(w_{\mathrm{d}}t)t\right).$
(25)

Such an approximation is legitimate because of extremely low value of velocity
variance mainly (see the inset in the Fig. 1). Note that the factors
$\sigma^{2\ell}/(2\ell)!!$ measured in real-road data rapidly vanishes since
the maximal value of velocity variance is $\approx 0.02$ in the main lane or
$\approx 0.06$ in the fast lane. The expansion (25) can be replaced (after the
scaling procedure) by the formula

$\eta(t)\approx\wp(t)+\sigma^{2}\wp(t)\left(\frac{\beta^{2}}{2t^{2}}+\frac{D^{2}t^{2}}{2}-D(t+\beta)\right).$
(26)

The influence of the low velocity-variance to TC-distribution is demonstrated
in the Fig. 3 where the curves (26) are compared to the zero-approximation

$\eta(t)\approx
A~{}\Theta(t)~{}\mathtt{e}^{-\frac{\beta}{t}}\mathtt{e}^{-Dt},$ (27)

while the relations (17), (18) hold true. We remind that the normalization and
re-scaling conditions
$\int_{\mathbb{R}}\eta(t)\,\mathtt{d}t=\int_{\mathbb{R}}t\,\eta(t)\,\mathtt{d}t=1$
have been applied.

Figure 3: Time-clearance distribution (analytically derived). The signs
represent the values of function (26) calculated for
$\beta\in\\{0,1/2,3/2,5/2\\},$ whereas the curves display the zero-
approximation (27) for the same $\beta.$ Small discrepancies between signs and
corresponding curves are caused by fact that statistical variance $\sigma^{2}$
of the modified velocities (measured on freeways) is very low.
Phenomenologically, this relation can be estimated (for the main-lane data) by
the inequality $\sigma^{2}\leq(45\beta/55)^{4}\exp[-45\beta/14].$

##  4 Mathematical derivation of multi-clearance distribution

Now, knowing the one parameter family of the probability densities (26) or
their zero-approximations (27) one can derive an analytical prediction for the
so-called _$n$ th multi-clearance distribution_ $\tau_{n}(t)$ which represents
the probability density for a clear time gap $t$ among $n+2$ neighboring
particles. Therefore, the main goal of this section is to quantify (by means
of theory of functional convolutions) the probability that time-period between
two following instants (the first one: the back bumper of the $k$th vehicle is
leaving the detector; the second one: the front bumper of the $(k+n+1)$th
vehicle is intersecting the detector line) is ranging in the interval
$[t,t+\mathtt{d}t).$ Using this notation we find that the probability density
for the time-clearance between two succeeding cars (derived in the previous
section) is $\eta(t)=\eta_{0}(t).$ Regarding the clearances as independent
(which is demonstrated in the figure 4) the $n$th probability density
$\eta_{n}(t)$ can be calculated via recurrent formula

$\eta_{n}(t)=\eta_{n-1}(t)~{}\star~{}\eta_{0}(t),$ (28)

where symbol $\star$ represents a convolution of the two probabilities, i.e.

$\eta_{n}(t)=\int_{\mathbb{R}}\eta_{n-1}(s)\eta_{0}(t-s)\,\mathtt{d}s.$ (29)

Figure 4: Graphical representation of correlation analysis of successive time-
clearances. The values $\omega_{tt}=\omega_{tt}(\varrho)$ describes the
standard correlation coefficient calculated for pairs of succeeding time-
clearances extracted from traffic data in various density regions.

As a first resort, we will concentrate our endeavor to studying the low-
density traffic regimes where the locations of cars are independent and
therefore the traffic flow corresponds (from mathematical point of view) to
the Poisson process of uncorrelated events. As published in [8, 11, 21] such a
situation is characterized by the negligible value of socio-physical
coefficient $\beta$ reflecting a driver’s mental strain. Thus, the associated
clearance distribution reads $\eta(t)=\Theta(t)\mathtt{e}^{-t}$ and
sequentially the multi-clearance distribution for free traffic looks like

$\eta_{n}(t)=\Theta(t)\,\mathtt{e}^{-t}\int_{0}^{t}\int_{0}^{s_{1}}\int_{0}^{s_{2}}\ldots\int_{0}^{s_{n-2}}\int_{0}^{s_{n-1}}\mathtt{d}s_{n}\mathtt{d}s_{n-1}\ldots\mathtt{d}s_{2}\mathtt{d}s_{1}=\Theta(t)\frac{t^{n}}{n!}\mathtt{e}^{-t}.$
(30)

