Content uploaded by Qian-Yong Chen
Author content
All content in this area was uploaded by Qian-Yong Chen
Content may be subject to copyright.
Analysis of LWR model with fundamental diagram
subject to uncertainties
Jia Li∗
, Qian-Yong Chen†
, Haizhong Wang∗
, and Daiheng Ni∗‡
Abstract
The LWR model is of interest since it is simple and can successfully reproduce some es-
sential features of traffic flow, such as the formation and propagation of traffic disturbances.
In this paper, we investigate the LWR model from an uncertainty perspective. We attempt to
analyze how reliable the LWR model prediction will be if the fundamental diagram (FD) in
use is not accurately specified. To fulfill this end, we postulate a flux function (equivalently a
fundamental diagram) driven by a random free flow speed, which accommodates the scatter-
ing feature observed in the speed-density data. We provide essential mathematical properties
and solution schemes of the LWR model with the probabilistic fundamental diagram. In case
studies, the approach to evaluate the uncertainty of traffic disturbance propagation with this
model is presented. We find that if fundamental diagram in a LWR model cannot be perfectly
specified, the uncertainty associated with the location of a traffic disturbance would increase
over time. In contrast, the magnitude of the traffic disturbance can still be accurately predicted.
Keywords: LWR, Fundamental diagram, Uncertainty, ENO
∗Department of Civil and Environmental Engineering, University of Massachusetts at Amherst,
01003, USA (lijia.ust@gmail.com, wang@engin.umass.edu, ni@ecs.umass.edu).
†Department of Mathematics and Statistics, University of Massachusetts at Amherst, 01003,
USA. (qchen@math.umass.edu).
‡Corresponding author.
1
1 Introduction
1.1 Background
The LWR model (Lighthill and Whitham, 1955; Richards, 1956), as the simplest kinetic wave
(KW) model, is central in many investigations associated with traffic dynamics and control. This is
probably because some essential features of traffic flow, such as wave formation and propagation,
can be qualitatively well reproduced with the LWR model. The reader is referred to (Daganzo,
1995; Zhang, 2001), and the references therein for a comprehensive discussion of this model and
its variations.
At the same time, it is well known that the LWR model has limited capability to capture certain
complicated traffic phenomena. For example, the original LWR model fails to generate capacity
drop, hysteresis, relaxation, platoon diffusion, spontaneous congestion, ghost jams and traffic os-
cillations (i.e., stop-and-go traffic of period 5 to 10 minutes), etc. The existence of these traits
have been confirmed in numerous field observations. The plots of speed-concentration or flow-
concentration demonstrate the complexity of traffic flow in a straightforward way, where scatter-
ings are usually structured (see, e.g. Fig. 1). In any realistic traffic flow model, the mass (i.e., the
number of vehicles) conservation always holds true and is unquestionable.
The major drawbacks of the original LWR model lie in the facts that: First, an equilibrium
bivariate relation between traffic variables, e.g., traffic flow and occupancy, is always assumed;
Second, the original LWR model does not consider the particular properties of the traffic flow
induced by the road geometry and associated latitudinal driving behavior, such as lane-changing.
Moreover, the LWR model assumes a uniform driving population, while in reality inhomogeneity
of drivers and vehicles is non-negligible. Accordingly, to obtain better description of traffic flow
based on the LWR model, one may either incorporate high-order effects into the fundamental
diagram, i.e. bivariate relation of the equilibrium traffic variables, or adopt the LWR as the building
block model in a more realistic setting and incorporate the above mentioned factors explicitly.
The high-order effects in each case have different meanings depending on the application sce-
2
narios. For example, it may indicate the relaxation and anticipation in the PW-like models. In
all the models where high-order effects are considered, ‘momentum’ equations often arise. Com-
pared with the LWR model, high-order models admit much more rich phase transition patterns
and possess metastable property (see Zhang et al. (2009) for a more thorough discussion, where
the physical boundedness of solution is also discussed). Traveling wave solutions of such models,
as a high-order counterpart of shock waves, provide an explanation to the scattering observation.
Interested reader is referred to (Greenberg, 2004; Zhang and Wong, 2006) for in-depth discussion
of this topic.
Another way of incorporating the high-order effects is less explicit, which relaxes the smooth-
ness and concavity constraints of the fundamental diagram (Koshi et al., 1983). These two ap-
proaches are essentially the same, and are unified in the ‘effective fundamental diagram’ frame-
work (Zhang, 2001), where they are named the ‘higher-order’ and ‘lower-order’ extensions of the
LWR model, respectively. In the second case, one considers the externalities such as network ge-
ometry (Jin, 2003), lane-changing (Laval and Daganzo, 2006), and traffic inhomogeneity (Wong
and Wong, 2002; Ngoduy and Liu, 2007). It is noteworthy that in some recent efforts, the funda-
mental diagram is modeled as a random function (Ngoduy, 2009; Wang et al., 2009) in the context
of multi-class modeling. Such modeling essentially captures the random variation of driving be-
havior. Phenomena such as capacity drop, platoon diffusion and traffic oscillation are successfully
reproduced in these models. Further, Ngoduy (2010) applied multi-class modeling to describe
traffic network operations based on a dynamic routing algorithm which determines turning move-
ments at nodes. Also considering network geometry, Sun et al. (2009) proposed a shock-fitting
algorithm to address traffic flow on inhomogeneous highways. By introducing an interface be-
tween homogeneous sections and considering entropy and physical conditions, inhomogeneous
highway problems can be solved by applying the shock-fitting algorithm to equivalent homoge-
neous sections.
The above discussion reveals that the fundamental diagram is indispensable in almost all exten-
sions of the LWR model. Therefore, a thorough discussion of the fundamental diagram is desirable.
