Probabilistic Breakdown Phenomenon at On-Ramp Bottlenecks in Three-Phase Traffic Theory

Abstract

A nucleation model for the breakdown phenomenon in freeway free traffic flow at an on-ramp bottleneck is presented. This model, which can explain empirical results on the breakdown phenomenon, is based on assumptions of three-phase traffic theory in which the breakdown phenomenon is related to a first-order phase transition from the "free flow" phase to the "synchronized flow" phase. The main idea of this nucleation model is that random synchronized flow nucleation occurs within a metastable inhomogeneous steady state associated with a deterministic local perturbation in free flow, which can be considered "deterministic vehicle cluster" in free flow at the bottleneck. This deterministic vehicle cluster in free flow is motionless and exists permanent at the bottleneck due to the on-ramp inflow. In the nucleation model, traffic breakdown nucleation occurs through a random increase in vehicle number within this deterministic vehicle cluster, if the amplitude of the resulting random vehicle cluster exceeds some critical amplitude. The mean time delay and the associated nucleation rate of speed breakdown at the bottleneck are found and investigated. The nucleation rate of traffic breakdown as a function of the flow rates to the on-ramp and upstream of the bottleneck is studied. The nucleation model and the associated results exhibit qualitative different characteristics than those found earlier in other traffic flow nucleation models. Boundaries for speed breakdown in the diagram of congested patterns at the bottleneck are found. These boundaries are qualitatively correlated with numerical results of simulation of microscopic traffic flow models in the context of three-phase traffic theory.

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arXiv:cond-mat/0502281v3 [cond-mat.stat-mech] 5 Jul 2005
Probabilistic Breakdown Phenomenon at
On-Ramp Bottlenecks in Three-Phase Traffic
Theory
Boris S. Kerner
DaimlerChrysler AG, REI/VF, HPC: G021, 71059 Sindelfingen, Germany
and
Sergey L. Klenov
Moscow Institute of Physics and Technology, Department of Physics, 141700
Dolgoprudny, Moscow Region, Russia
Abstract
A nucleation model for the breakdown phenomenon in an initial non-homogeneous
free traffic flow that occurs at an on-ramp bottleneck is presented. This model is
in the context of three-phase traffic theory. In this theory, the breakdown phe-
nomenon is associated with a first-order phase transition from the “free flow” phase
to the “synchronized flow” phase. In contrast with many other nucleation models for
phase transitions in different system of statistical physics in which random precluster
emergence from fluctuations in an initial homogeneous system foregoes subsequent
cluster evolution towards a critical cluster (critical nuclei), random precluster oc-
currence in free flow at the bottleneck is not necessary for traffic breakdown. In
the model, the breakdown phenomenon can also occur if there were no fluctuations
in free flow. This is because there is a permanent and motionless non-homogeneity
that can be considered a deterministic vehicle cluster localized in a neighborhood
of the bottleneck. The presented nucleation model and a nucleation rate of traffic
breakdown that follows from the model exhibit qualitatively different features in
comparison with previous results. In the nucleation model, traffic breakdown nucle-
ation occurs through a random increase in vehicle number within the deterministic
vehicle cluster, if the amplitude of the resulting random vehicle cluster exceeds some
critical amplitude. The mean time delay and the associated nucleation rate of traf-
fic breakdown at the bottleneck are found and investigated. The nucleation rate of
traffic breakdown as a function of the flow rates to the on-ramp and upstream of the
bottleneck is studied. Boundaries for traffic breakdown in the diagram of congested
patterns at the bottleneck are found. These boundaries are qualitatively correlated
with numerical results of simulation of microscopic traffic flow models in the context
of three-phase traffic theory.
Preprint submitted to Elsevier Science2 February 2008

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1 Introduction
Empirical observations of freeway traffic made in various countries show that
the onset of congestion in an initial free flow is associated with an abrupt
decrease in vehicle speed. This traffic breakdown called the “breakdown phe-
nomenon” occurs mostly at freeway bottlenecks, in particular on-ramp bottle-
necks. The traffic breakdown is accompanied by a hysteresis effect (see refer-
ences in the reviews [1,2], the book [3], and the conference proceedings [4,5,6]).
The breakdown phenomenon has a probabilistic nature [7,8,9]: At the same
on-ramp bottleneck, traffic breakdown is observed at different flow rates in
different realizations (days). The probability of the breakdown is a strong
increasing function of flow rate downstream of the bottleneck [8,9].