As a second resort, we aim to derive a general formula, i.e. formula for non-
zero $\beta.$ For these purposes we use the zero-approximation (27). Applying
the method of mathematical induction and an approximation of the function

$g_{n}(t,s)=\mathtt{e}^{-\beta\bigl{(}\frac{n^{2}}{s}+\frac{1}{t-s}\bigr{)}}\approx\mathtt{e}^{-\frac{\beta}{t}(n+1)^{2}}$
(31)

in the saddle point one can obtain

$\eta_{n}(t)=\Theta(t)\int_{0}^{t}A_{n-1}As^{n-1}\mathtt{e}^{-\beta\frac{n^{2}}{s}}\mathtt{e}^{-Ds}\mathtt{e}^{-\frac{\beta}{t-s}}\mathtt{e}^{-D(t-s)}\,\mathtt{d}s~{}=\\\
=\Theta(t)A_{n-1}A\mathtt{e}^{-Dt}\int_{0}^{t}s^{n-1}g_{n}(t,s)\,\mathtt{d}s~{}\approx\\\
\approx\Theta(t)A_{n-1}A\mathtt{e}^{-\frac{\beta}{t}(n+1)^{2}}\mathtt{e}^{-Dt}\int_{0}^{t}s^{n-1}\,\mathtt{d}s\approx\Theta(t)A_{n-1}An^{-1}t^{n}\mathtt{e}^{-\frac{\beta}{t}(n+1)^{2}}\mathtt{e}^{-Dt}.$
(32)

Hence

$\eta_{n}(t)\approx\Theta(t)A_{n}t^{n}\mathtt{e}^{-\frac{\beta}{t}(n+1)^{2}}~{}\mathtt{e}^{-Dt},$
(33)

where (after applying the re-normalization procedure)

$A_{n}^{-1}=2\left(\sqrt{\frac{\beta}{D}}(n+1)\right)^{n+1}\mathcal{K}_{n+1}\bigl{(}2(n+1)\sqrt{D\beta}\bigr{)}.$
(34)

This fixes the proper normalization
$\int_{\mathbb{R}}\eta_{n}(t)\,\mathtt{d}t=1.$ In addition to that the mean
$n$th spacing equals to

$\int_{\mathbb{R}}t\,\eta_{n}(t)\,\mathtt{d}t=n+1.$ (35)

Note, that (33) holds true also for the limiting case $\beta=0.$ Really, with
help of formulas $\lim_{x\rightarrow 0}x^{n+1}\mathcal{K}_{n+1}(x)=(2n)!!$ and
$\lim_{\beta\rightarrow 0_{+}}D(\beta)=1$ we easily deduce that

$\lim_{\beta\rightarrow
0_{+}}2\left(\sqrt{\frac{\beta}{D}}(n+1)\right)^{n+1}\mathcal{K}_{n+1}\bigl{(}2(n+1)\sqrt{D\beta}\bigr{)}=n!.$
(36)

Therefore it holds $A_{n}=1/n!,$ which is in a full consonance with the
relation (30). Thus, the relation (33) constitutes a zero-approximation for
the distribution of time multi-clearances. Since we have supposed (in the
previous deductions) that the variance of modified velocities is negligible
(in a local sense), the form of the distribution (33) is a direct consequence
of the distribution for spatial clearances and the fact, that all cars have
practically the same velocity (in a local sense, again).

Because the empirical measurements show low (but not negligible) variances in
modified velocities (see the inset in the Fig. 1), it seems more realistic to
derive the time multi-clearance distributions under the conditions
$\sigma^{2}>0$ and $\sigma^{2n}\approx 0$ for $n=2,3,4,\ldots.$ Thus, we
suppose that clear time-intervals among succeeding cars are distributed
according the rule (26). The detailed analysis of such a relations vindicates
that the dominating term in the last summand of (26) is $\sigma^{2}Dt\wp(t).$
Hence, we surmise that the time clearance follows the law

$\eta_{0}^{(\sigma)}(t)\approx\wp(t)-\sigma^{2}Dt\wp(t).$ (37)

Here we remark (for mathematical correctness) that this function has to be
understood as an approximative probability density, since the perturbation
term $\sigma^{2}Dt\wp(t)$ causes that $\eta_{0}^{(\sigma)}(t)$ does not fulfil
the exact mathematical definition.