3
In both theoretical and practice point of view, the existence and specification of the fundamental
diagram is of critical concern. In particular, whether the scattered observations are inherent and
how they should be treated attract special attentions. Castillo and Ben´
ıtez (1995), Coifman (1996),
and Cassidy (1998) addressed the problem of the fundamental diagram fitting. They demonstrated
that well-defined and reproducible relations of the traffic variables, such as flow-occupancy, can
be obtained if the stationary traffic data are carefully identified and used. Zhang (2001) indicated
that such well-behaved fundamental diagram is obtained at the price of lowering data resolution.
In another word, the scattered observations associated with the empirical fundamental diagram are
eliminated when transient traffic states are smoothed out or discarded.
Some other researchers attempt to interpret the scattered observations directly. For example,
the hysteresis phenomenon has been well known (Treiterer and Myers, 1974; Zhang, 1999), which
indicates that the speed-occupancy curve consists of hysteresis loops. Meanwhile, Newell’s ex-
tension of the fundamental diagram (Newell, 1965) and the reverse-λshaped the fundamental
diagram (Koshi et al., 1983) have been reported in literature for a long time. Also, Treiber and
Helbing (2003) show that conventional measurement methods with the delay of driving behav-
ior may result in the scattering of flow-density data in ‘synchronized’ congested traffic. In short,
the recurrent non-equilibrium transitions associated with the fundamental diagram seems to be a
versatile yet puzzling problem, and continues triggering discussions of different dimensions in re-
cent years. Nonetheless, it is evident that the exact nature of the fundamental diagram, is hardly
unequivocal till present. In this sense, uncertainty with the fundamental diagram persists.
The aim of this paper, rather than further develop or unify the existing models that employ
or describe the fundamental diagrams, is to quantify the potential influences brought up by the
ambiguous specification of the fundamental diagram in a kinematic wave setting. To fulfill this
end, we first analyze the sources of uncertainties associated with the fundamental diagram. Then
following the methodology of uncertainty analysis, we develop a numerical procedure to evaluate
how the uncertainty of the fundamental diagram would affect the output of the LWR model. We
consider the scenario where the LWR model is used to predict the location and magnitude of a
4
traffic disturbance. This scenario is critical to more complicated applications, such as the queue
length estimation at a signalized intersection (see, e.g. (Yi et al., 2001) for illustration) and ramp
metering of a freeway system.
The uncertainty analysis of kinematic wave model is rare. Our study reveals an important
yet unknown quantitative feature of the LWR model, which is of great practical significance and
theoretical interests, as the LWR model is extensively employed in various applications, such as
queue length estimation, ramp metering, etc. We caution the use of the LWR model in applications
where the location of traffic disturbance is of critical concern. Otherwise, the LWR model is robust.
1.2 Paper organization
The remainder of this paper is organized as follows. In Section 2, we first present a brief account
for uncertainty analysis and differentiate the source of uncertainties associated with the speed-
density plot. In this context, we discuss the uncertain representation of the fundamental diagram
as a random function, which accommodates a distribution of flow for any given density value.
Then we propose a specific form of the random fundamental diagram which assumes that the
random nature is dominated by the uncertain determination of free flow (Section 3). Reasons and
implication of introducing such assumption are detailed therein, in a perspective of interpolation.
With the proposed random fundamental diagram, we conduct a numerical investigation (Section 4).
The model to be solved is the LWR model with the proposed random fundamental diagram. An
algorithm based on the essentially non-oscillatory (ENO) finite difference scheme is implemented
to evaluate the uncertainty associated with propagating traffic disturbance. In Section 5, we briefly
summarize our findings. Though not attempting to be explanatory, our study essentially unveils
the robustness of the LWR model to the imperfect specification of the fundamental diagrams. It is
observed that, mathematically, the magnitude of traffic disturbance predicted by the LWR model
is insensitive to the initial specification of the fundamental diagram. The initial uncertainty of the
fundamental diagram is damped as time goes by. However, the error of the predicted location of
traffic disturbance may deteriorate.
5
2 An analysis of uncertainty
As discussed in the preceding section, multiple reasons account for the scattered observation as-
sociated with the speed-density relationship of traffic, i.e., the fundamental diagram. A thorough
investigation of the scattering phenomenon serves as the basis for further model development and
analysis.
The uncertainty is commonly understood as the factors related to imperfect knowledge of the
system under concern, especially those being random in nature. It is closely related to hetero-
geneity, which denotes the state that entities within a given system are of non-uniform character.
For example, when the heterogeneity is not faithfully recognized, the uncertainty increases. Con-
versely, decrease of uncertainty means that the system is better understood, thus the heterogeneity
is better characterized. In the context of traffic flow theory, the issue of uncertainty is more compli-
cated, because even for a uniform freeway traffic system, the understanding of its behavior seems
still incomplete. In particular, empirical recurrent transition patterns of traffic states have not yet
been adequately identified, as also implied by the discussion in preceding section. Moreover, there
is unlikely to be any unquestionable physical law for the traffic flow except the conservation of ve-
hicles. As a result, no traffic flow model can literally exclude all the others. That means, whatever
traffic flow model is employed, the uncertainty associated with the modeling is inevitable.
In system modeling literature, the uncertainty is formally defined as “any deviation from the
unachievable ideal of completely deterministic knowledge of the relevant system” (Walker et al.,
2003). Therefore, uncertainty could indicate any factor that is not appropriately captured by rele-
vant model. Mathematically, it is usually modeled as a set of random variables. The uncertainty
analysis is to systematically model the uncertainties associated with a system and to understand
the related influences to the system output. Risk and reliability assessment are among the most
significant examples of uncertainty analysis, and find their applications in hydrology, structure en-
gineering and economics, etc. The uncertainty analysis in the general context of conservation laws
has been reported in literature for over two decades. Many of them are in the same spirit of ran-
domizing the flux function of a conservation system, either spatially or temporally. For example,
6
in Holden and Risebro (1991); Holden (2000), the authors discuss the expectation and conver-
gence of a stochastic Buckley-Leverett equation. In Wehr and Xin (1997) the large-time property
of Burger’s equation is analyzed. More recently, Bale et al. (2002) presented a general solution
approach, without directly addressing the issue of randomness, for the conservation equation with
flux function admitting spatial variation. All of them have focused on the spatial variability of the
flux function.