Most microscopic, macroscopic, probabilistic, and other models of freeway
traffic explain the onset of congestion in free flow by moving jam emergence
(see, e.g. [10,11,12,13,14,15] and references in the reviews [16,17,18,19], as
well as the conference proceedings [20,21,22,23,24]). In particular, in mod-
els of an on-ramp bottleneck moving jam(s) occurs spontaneously in free
flow at the bottleneck when the flow rate upstream of the bottleneck is high
enough and the flow rate to the on-ramp increases gradually beginning from
zero [17,18,25,26,28,27]. However, the fundamental model result that the onset
of congestion in free flow on a homogeneous road and at freeway bottlenecks is
associated with spontaneous moving jam emergence [10,13,14,15,25,26,28,27,16,17,18,19]
is in a serious conflict with empirical evidence [29,30,3].
Consequently, in 1996–1999 Kerner introduced three-phase traffic theory (see [3]
for a review). In this theory, there are three traffic phases: free flow, synchro-
nized flow, and wide moving jams. In accordance with empirical investigations
of phase transitions, in this theory moving jams do not emerge spontaneously
in free flow. Rather than moving jam emergence, a phase transition from free
flow to synchronized flow (F→S transition for short) governs the onset of con-
gestion in free flow [29,30]. A first-order F→S transition postulated in three-
phase traffic theory [29] discloses the nature of the breakdown phenomenon at
freeway bottlenecks found in empirical observations [1,2]. In other words, the
terms “F→S transition”, “breakdown phenomenon”, “speed breakdown”, and
“traffic breakdown” are synonyms. The first microscopic models in the con-
text of three-phase traffic theory are stochastic models [31,32]. These models
exhibit phase transitions as well as all types of congested patterns found in
empirical observations [31,32,33,34,3]. Recently, some new microscopic models
based on three-phase traffic theory have been developed [35,36,37], which can
show some congested pattern features found earlier in [31,32].
One of the important methods for a study of phase transitions in non-linear
distributed multiple-particle systems is a probabilistic theory based on an
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analysis of a master equation (e.g., [38,39,40]). First probabilistic theories for
traffic flow based on a master equation for a random vehicle cluster have
been introduced by Mahnke et al. [13,14] and further developed by K¨ uhne et
al. [15,41] (see the recent review by Mahnke et al. [40]). As usual for other
metastable systems of statistical physics, for an initial metastable traffic flow
a well-known two-well potential nucleation problem arises from the master
equation, which analysis is made based on general well-known methods of sta-
tistical physics [38,39]. One of the main results of this analysis is the nucleation
rate for the critical vehicle cluster (critical nuclei) whose occurence leads to a
phase transition within the initial metastable state of traffic flow. As usual for
other metastable systems of statistical physics [38,39], in the metastable traffic
flow the nucleation rate for a phase transition is an exponential function of a
control parameter, flow rate (or vehicle density) for traffic flow [13,14,15,41,40].
Rather than these common well-known results, an interest for the field has a
nucleation model for a metastable traffic flow. Both the model and associated
dependencies of the nucleation rates for phase transitions on traffic variables
and control parameters should be adequate with real traffic flow features.
In [13,14,15], models for vehicle cluster nucleation in an initial spatially ho-
mogeneous traffic flow on a homogeneous circular road are introduced (see,
however, Sect. 4.2). One of the basic assumptions of these models is that in
an initial homogeneous free flow firstly random precluster should emerge from
fluctuations. In other words, this precluster foregoes subsequent cluster evolu-
tion towards a critical cluster (critical nuclei) whose growth leads to a phase
transition. The rate of precluster emergence w+(0) in traffic, in which initially
no vehicle cluster exists, can be different from the attachment rate of cluster
evolution, when a random cluster with n ≥ 1 vehicles already exists at the
road [13,14,15,40].
A first nucleation model based on the master equation for traffic breakdown
at an on-ramp bottleneck, i.e., in an open traffic system has been suggested
by K¨ uhne et al. [41] and Mahnke et al. (Chap. 17 of Ref. [40]). As in the
models of homogeneous road [13,14,15], in this model at given flow rates to
the on-ramp qonand on the main road upstream of the on-ramp qinrandom
vehicle cluster occurrence and evolution that can lead to traffic breakdown
are realized only after a vehicle precluster consisting of one vehicle randomly
appears at the bottleneck. The rate of precluster emergence is equal to the
flow rate to the on-ramp [41,40]: w+(0) = qon. This idea about foregoing
precluster emergence is associated with a basic model assumption that at qon=
0 no vehicle cluster can randomly appear, specifically no traffic breakdown is
possible [41,40]. Probably, this basic model assumption leads to the nucleation
rate for traffic breakdown that is proportional to qon[41,40]. However, in real
traffic flow both flow rates qonand qinexhibit random fluctuations that can
cause cluster emergence.