Using the method of mathematical induction we prove below that (under the
previous surmises) the multi-clearance distribution of order $n$ reads

$\eta_{n}^{(\sigma)}(t)\approx\Theta(t)\frac{A^{n+1}}{n!}t^{n}\mathtt{e}^{-(n+1)^{2}\frac{\beta}{t}}\mathtt{e}^{-Dt}\bigl{(}1-\sigma^{2}Dt\bigr{)}.$
(38)

For completeness, we remark that in the following mathematical operations

$\eta_{n+1}^{(\sigma)}(t)=\eta_{n}^{(\sigma)}(t)\star\eta_{0}^{(\sigma)}(t)\approx\Theta(t)\frac{A^{n+2}}{n!}\mathtt{e}^{-Dt}\int_{0}^{t}\mathtt{e}^{-(n+1)^{2}\frac{\beta}{s}}s^{n}\mathtt{e}^{-\frac{\beta}{t-s}}~{}\mathtt{d}s~{}+\\\
+D^{2}\sigma^{4}\Theta(t)\frac{A^{n+2}}{n!}\mathtt{e}^{-Dt}\int_{0}^{t}\mathtt{e}^{-(n+1)^{2}\frac{\beta}{s}}(t-s)s^{n+1}\mathtt{e}^{-\frac{\beta}{t-s}}~{}\mathtt{d}s~{}-\\\
-D\sigma^{2}\Theta(t)\frac{A^{n+2}}{n!}\mathtt{e}^{-Dt}\int_{0}^{t}\mathtt{e}^{-(n+1)^{2}\frac{\beta}{s}}s^{n+1}\mathtt{e}^{-\frac{\beta}{t-s}}~{}\mathtt{d}s~{}-\\\
-D\sigma^{2}\Theta(t)\frac{A^{n+2}}{n!}\mathtt{e}^{-Dt}\int_{0}^{t}\mathtt{e}^{-(n+1)^{2}\frac{\beta}{s}}(t-s)s^{n}\mathtt{e}^{-\frac{\beta}{t-s}}~{}\mathtt{d}s~{}=\\\
=\Theta(t)\frac{A^{n+2}}{(n+1)!}t^{n+1}\mathtt{e}^{-(n+2)^{2}\frac{\beta}{t}}\mathtt{e}^{-Dt}\bigl{(}1-\sigma^{2}Dt\bigr{)}$
(39)

there have been used the approximation (31) and the surmise $\sigma^{4}\approx
0.$

##  5 Empirical multi-clearance distribution vs. analytical prediction

In this section we will balance the theoretical prognoses deduced in the
sections 3 and 4 against the empirical multi-clearance distributions of
freeway data analyzed in the section 2 (two-lane freeway A9 - Netherland, two-
lane freeway D1 - Czech Republic). In the following part of the text we will
consider the multi-clearances (here: cumulated clearances between $n=5$ or
$n=8$ succeeding cars)

$\varkappa_{i}^{(j)}=\sum_{k=(j-1)m+i}^{(j-1)m+i+n}\frac{t_{k\ell}}{\bar{t}_{j}},\quad(\ell=1,~{}j=1,2,\ldots,M_{1},~{}i=1,2,\ldots,m-n)$
(40)

enumerated for data sets (5). These multi-clearances are associated with the
specific traffic density through the formula (8) and re-scaled so that the
average multi-clearance (in the given data-samples $S^{(\mathrm{main})}_{j}$)
is equal to $n+1.$ Now, the multi-clearances $\varkappa_{i}^{(j)}$ represent a
statistical realization of random variable $t$ considered in the section 4 and
can be therefore confronted with the theoretical probabilities.

Figure 5: Statistical distribution of time multi-clearances for low density
regimes. The bars shows the empirical probability density for clear time-
interval among five (or eight) succeeding vehicles moving in main lane (for
various density regimes – see legend for details). The blue dashed curves
represent the prediction (33) plotted for $n=3$ (top plot) or $n=6$ (bottom
plot), and for the fitted value of coefficient $\beta=\beta_{\mathtt{fit}}$
obtained by minimizing the weighted error-function (42). Continuous curves
(green) display the analytical approximation (41) plotted for the fitted
values of $\beta=\beta_{\mathtt{fit}}$ and
$\varepsilon=\varepsilon_{\mathtt{fit}}$ specified by minimizing the weighted
error-function (43). For clearness, we also plot the dash-dotted curves
(magenta) visualizing the multi-clearance distribution (30) valid for an
occurrence of independent events. Plus signs illustrates how the time multi-
clearances (gauged by the fast-lane-detectors) differ from those detected in
the main lane.