In the traffic field, Jou and Lo (2001) considered a nonlinear macroscopic traffic flow equation
perturbed by a Brownian motion. Zhang et al. (2008) address the spatial inhomogeneity in multi-
class traffic flow models, which essentially is a conservation law problem with a spatially varying
flux. The multi-class model with temporally varying flux, namely the fundamental diagram with
randomness over time, has also been investigated (Ngoduy, 2009). As a side note, the latter two
studies did not stress the uncertainty issue directly. Rather, the efforts are mainly to explain the pre-
viously unexplained phase transitions or wave patterns as a result of system inherent randomness
and inhomogeneity. But such formulations are mathematically equivalent to that of an uncertainty
analysis. The key difference lies in that uncertainty analysis is more general and does not mean
to build up any specific causality relation. The fundamental focus of uncertainty analysis is to
quantify the influence due to imperfect modeling.
In short, both this paper and the work in Ngoduy (2009) deal with the randomness associated
with traffic flow processes, which is believed to result in the wide scattering with fundamental
diagram. However, Ngoduy (2009) try to model the random capacity, and reproduce the wide
scattering. Our work investigates how the uncertainty may influence the prediction of the LWR
model. That is, if the fundamental diagram cannot be precisely specified due to the various sources
of uncertainties that we enumerate in the paper, can we still trust the prediction result of the LWR
model? In other words, Ngoduy’s work is modeling, and we are dealing a dual problem of model-
ing, namely, uncertainty analysis.
Before unfolding our investigation, empirical evidence is presented. For any empirical speed-
density relation, the mapping from density to speed is always multi-valued after rounding off the
7
density data. This enables us to calculate the mean and standard deviation of the speed indexed
by density. In Figure 1 the mean and standard deviation of the speed versus density at one station
from GA-400 ITS dataset (which are collected by virtual loop detectors on freeway GA-400) are
shown. In this plot, one can see that the standard deviation is comparable to the mean of speed
in quite wide range, indicating the existence of non-ignorable variability in speed-density relation.
Moreover, the shape of the standard deviation is somewhat interesting, exhibiting a peak around
50 veh/mi, with magnitude 15 mph.
0 50 100 150 200 250 300 350 400
0
10
20
30
40
50
60
70
density k (veh/mi)
velocity v (mi/hr)
mean
std
Figure 1: First and second order speed-density relation at one typical site based on GA400 ITS
data
In our investigation, we focus on how such variability, i.e. uncertainty (we explain their con-
nection in the next paragraph), influences the output of a LWR model. The LWR model assumes
that the speed-density (v-k) relation is time independent, i.e., the system equilibrium is already
achieved (by equilibrium, we mean that only transition between stationary traffic states is allowed).
The LWR model says that the density k(x, t)of traffic flow is the solution to the following equa-
tions: kt+ (kv)x= 0
v=v(k)
k(x, 0) = k0(x).
(1)
The first equation is the conservation law with the assumption of enough regularities of involved
functions, the second equation is the fundamental relation which holds under the equilibrium as-
sumption, and the last equation provides the initial state of the solution. Flow q=q(k)≡kv(k)is
8
known as the flux function of k, which depicts the dynamics of the concerned quantity, i.e., traffic
flow density.
The structured variability of the empirical speed-density relation, induces uncertainty whenever
a fundamental diagram is used. To correctly incorporate the exhibited randomness into Eq. (1), we
first need to analyze the possible sources of randomness. The variability in Fig. 1 is likely to be
the result of multiple factors, e.g., transient traffic, multi-class issue, lane-changing, measurement,
etc. Namely, the models appearing in the preceding discussions may all explain the variability to a
certain extent. But we suspect that no single model would be adequate.
The involved uncertainties fall into two categories. The first type of uncertainty is the ran-
domness during data collection and processing, e.g., inaccurate reading and data roundoff that is
reflected in the initial data setting and scatter plot of the v-krelation. In this case, the collective
small additive errors would altogether obey the normal distribution law, according to the central
limit theorem (CLT); alternatively, if the errors are small and multiplicative, they jointly behave by
the lognormal law. This type of uncertainty is statistical and relatively well studied. Another type
of uncertainty pertains to the dynamics of traffic. For example, the drivers’ behaviors vary from
one driver to another, thus the group is best described in distributional terms rather than determin-
istically. Our discussions in this and the preceding sections imply that the imperfect modeling of
the real behavior of traffic leads to this kind of uncertainty.
In the case of the LWR model, the fundamental diagram itself cannot be perfectly specified,
because the empirical fundamental diagram is the result of more complex dynamics that are un-
explained in the scope of the LWR model. Therefore, uncertainty arises. This is exactly the un-
certainty that we are interested in and want to study. We argue that the second type of uncertainty
is essential and dominant, since the first type can usually be controlled reasonably well, through
improving the measuring techniques. At last, we provide an intuitive interpretation of Fig. 1, in
a manner analogous to that of the Brownian bridge. That is, taking the vas a random process
indexed by k, at two points 0and jam density kjam the knowledge of vis relatively complete since
the constraints are imposed by the definition of the two states. This explains the smaller variances
9
at two ends as shown in the plot.