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Moreover, whereas the basic assumption about the necessity of precluster
emergence is physically justified for a homogeneous road [13,14,15], this is
not the case for for traffic breakdown at the bottleneck. This is because in
contrast with the model of a homogeneous road [13,14,15], an initial state of
free flow at the bottleneck at qon > 0 and qin > 0 is non-homogeneous re-
gardless of fluctuations [3]. This means that even there were no fluctuations
in free flow at the bottleneck, nevertheless free flow is non-homogeneous in a
neighborhood of the bottleneck. This is because two different flows permanent
merge within the merging region of the on-ramp – the flow with the rate qon
and the flow with the rate qin.
It has been shown in three-phase traffic theory [3] that due to the merging of
these flows a steady state of free flow at the bottleneck is non-homogeneous:
A permanent and motionless local perturbation (deterministic perturbation)
occurs at the bottleneck. Within this perturbation that can be considered a de-
terministic vehicle cluster the speed is lower and density is greater than down-
stream of the cluster. This deterministic cluster can lead to traffic breakdown
even if there were no random fluctuations at the bottleneck [42,3]. However, in
the nucleation model [41,40] the master equation is formulated for probability
of a vehicle cluster, which can occur due to fluctuations only. We see that
for an initially non-homogeneous traffic flow, which occurs at the bottleneck,
another physical approach should be applied. This is due to at least two rea-
sons mentioned above: (i) Regardless of fluctuations, there is already a vehicle
cluster at the bottleneck that growth can lead to traffic breakdown: No vehicle
precluster emerging from fluctuations is necessary for traffic breakdown. (ii)
When influence of fluctuations on traffic breakdown is considered, fluctuations
caused by both flow rates qonand qinshould be taken into account.
Thus, in contrast with the nucleation model at the bottleneck of Ref. [41,40]
a nucleation model that is adequate with empirical results for traffic break-
down at the bottleneck should consider vehicle cluster evolution in an initially
non-homogeneous free traffic. This random vehicle cluster should include both
the total vehicle number within an initial deterministic cluster and addition
vehicles within the cluster, which occur due to fluctuations.
However, there are no nucleation models and associated probabilistic the-
ories for an initially spatially non-homogeneous free flow. Moreover, struc-
ture non-homogeneities play also a very important role for phase transitions
in many other multiple-particle systems of statistical physics like granular
flow, biological physics, reaction-diffusion systems, etc. (see references, e.g.
in [20,21,22,23,24,43,44]). Thus, it seems important to derive a probabilistic
theory for traffic breakdown in non-homogeneous traffic flow.
In this paper, a nucleation model for the probabilistic breakdown phenomenon
in an initially spatially non-homogeneous traffic flow at an on-ramp bottleneck
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is presented. The article is organized as follows. In Sect. 2, the nucleation model
is considered. Nucleation rate and the mean time delay for traffic breakdown
that result from this model are studied in Sect. 3. Results of the paper as well
as their comparison with earlier nucleation models for traffic breakdown of
Ref. [13,14,15,41,40] are discussed in Sect. 4.
2 Nucleation model of traffic breakdown in initially non-homogeneous
free flow at on-ramp bottleneck
2.1 Non-homogeniety in free flow at bottleneck as reason for vehicle cluster
In accordance with three-phase traffic theory [42,3], in a nucleation model we
assume that the breakdown phenomenon at an on-ramp bottleneck is asso-
ciated with an initial non-homogeneity of free flow at the bottleneck. This
non-homogeneity exists at the bottleneck even in a hypothetical case in which
there were no fluctuations in traffic flow. In this hypothetical case, this non-
homogeneity can be considered a deterministic (permanent) local perturba-
tion localized at the bottleneck. The non-homogeneity of an initial free flow
at the bottleneck is caused by two flows, which merge at the bottleneck: (i)
An on-ramp inflow with the rate qon. (ii) A flow on the main road upstream
of the bottleneck with the rate qin. This flow merging occurs permanent and
on the same freeway location (within an on-ramp merging region). For this
reason, the non-homogeneity of free flow at the bottleneck is motionless and
permanent (Fig. 1 (a)).
To explain features of this non-homogeneity, note that at a given high enough
flow rate qinin free flow on the main road upstream of the bottleneck, vehicles
that merge from the on-ramp onto the main road force the vehicles on the main
road to decelerate in the vicinity of an on-ramp merging region. This deceler-
ation leads to a local decrease in speed and consequently to a local increase
in density in the vicinity of the bottleneck. As a result, a local perturbation,
i.e., non-homogeneity in free flow appears.
If no fluctuations in free flow are considered, as mentioned above the non-
homogeneity in free flow in the form of a deterministic local perturbation
occurs at the bottleneck. The speed v(B)
ministic perturbation correspond to the conditions
freeand density ρ(B)
freewithin this deter-
v(B)
free< v(free), ρ(B)
free> ρ(free), (1)
where v(free)and ρ(free)are the average vehicle speed and density in homo-
geneous free flow on the main road downstream of the perturbation (Fig. 1
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