Figure 6: Statistical distribution of time multi-clearances for high density
regimes. The bars shows the empirical probability density for clear time-
interval among five (or eight) succeeding vehicles moving in main lane (for
various density regimes – see legend for details). The blue dashed curves
represent the prediction (33) plotted for $n=3$ (top plot) or $n=6$ (bottom
plot), and for the fitted value of coefficient $\beta=\beta_{\mathtt{fit}}$
obtained by minimizing the weighted error-function (42). Continuous curves
(green) display the analytical approximation (41) plotted for the fitted
values of $\beta=\beta_{\mathtt{fit}}$ and
$\varepsilon=\varepsilon_{\mathtt{fit}}$ specified by minimizing the weighted
error-function (43). For clearness, we also plot the dash-dotted curves
(magenta) visualizing the multi-clearance distribution (30) valid for an
occurrence of independent events. Plus signs illustrates how the time multi-
clearances (gauged by the fast-lane-detectors) differ from those detected in
the main lane.

To be factual, we divide the entire interval of traffic densities into the
subintervals $[0,3),$ $[1,4),$ $[2,5),$ and so on and analyze the freeway
multi-clearance distribution $p(\varkappa)$ separately in each subinterval.
Similar approach has been examined in the articles ([8, 11, 9, 21]) and
reflects the known fact that the headway distributions are substantially
influenced by the changing location of the traffic ensemble in the phase
diagram. To prevent the mixing of the states with different vigilance of car
drivers (or with different temperature of associated heat bath – in the
thermodynamic interpretation of vehicular traffic), we inspect the empirical
multi-clearances separately in each density interval. Such an approach permits
the investigation of density-dependent evolution of multi-clearance
distributions.

In the Fig. 5 and 6 there are plotted the empirical multi-clearance
distributions (histograms) against the poissonian distribution (30), zero
approximation (33) calculated for $n=3$ or $n=5,$ and the final analytical
prediction

$\eta_{n}^{(\varepsilon)}(t)=\Theta(t)\frac{A^{n+1}}{n!}t^{n}\mathtt{e}^{-(n+1)^{2}\beta/t}\mathtt{e}^{-Dt}\bigl{(}1-\varepsilon
t\bigr{)}.$ (41)

The values of the fitted parameters ($\beta$ in (33), and
$\beta,~{}\varepsilon$ in (41)) have been determined by means of formulae

$\beta_{\mathtt{fit}}^{\bullet}=\mathrm{argmin}_{\beta\in[0,\infty)}\int_{0}^{\infty}\bigl{|}\eta_{n}(t)-p(t)\bigr{|}^{2}t\mathtt{e}^{-t/4}\,\mathtt{d}t,$
(42)
$\bigl{(}\beta_{\mathtt{fit}},\,\varepsilon_{\mathtt{fit}}\bigr{)}=\mathrm{argmin}_{\beta\in[0,\infty),\varepsilon\in[0,\infty)}\int_{0}^{\infty}\bigl{|}\eta_{n}^{(\varepsilon)}(t)-p(t)\bigr{|}^{2}t\mathtt{e}^{-t/4}\,\mathtt{d}t,$
(43)

i.e. minimizing the statistical distance
$\chi_{n}(\beta,\varepsilon)=\int_{0}^{\infty}|f_{n}(t)-p(t)|^{2}t\mathtt{e}^{-t/4}\,\mathtt{d}t$
(weighted by the factor $t\mathtt{e}^{-t/4}$) cumulating the weighted
deviations between theoretical prediction $f_{n}(t)$ and empirical frequency
$p(t).$