3 A descriptive model of fundamental diagram
With the above analysis, we only consider the second type uncertainty when reformulating the
LWR model since it is dominant and inherent. Most generally, to account for this randomness, we
express the traffic speed as a positive valued multivariate function,
v=v(k, x, t, ω) : (R+,R,R+,Ω) 7→ R+,(2)
where ωis an appropriately defined set on Ω, the probability space equipped with measure Px,t(·).
Definition of such form means that the traffic speed is not only a function of density k. The
inclusion of xand tenables us to model inhomogeneous roads and as well as any road whose
condition changes over time (e.g. by car accident). The dependence of von ωis included to model
the previously discussed uncertainty. So even when k, x, t are fixed, the traffic speed is not known
for sure due to its probabilistic nature.
This definition leads to a stochastic flux function, which is the product of traffic speed and
density,
f≡kv =kv(k, x, t, ω) : (R+,R,R+,Ω) 7→ R+.(3)
Substituting the Formula (3) back into Eq. (1), we obtain the stochastic formulation of the LWR
model as,
kt+ (kv(k, x, t, ω))x= 0.(4)
Equation (4) in general admits a random function as solution if it exists. Before attempting to
obtain useful calculation results, it is desirable to justify the validity of this equation. The mini-
mum requirement is that with trivial probability measure it is consistent with usual deterministic
equation. A detailed discussion of the general form of Eq. (4) is beyond the scope of current paper.
To make the analysis and computation tractable, we restrict our focus to some specific form of
flux function in this paper. This is equivalent to treat a specific form of Equation (3). In particular,
10
we assume that the stochastic flux function is independent of both space and time, i.e., we are
dealing with homogeneous road with no accident in the course of consideration,
f=f(k, ω).(5)
The assumption in the form of (5) greatly simplifies the matter since each realization of fwould
be a function of one variable konly, which is just the deterministic case we usually deal with.
To address the observed pattern of uncertainty in Fig 1, in this paper we attribute the uncertainty
of f(·)to the uncertainty of the free flow speed. This decision is out of the following consideration.
In Castillo and Ben´
ıtez (1995), it is argued that the free flow speed, jam density and shock wave
speed at jam density fully characterize a flow-density relationship under very modest assumptions
regarding its smoothness and convexity. Therefore, the uncertainties of the fundamental diagram
are mostly inherited from these three parameters. The jam density is almost constant. Moreover,
various research suggests a very consistent value for the shock wave speed at jam density (about
−20 km/h), when using different data sets (Castillo and Ben´
ıtez, 1995). Thus, it is appropriate
to assume, in our first study of the uncertainty effect, that the free flow speed introduces most
uncertainties in the modeling of the fundamental diagram. However, as shown in Li (2008), the
constant wave speed is not well-defined for certain cases. In addition, the capacity appears as a
parameter for the speed-density relation proposed in Li (2008), which makes it possible to directly
model the randomness of the capacity. The advantage of such possibility needs to be further
investigated.
Based on the above analysis, we write the free flow speed in the following form,
vf≡¯vf+ (sk +r)ε, (6)
where ¯vfis a constant, and εrepresents the imperfect knowledge of actual vf, with (sk +r)being a
scaling coefficient. Adopting this scaling coefficient implicitly assumes that vfdepends on density
k. One may write ε=λ˜ε, where E˜ε= 0,σ˜ε= 1 and λcontrols the variance of ε. The reason
of using a function of kas the scaling coefficient is as follows. If we adopt the following v-k
relation (this relation can be derived from car-following models and reduces to a series of classical
11
fundamental diagram models, see (Castillo and Ben´
ıtez, 1995) and (Yi et al., 2001)),
(v(k)
vf
)α+ ( k
kjam
)β= 1,(7)
then after substituting Equation (6) in, we have,
v(k)d
=vf(1 −(k
kjam )β)1/α
= (¯vf+ (sk +r)ε)(1 −(k
kjam )β)1/α.(8)
Combined with the definition of flux, we get the formula for the flux:
f(k)d
=k(¯vf+ (sk +r)²)(1 −(k
kjam
)β)1/α.(9)
The meaning of sk +ris interpreted as follows. Letting s > 0, then as kincreases, the variance
of the first term in the product of formula (8) becomes larger. This reflects the common perception
that the free flow speed is less informative for the inference of v(k)when density kincreases, as in
this case, the system state represented by (k, v)is moving away from the state (0, vf). Since vfand
kjam are symmetric in Equation (7), we may have a similar argument and term similar to (sk +r)ε
accompanying kjam. However, this will make it difficult to explicitly express vas a function of
k. Though we can numerically solve vfor a given kby utilizing an iterative procedure, we would
rather put this issue aside at the moment and restrict the focus on the formula (8) to obtain some
closed-form results.
From formula (8), taking the expectation at both sides, we obtain,
E(v(k)) = ¯vf(1 −(k
kjam
)β)1/α,(10)
and by calculating the second-order moment, we obtain,
V ar(v(k)) = (sk +r)2(1 −(k
kjam
)β)2/α.(11)
The v-krelation depicted by formula (8) is not completely based on the microscopic traffic
behaviors. It, however, has three advantages. First, it is constructed in a heuristic manner as
previously mentioned. Its interpretation is sound and straightforward. Second, the postulated v-k
relation is consistent with the observation that a peak of standard deviation exists between 0and
12
kjam, which also takes a maximum value in this region. Third, this relation leads to an easy-
to-sample random flux function (9), which is actually a family of curves controlled by only one
random parameter ε. The essential mathematical properties of (9) is provided in the appendix,
which are the basis for the solution of the LWR model and uncertainty evaluation.
4 Numerical investigation
When the LWR model is embedded with uncertainty, the predicted propagation of a traffic distur-
bance shall be uncertain, in terms of its location and magnitude. Purpose of this section is to reveal
the uncertain feature of the LWR model through a set of numerical examples.
4.1 Problem setting
The propagation of local traffic disturbance on a one-lane freeway is studied. Traffic disturbance is
either a local hump or a local vacuum of the otherwise constant traffic density profile. The density
profile at time zero is denoted as k(x) = k0+ε˜
k(x), where xis spatial coordinates, k0is a constant
representing the dominant density on the road, ˜
k(x)is some non-constant function representing
the disturbance to k0, and εgoverns the magnitude of disturbance. When no uncertainty of the
fundamental diagram is taken into consideration and εis small, the problem is relatively well
understood in literature. The LWR model in this situation is approximated by its linearized form,
kt+Q0(k0)kx= 0 (12)
solution to which is k(x−ct), where c=Q0(k0). This implies that roughly the disturbance prop-
agates at a speed of Q0(k0)with the shape of disturbance unchanged. It is, however, curious to
know how the uncertain specification of the fundamental diagram influences the propagation of
the traffic disturbance. Normally one would expect that both the location and magnitude of traffic
disturbance be uncertain in this case. But whether the uncertainties will diminish or increase is
largely unknown and needs to be explored. From a practical standpoint, the diminishing uncer-
tainty is favorable, because it means that the initial possibly mis-specified fundamental diagram
13
does not deteriorate the predictions of the LWR model in the long run. In mathematical terms, we
are interested in the following mapping:
φ: (UF D, t)7→ (Uloc, Umag )(13)
where UF D, Uloc, Umag denotes the uncertain levels of the fundamental diagram, location and mag-
nitude of traffic disturbance at time t, respectively. The particular measures of uncertain level are
discussed in the coming subsection.
4.2 Uncertainty measures
The uncertainty can be measured in multiple ways, depending on the circumstances it is used.
Assume the concerned quantity is X, which is mathematically represented by a random variable.
The uncertainty of X, intuitively the dispersion of values of X, is commonly measured as follows,
1. Standard deviation: σX;
2. Coefficient of Variation (CoV): γX=σX/µX;
3. L2-moment.
In our investigation, the location and magnitude of a traffic disturbance are of primary interests.
The uncertainty of two quantities Uloc, Umag are best described the coefficient of variation, or the
CoV, since it essentially normalizes the variability of a quantity to its absolute magnitude. This
concepts is in analogy to the ratio of noise and signal, and provides useful insight into the potential
relative error when one adopts a deterministic model (e.g., the LWR model in its original form) to
make predictions. Moreover, the coefficient of variation is dimensionless and invariant with respect
to the change of units of the system. In this sense, its numeric value can be easily compared with
the uncertainty of other models and its interpretation keeps consistent. In addition, the uncertain
level of the fundamental diagram is due to the randomness of vfand it assumes a parametric form,
as we postulated above. Therefore, its uncertain level can be simply represented by the standard
deviation associated with the parameter vf.
14
4.3 ENO-based numerical algorithm
We adopt the finite difference methods with an essentially non oscillatory (ENO) reconstruction
to obtain the numerical solution of (1). Roughly speaking, the ENO method reconstructs the cell
boundary values through adaptively utilizing the local stencil information. In particular, the stencil
with the minimal non-smoothness measure is selected. With ENO, high order finite volume or
finite difference methods are immediately available. The order of accuracy depends on the size of
the adopted stencil. The effectiveness of this method to solve the conservation equations of traffic
flow (i.e., the LWR model) has been examined, e.g., in Zhang et al. (2003) and Zhang et al. (2006).
Using an ENO-based algorithm enables convenient extension of current analysis to more complex
problems, such as multi-class models. For a detailed description of this method, the reader is
referred to Shu (1998) and references therein. In particular, we use the 3rd order ENO in our
investigation.
Based on the ENO scheme, an algorithm can be devised to compute the uncertainty associated
with the traffic propagation. The Monte-Carlo simulation (MCS) is implemented in this paper.
Its core building block consists of two sequential steps: initialization and time-marching. In the
initialization step, initial data are loaded and random fundamental diagram is sampled. In the time-
marching step, temporal evolution of the density profile is computed with the ENO scheme. By
repeating the two steps, the statistics of density profile which is subject to the randomness of the
fundamental diagram can be obtained. Accordingly the uncertainty measure as mentioned above
can also be calculated.
The notations to employ are explained as below: Course of simulation, [0, T ]; Simulation
range, [0, N ∆x]; Time step, ∆t; Grid length ∆x; Time index, j, with j∈Z+, j ≤T/∆t; Space
index, i= 1, . . . , N. Simulated average traffic density of i-th cell at time j,kj
i.
* Initialization:
1. Load initial data {k0
i, i = 1, . . . , N }, where k0
i=k0(xi)is the grid point value. Moreover,
set j= 1,ε0= 1;
15
2. Generate εj, independent of ε1, . . . , εj−1and follows the same prescribed distribution, e.g.
εj∼U(−√3,√3), for all j. Generate the random flux function f(k)following Eq. (9);
3. Split the flux function f(k) = f+(k) + f−(k), following Eq.(23) in Appendix;
* Time Marching:
4. Identify {f+(kj
i), i = 1 . . . , N}as cell averages and obtain v−
i+1/2=ˆ
f+
i+1/2by ENO recon-
struction. Similarly, get v+
i+1/2=ˆ
f−
i+1/2;
5. Update the cell values,
kj
i=kj−1
i−∆t
∆x(ˆ
fi+1/2−ˆ
fi−1/2)(14)
Check whether j+ 1 ≤T /∆t. If so, let j=j+ 1, and go back to 2; otherwise, stop.