As visible in the Fig. 5 and 6 the freeway multi-clearance distribution
delineated for low-density states coincides with the distribution (30), which
confirms the surmise that vehicles in free traffic regime are moving as
independent elements. Thus, their statistics is purely poissonian. As the
traffic density rises one can detect larger deviations from (30), which
demonstrates stronger interactions among the cars. Roughly speaking, the
fitted thermal parameter $\beta$ is increasing (in both cases: zero-
approximation and also the final probability distribution) with the traffic
density (see the Fig. 7). The regions of the temporal descent in $\beta$ value
agree with the critical regions in the fundamental diagram. To be precise, the
transmission between traffic regimes (from free to congested regime) causes
the transient consolidation of traffic, which leads to a reduction of driver’s
mental-strain. Since (see the [21]) the thermal parameter reflects a level of
mental pressure, the detected drop in course of $\beta=\beta(\varrho)$ is
expectable. We add that the evolution of thermal parameter $\beta$ corresponds
to the behavior of the associated quantity investigated within the scope of
the articles [8, 11, 21, 26]. Moreover, in all traffic states the distribution
(41) fits the real-road data more impressively than the original approximation
(33), as comprehensible. Such a fact is clearly visible in the Fig. 9 where
the statistical distance $\chi(\beta,\varepsilon)$ between theoretical and
empirical distributions are outlined.

Figure 7: Optimal values of the parameter $\beta$ in theoretical
distributions. Signs (circles for $n=3$ and squares for $n=7$) visualize the
value of the fitted parameter $\beta_{\mathtt{fit}}$ (see the relation (43))
that minimizes the statistical distance between theoretical curve (41) and
empirical histogram $p(t)$ (enumerated for main-lane data only). In the inset
there is plotted the value of the thermal parameter
$\beta_{\mathtt{fit}}^{\bullet}$ considered in (42).  Figure 8: Optimal value
of the parameter $\varepsilon$ in the theoretical distribution (41). Signs
(circles for $n=3$ and squares for $n=7$) visualize the value of the fitted
parameter $\varepsilon_{\mathtt{fit}}$ (see the relation (43)) that minimizes
the statistical distance between theoretical curve (41) and empirical
histogram $p(t)$ (enumerated for main-lane data only).

The important component of our research is a discussion about stability of the
above-introduced fitted procedure, i.e. stability of the detected interaction-
parameter $\beta.$ The explorations of the dependency between the estimated
interaction-parameter $\beta_{\mathtt{fit}}$ and number $n$ of cumulated
clearances shows only insignificant fluctuations in
$\beta_{\mathtt{fit}}=\beta_{\mathtt{fit}}(n).$ Such a fact is illustrated
comprehensibly in the Fig. 7. It means that the value of
$\beta_{\mathtt{fit}}$ is only slightly influenced by the order of the multi-
clearance distribution $\eta_{n}^{(\varepsilon)}(t).$ Therefore the estimated
interaction-parameter $\beta_{\mathtt{fit}}$ represents a alternative traffic
quantity whose values reflect the traffic state.

##  6 Three-phases traffic – evolution of multi-clearance distribution

Finally, we try to connect the presented results with the three-phase traffic
theory (see [4]). The aim is to explore the essence of changes in multi-
clearance distributions and detect a possible phase changes in the phase
surface $\varrho\times J.$ For these purposes we divide the entire phase space
into the separate two-dimensional subregions
$[\varrho,\varrho+\Delta\varrho)\times[J,J+\Delta J).$ By analogy to the
previous approach we investigate empirical distributions separately in each
subregion. Surprisingly, in spite of the existing opinion, in all traffic
phases the multi-clearances belong to the same family of distributions (38).
However, the resulting value $\beta^{\bullet\bullet}_{\mathtt{fit}}$ (obtained
with help of the formula (43)) depends on density and flow, i.e.
$\beta^{\bullet\bullet}_{\mathtt{fit}}=\beta^{\bullet\bullet}_{\mathtt{fit}}(\varrho,J).$

Figure 9: Statistical distance between theoretical and empirical
distributions. We plot the values of statistical distances
$\chi_{n}(\beta_{\mathtt{fit}}^{\bullet})$ and
$\chi_{n}(\beta_{\mathtt{fit}},\varepsilon_{\mathtt{fit}})$ between main-lane
multi-clearance distribution $p(t)$ and zero-approximation (33) (pale bars) or
final formula (41) (dark bars), respectively.

As comprehensible from the Fig. 10 the basic trends in the partial dependency
$\beta^{\bullet\bullet}_{\mathtt{fit}}=\beta^{\bullet\bullet}_{\mathtt{fit}}(\varrho)$
fully corresponds to the behavior
$\beta_{\mathtt{fit}}=\beta_{\mathtt{fit}}(\varrho)$ obtained in the section
5. Moreover, there is distinctly visible how the thermal parameter is adapted
to the actual position in the phase space. The phase transition of the traffic
system from free flows to synchronized flows (and vice versa) is accompanied
by the corresponding changes in $\beta.$ Note that in the transitional region
(between 35 and 55 $veh/km/lane$) the stable growth of thermal-like
coefficient $\beta$ is temporarily attenuated (in fact, $\beta$ falls), which
endorses the hypothesis on phase transition. Also, there is transparently
demonstrated how the different traffic phases (i.e. drivers under a different
mental pressure) influence the statistics of vehicular interactions.