4.4 Case studies
Throughout the following calculations, we assume traffic disturbance at time zero be as follows,
˜
k(x) = sin((x−2)π)I(2,3) (15)
and dominant density,
k0= 50 (16)
where the unit of xis mile, unit of density is veh/mile. The density profile described by k0+γ˜
k
has a disturbance of 1 mile, located at 2.5 mile and with magnitude γ. We set α= 1,β= 1,
s= 0.05 and r= 3. Moreover, let ¯vf= 60 mile/hour and kjam = 200 veh/mile, then the random
flux function takes the form of
f(k)d
=k(60 + (0.05k+ 3)ε)(1 −k
200)(17)
where as before, ε=λ˜ε,˜εis a random scalar with zero mean and unit variance. λis the parameter
controlling the uncertainty level of the fundamental diagram.
The ˜εcan be conveniently selected to follow the uniform distribution law U(−√3,√3). Let
λ= 1, the v-krelation and 20 realizations of the flux in Eq. (17) are shown in Figure 2. The
16
computation domain is 10 mile, discretized with time step ∆t= 1 second, and grid length ∆x=0.1
mile. The free boundary conditions are imposed, i.e., ∂k/∂x|L,R = 0, where L, R indicates the left
and right boundaries of the computation domain.
0 50 100 150 200 250 300 350 400
0
10
20
30
40
50
60
70
density k (veh/mi)
velocity v (mi/hr)
mean
std
0 50 100 150 200
0
500
1000
1500
2000
2500
3000
3500
4000
density k (veh/mi)
flow q (veh/hr)
Figure 2: Hypothetical random speed-density relation (left panel) and 100 realizations of corre-
sponding random flux function (right panel)
Case I Deterministic fundamental diagram: Let λ= 0, which reduces the model depicted by
Equation (17) to a deterministic model. With the initial conditions,
Initial condition a: (kl, kr) = (110,30)
Initial condition b: (kl, kr) = (30,110),(18)
the so-called Riemann problem is to be solved. Consistent with the theoretical prediction, rarefac-
tion wave and shock wave are observed, which are shown in Fig. 3. This indeed shows that the
ENO type finite difference scheme does preserve the shape of shock wave well, with almost no nu-
merical oscillations. Such feature is desirable and essential in our study to rule out the possibility
of artificial traffic oscillations.
Case II Dispersion of Density Profile at a Fixed Time: In this case, the dispersion of density
profiles at a fixed time is computed subject to two initial conditions: local congestion and local
vacuum. The time is set to 60 sec and λ= 1. The simulation results are shown in Figure 4. One
can see that, due to the uncertainty of the fundamental diagram, the output exhibits significant
dispersions, in terms of their locations. It indicates that the predictability of location of traffic
disturbances deteriorates. Meanwhile, the magnitude of traffic disturbance is also uncertain but
17
0
5
10 0 50 100 150 200 250
20
40
60
80
100
120
time t (sec)
distance x (mi)
density k (veh/mi)
0
5
10 0 50 100 150 200 250
20
40
60
80
100
120
time t (sec)
distance x (mi)
density k (veh/mi)
Figure 3: Temporal development of density kwith deterministic the fundamental diagram: rar-
efaction waves (left panel) under Riemann initial condition (kl, kr) = (110,30), and shock waves
(right panel) under Riemann initial condition (kl, kr) = (30,110).
Mean of k∗Std of k∗Mean of x∗Std of x∗
Under initial condition a 16.09 0.50 5.99 0.75
Under initial condition b 26.46 1.92 8.63 0.63
Table 1: The statistic of numerical solution under initial condition a and b.
with much less variation across all the realizations. The statistics of two quantities, the maximum
local fluctuation and its location, are listed in Tab. 1. They represent the magnitude of traffic
disturbance which is defined as
k∗= max
0≤xi≤10 |k(xi)−50|,(19)
and the location of traffic disturbance
x∗=argmaxxi|k(xi)−50|.(20)
The measure of uncertainty level of k∗, x∗, i.e., the CoV’s, are calculated using the values in Tab. 1.
It turns out that CoVa,k∗>CoVb,k∗and CoVa,x∗>CoVb,x∗. This implies that the local vacuum is
somewhat more predictable than the local jam.
Case III Evolution of Uncertainties: The uncertainty of the fundamental diagram is transferred
to the solution of the LWR model and thus propagates over time. The evolution trend of this
uncertainty would be of interests. Information of this type provides important hints regarding the
18
0246810
0
20
40
60
80
100
120
140
160
180
200
distance (mi)
density k (veh/mi)
0246810
0
20
40
60
80
100
120
140
160
180
200
distance (mi)
density k (veh/mi)
Figure 4: Propagation of local disturbance (Left: local jam; Right: local vacuum) with λ= 1,
t= 600 sec, 20 realizations. The dotted line is initial data and solid lines are simulation results.
extent to which the prediction of the LWR model is still acceptable. Normally, prediction is less
useful when uncertainty associated with the prediction becomes too ’big’.
Following the general setting of the last case but considering a time span, we run the simulation
with the initial condition being a local jam. The results are shown in Fig. 5. In these figures, each
solid line corresponds to a random fundamental diagram with a fixed uncertainty level, i.e., a fixed
value of λ(which is defined in Sec. 3). In particular, λranges from 0.5to 1.5and distributes
evenly.
The interesting finding is: the uncertainty of the location and the magnitude of a traffic distur-
bance exhibits an opposite trend over time evolution. For all the uncertainty levels, the uncertainty
of the location keeps increasing. While under the same circumstance, the uncertainty of the mag-
nitude of traffic disturbance is suppressed over the time.
Moreover, we observe that Umag is at the order of 0.1and Uloc is at the order of 0.01. Here Umag
and Uloc are the uncertainty level of the magnitude and location of the disturbance, respectively.