For completeness, we emphasize again that all the statistical test discussed
in this article have been executed for the main lane traffic data only. As
demonstrated in the Fig. 5 and 6 the time multi-clearance distributions
detected in the fast-lane data show the well-known discrepancies with the
main-lane data (predominantly in the regions of low densities). For denser
streams the fast-lane distributions converge to the main-lane distributions,
which endorses the belief of scientists that in congested traffic states the
correlations among the vehicles in different traffic lanes are much stronger
than in free-flow regimes.

Figure 10: Thermal parameter as a function of traffic density and traffic
flux. We plot the dependency of the thermal parameter
$\beta^{\bullet\bullet}_{\mathtt{fit}}$ occurring in the formula (43) for
various subregions in the phase diagram $\varrho\times J.$

##  7 Summary, conclusion, and future prospects

To conclude, this article deals with cumulative time clearances among several
subsequent vehicles passing a given point of an expressway. More specifically,
we are concentrated predominantly on time intervals between two following
occurrences: 1. the rear bumper of $k$th car has intersected the detector
line, 2. the front bumper of $(k+n)$th car has intersected the detector line.
Furthermore, we subtract the time-periods when vehicles are occupying the
detectors, which means that a fundamental quantity for this research is the
time multi-clearance. Using the local thermodynamical traffic model
(originally introduced in [8] and solved analytically in [11]) with short-
range repulsive potential among the elements we have derived (applying theory
of functional convolutions) a mathematical formula for time multi-clearance
distribution. This final theoretical distribution represents an one-parametric
family of functions, where the one and only fitting parameter $\beta$ reflects
the relevant flow and density. The obtained analytical predictions have been
successfully compared with statistics of freeway data. The detected
correspondence between theoretical and empirical distributions allows an
elaborated insight into changes of vehicular-traffic microstructure influenced
by the momentary traffic state.

In spite of the existing conviction that time-headway distributions in the
different phases are markedly different, the stochastic analysis of multi-
clearances reveals a different view: the statistical distribution of traffic
clearances (and multi-clearances) in all traffic phases belongs to the same
one-parametric family of distributions. Those distributions are varying from
the Poisson distribution (detected for low-density states) to the low-variance
distribution (41) that ascertains a presence of stronger correlations among
neighboring vehicles, i.e. stronger vehicular synchronization than in free-
flow states. The existing conviction on two types of traffic headway-
distributions is probably a consequence of insufficient sorting of freeway
data. In the earlier scientific papers (dealing with traffic headways) authors
usually separate vehicular data into two/three parts only (according a traffic
phase) and in fact they combine all the congested states into a single
ensemble. But, as apparent from this research, the congested-phase-
distributions are significantly varying according to the traffic flow and
density. For that reason the binary data-separation seems to be deficient.

Another substantial outcome of these considerations is the fact that the
interaction-parameter $\beta$ (quantifying a measure of a vehicular
synchronization) represents an alternative traffic quantity which reflects an
actual traffic state. Moreover, the value $\beta$ is (in view of the fact that
all clearances are scaled) directly connected to the statistical variance of
traffic clearances, and its estimated value can be therefore obtained by a
simple comparison of the empirical and theoretical variances, which is quite
effortless.

However, the open problem remains how to approximate the multi-clearance
distribution for free-flow vehicles moving in fast lanes. Presence of large
amount of vehicular _leaders_ in free-traffic regimes, i.e. comparable
percentage of leaders and _followers_ , causes that the clearance distribution
is probably a compound of two partial distributions (first one for leaders,
second one for followers). Thus, the expected approach leading to analytical
predictions for fast-lane clearances can be found in the theory of finite
mixture distributions (similarly to semi-poissonian model discussed in [5]).

### Acknowledgement

The author would like to thank Cecile Appert-Rolland (Laboratoire de Physique
Théorique Université de Paris-Sud, Orsay) for valuable remarks which have been
conductive to the presented research. This work was supported by the Ministry
of Education, Youth and Sports of the Czech Republic within the project MSM
6840770039 and by the Czech Technical University within the project
SGS12/197/OHK4/3T/14.

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