This implies that in the LWR model, the location of predicted density disturbance is more sensitive
to the uncertainty of the fundamental diagram. Even worse, the error of the location prediction
due to mis-specification of the fundamental diagram increases over the time. One may roughly
estimate from the left panel of Fig. 5 that the Uloc roughly increases at a rate of 1%/min. Suppose
19
that a traffic disturbance travels at 10mi/hr, one can roughly calculate that after 30 minutes, the
absolute prediction error is 10mi/hr ×0.5hr ×30% = 1.5mile. Though the calculation needs
more empirical justification, from a practical standpoint, it may pose a challenging problem for the
traffic control on long arterial roads, where the location of congestion is a major concern.
0 100 200 300 400 500 600
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time (sec)
Uloc
0 100 200 300 400 500 600
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Time (sec)
Umag
Figure 5: Propagation of uncertainty over time (Left: uncertainty of location Uloc; Right: uncer-
tainty of magnitude Umag). The uncertainty level is calculated based on 20 trials in each scenario.
Multiple conclusions can be drawn from the above case studies. The LWR model is very suc-
cessful on predicting the propagation of first-order waves. However, our example indicates that
caution should be taken with such an application when the specification of fundamental diagram
is uncertain. Relatively small randomness of the fundamental diagram could result in significant
dispersion of the location of traffic disturbance. In addition, we observe an increasing trend of
the uncertainty level of the location of traffic disturbances. This is consistent with the common
recognition that a model deteriorates when applied for a long time prediction. But surprisingly,
our investigation shows that the uncertainty level of the magnitude actually decays. Another ob-
servation is that, the propagation of local vacuums seems more predictable than the propagation of
local jams when all other conditions are the same. Overall, when the fundamental diagram cannot
be perfectly specified, the corresponding prediction of the LWR model should be limited to rela-
tively short range of time, as the uncertainty level of the location of propagating traffic disturbance
is always increasing over the time.
20
5 Concluding remarks
Motivated by empirical observations from freeway traffic data, in particular by the scattering of the
empirical the fundamental diagram, we investigate the LWR model with the fundamental diagram
involving uncertainties. We discuss the general concept of uncertainty and analyze the sources
of randomness associated with the fundamental diagram. A specific form of the stochastic flux
function is postulated, which is driven by the random free flow speed. We provide essential math-
ematical analysis of the proposed fundamental diagram. An ENO-based finite difference method
combined with the Monte-Carlo method is implemented to compute the uncertainty level asso-
ciated with propagating traffic disturbance. Our case studies imply that that if the fundamental
diagram cannot be undoubtedly specified, the prediction of the location of traffic disturbance may
become problematic.
There are various sources of uncertainty associated with the fundamental diagram in the LWR
model for traffic system. It is realized that the fundamental diagram is uncertain due to multiples
reasons and a detailed account of all causes could be very complicated. The assumption that the
uncertainty is from a single parameter (namely, the free flow speed), is the first step to study
the uncertainty intrinsic to traffic flow. However, one may expect that similar results hold when
other more realistic the fundamental diagram is adopted. The merit of this study is to initiate the
investigations of uncertainty associated with the fundamental diagram, which is a key component
of the LWR model. We plan to further investigate the influence of uncertainty on other traffic flow
models and make comparison with empirical observations in the future.
Appendix
Properties of the proposed fundamental diagram
At last, we provide the following properties that are useful for developing the numerical scheme
of uncertainty evaluation. When α=β= 1, the function defined by Eq. (9) has the following
properties:
21
1. For each ε, function f(k)is smooth.
2. Upper bound of absolute value of the derivative:
sup
0≤k≤kjam |f0(k)| ≤ 5kjam|εs|+ 3|εr|+ 3¯vf(21)
This is obtained by utilizing the triangle inequality. For technical convenience, we take
²∼U(−√3,√3), making the right hand side of (21) a finite value, denoted as αf,
αf≡α(kjam,¯vf, r, s) = 5√3kjams+ 3√3r+ ¯vf(22)
3. There exists decomposition of f(k)(namely, flux splitting):
f(k) = f+(k) + f−(k),(23)
where f+(k)=(f(k) + αfk)/2and f−(k) = (f(k)−αfk)/2, satisfying df+(k)/dk > 0
and df−(k)/dk < 0.
Acknowledgment
The authors wish to thank the three anonymous reviewers for their valuable comments.
22
References
D.S. Bale, R.J. LeVeque, S. Mitran, and J.A. Rossmanith. A Wave Propagation Method for Con-
servation Laws and Balance Laws with Spatially Varying Flux Functions. SIAM J. Sci. Comput,
24(3):955–978, 2002.
M.J. Cassidy. Bivariate Relations in Nearly Stationary Highway Traffic. Transportation Research
Part B, 32(1):49–59, 1998.
J.M.D. Castillo and F.G. Ben´
ıtez. On the Functional Form of The Speed-Density Relationship I:
General Theory. Transportation Research Part B, 29(5):373–389, 1995.
B.A. Coifman. New Methodology for Smoothing Freeway Loop Detector Data: Introduction to
Digital Filtering. Transportation Research Record: Journal of Transportation Research Board,
1554(1):142–152, 1996.
C.F. Daganzo. Requiem for Second-Order Fluid Approximations of Traffic Flow. Transportation
Research Part B, 29(4):277–286, 1995.
J. M. Greenberg. Congestion Redux. SIAM Journal on Applied Mathe-
matics, 64(4):1175–1185, 2004. doi: 10.1137/S0036139903431737. URL
http://link.aip.org/link/?SMM/64/1175/1.
H. Holden and N.H. Risebro. Stochastic Properties of The Scalar Buckley-Leverett Equation.
SIAM Journal on Applied Mathematics, 51(5):1472–1488, 1991.
L. Holden. The Buckley-Leverett Equation with Spatially Stochastic Flux Function. SIAM Journal
on Applied Mathematics, 57(5):1443–1454, 2000.
W.L. Jin. Kinematic Wave Models of Network Vehicular Traffic. PhD Thesis, 2003.
Y.J. Jou and S.C. Lo. Modeling of Nonlinear Stochastic Dynamic Traffic Flow. Transportation
Research Record, 1771(-1):83–88, 2001.
23
M. Koshi, M. Iwasaki, and I. Ohkura. Some Findings and an Overview on Vehicular Flow Charac-
teristics. In Proceedings of the Eighth International Symposium on Transportation and Traffic
Theory: June 24-26, 1981, Toronto, Canada, page 403. University of Toronto Press, 1983.
J.A. Laval and C.F. Daganzo. Lane-changing in traffic streams. Transportation Research Part B,
40(3):251–264, 2006.
Michael Z. F. Li. A Generic Characterization of Equilibrium Speed-Flow Curves. Transportation
Science, 42(2):220–235, 2008.
M.J. Lighthill and G.B. Whitham. On Kinematic Waves. II. A Theory of Traffic Flow on Long
Crowded Roads. Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences (1934-1990), 229(1178):317–345, 1955.
G.F. Newell. Instability in Dense Highway Traffic: A Review. Proceedings of the Second Interna-
tional Symposium on the Theory of Road Traffic Flow, pages 73–85, 1965.
D. Ngoduy. Multiclass First-Order Modelling of Traffic Networks Using Discontinuous Flow-
Density Relationships. Transportmetrica, 6(2):121 – 141, 2010.
D. Ngoduy. Multiclass First-Order Traffic Model Using Stochastic Fundamental Diagrams. Trans-
portmetrica, pages 1–15, 2009.
D. Ngoduy and R. Liu. Multiclass First-Order Simulation Model to Explain Non-Linear Traffic
Phenomena. Physica A: Statistical Mechanics and its Applications, 385(2):667–682, 2007.
P.I. Richards. Shock Waves on the Highway. Operations Research, 4(1):42–51, 1956.
C.W. Shu. High Order ENO and WENO Schemes for Computational Fluid Dynamic. RTO educa-
tional notes, 5:439–582, 1998.
W. Sun, S. C. Wong, P. Zhang, and C-W. Shu. A Shock-Fitting Algorithm for the
Lighthill-Whitham-Richards Model on Inhomogeneous Highways. Transportmetrica. DOI:
10.1080/18128600903313936, 2009.
24
M. Treiber and D. Helbing. Memory Effects in Microscopic Traffic Models and Wide Scattering
in Flow-Density Data. Physical Review E, 68(4):46119, 2003.
J. Treiterer and J.A. Myers. The Hysteresis Phenomenon in Traffic Flow. Transportation and
Traffic Theory: Proceedings of the Sixth International Symposium on Transportation and Traffic
Theory, pages 13–38, 1974.
W.E. Walker, P. Harremo¨
es, J. Rotmans, J.P. van der Sluijs, M.B.A. van Asselt, P. Janssen, and
M.P.K. von Krauss. Defining Uncertainty: A Conceptual Basis for Uncertainty Management in
Model-Based Decision Support. Integrated Assessment, 4(1):5–17, 2003.
H. Wang, J. Li, Q.Y. Chen, and D. Ni. Speed-Density Relationship: From Deterministic to Stochas-
tic. In Proceedings of the 88th Annual Meeting of Transportation Research Board: Washington
DC, US, 2009.
J. Wehr and J. Xin. Front Speed in the Burgers Equation with a Random Flux. Journal of Statistical
Physics, 88(3):843–871, 1997.
G.C.K. Wong and S.C. Wong. A Multi-Class Traffic Flow Model–An Extension of LWR Model
with Heterogeneous Drivers. Transportation Research Part A, 36(9):827–841, 2002.
P. Yi, C. Xin, and Q. Zhao. Implementation and Field Testing of Characteristics-Based Intersection
Queue Estimation Model. Network and Spatial Economics, 1(1):202–222, 2001.
H.M. Zhang. A Mathematical Theory of Traffic Hysteresis. Transportation Research Part B, 33
(1):1–23, 1999.
H.M. Zhang. New Pperspectives on Continuum Traffic Flow Models. Networks and Spatial Eco-
nomics, 1(1):9–33, 2001.
M. Zhang, C.W. Shu, G.C.K. Wong, and S.C. Wong. A Weighted Essentially Non-Oscillatory
Numerical Scheme for a Multi-Class Lighthill–Whitham–Richards Traffic Flow Model. Journal
of Computational Physics, 191(2):639–659, 2003.
25
P. Zhang and S.C. Wong. Essence of Conservation Forms in the Traveling Wave Solutions of
Higher-Order Traffic Flow Models. Physical Review E, 74(2):26109, 2006.
P. Zhang, S.C. Wong, and C.W. Shu. A Weighted Essentially Non-Oscillatory Numerical Scheme
for a Multi-Class Traffic Flow Model on an Inhomogeneous Highway. Journal of Computational
Physics, 212(2):739–756, 2006.
P. Zhang, SC Wong, and Z. Xu. A Hybrid Scheme for Solving a Multi-Class Traffic Flow Model
with Complex Wave Breaking. Computer Methods in Applied Mechanics and Engineering, 197
(45-48):3816–3827, 2008.
P. Zhang, S.C. Wong, and S.Q. Dai. A Conserved Higher-Order Anisotropic Traffic Flow Model:
Description of Equilibrium and Non-Equilibrium Flows. Transportation Research Part B:
Methodological, 43(5):562–574, 2009.
26
