Hybrid Models of Traffic Flow
Hybrid Models of Traffic Flow: \Impacts of Bounded Vehicle Accelerations

Hybrid Models of Traffic Flow:
Impacts of Bounded Vehicle Accelerations

Jorge Andrés Laval

Abstract

This dissertation develops extensions to the kinematic wave theory of moving bottlenecks that allow the formulation of parsimonious traffic flow models that turn out to match field data with unprecedented accuracy. These "hybrid models" explicitly incorporate the effects of a stream of underperforming vehicles, whose trajectories are computed endogenously using realistic accelerations. Hybrid models exhibit important practical advantages stemming from their simplicity and mathematical stability. They do not amplify errors, they require few easily observable parameters and they can be validated by parts.
Two practical applications are included: the effects of heavy vehicles on uphill bottlenecks, and the effects of lane-changing maneuvers. In the first case, we found an important disagreement with the Highway Capacity Manual (HCM) formulas, possibly because the HCM formulas are based on microsimulation. Truck management schemes near bottlenecks caused by geometry can now be systematically devised; eg, truck-only lanes or truck metering. An approximate expression for the capacity of an upgrade is also developed, which matches simulation results accurately under all conditions.
The second application shows that the capacity losses imposed by the bounded accelerations of lane-changing vehicles is able to explain the relationship between the discharge rate of a moving bottleneck and its speed, as observed in the field. This is arguably the most important result of this dissertation, since it strongly suggests that lane-changing maneuvers may be the main cause for traffic instabilities on other bottleneck types (eg, merge, diverge, incidents).
The results of this dissertation allows for devising ITS solutions that maximize freeway capacity by controlling lane-changing and the arrival pattern of heavy vehicles to the bottleneck. The simulation of such strategies should be straightforward using hybrid models, and real-time applications should be possible for isolated bottlenecks.
To Viviana and Sofía with all my love.

Contents

1  Introduction
2  Background
    2.1  The KW model
        2.1.1  Fundamental Diagrams
        2.1.2  Godunov's method
    2.2  N-curves
    2.3  Theory of Moving Bottlenecks
        2.3.1  Discretization of the KW-MB model
    2.4  Extensions to the KW model
        2.4.1  Second-order models
        2.4.2  Single-pipe multi-class models
        2.4.3  Multi-lane single-class models
        2.4.4  Multi-lane multi-class models
        2.4.5  Bounded accelerations
    2.5  Lane changing
    2.6  Free motion models
        2.6.1  Crawl speed
3  Single-Pipe Formulation
    3.1  Introduction
    3.2  The CFM rule
    3.3  Implementation
    3.4  Numerical demonstration
        3.4.1  Case A: truck is slowed by a queue
        3.4.2  Case B: moving bottleneck joins a fast queue without slowing
        3.4.3  Realistic accelerations
    3.5  Discussion
4  Multi-Pipe Formulation
    4.1  Introduction
    4.2  The continuum formulation
        4.2.1  Solution method
    4.3  The discrete formulation
    4.4  Convergence
        4.4.1  The numerical experiment
        4.4.2  Stability
        4.4.3  Consistency
    4.5  Moving bottlenecks in multi-lane models
5  Applications
    5.1  Upgrade bottleneck
        5.1.1  Simulation
        5.1.2  Analytical solution
        5.1.3  Comparison with HCM
    5.2  Disruptive lane-changing (DLC)
        5.2.1  Formulation
        5.2.2  Flow downstream of a moving obstruction
6  Conclusions and extensions
A  Complements for the multi-lane model
    A.1  The IT principle
    A.2  Examples of moving bottlenecks in the multi-lane model
        A.2.1  A single moving bottleneck join a queue
        A.2.2  Passing
B  Analytical Solution of the Upgrade Bottleneck

List of Figures

    2.1  (a) Demand and supply functions; (b) triangular FD. Dots depict empirical data for (a) 30-sec and (b) 3-min intervals for loop-detector # 7 on I-80 @ University Av. (Berkeley, CA) on 09/28/02 from 6 am to 9 pm, westbound.
    2.2  (a) Numerical domain of dependence and stencil for Godunov's method with triangular FD; (b) exact solution of a Riemann problem when ki+1 ki are congested.
    2.3  (a) Representation of a moving bottleneck in a triangular FD; (b) solution of a simple example in the time-space plane.
    2.4  (a) Approximate trajectories; (b) appearance of the solution.
    2.5  (a) Accelerations predicted by (2.15-2.16); (b) crawl speed predicted by (2.18).
    3.1  Schematic representation of the hybrid model.
    3.2  (k,q)-diagram showing Cases A and B.
    3.3  (a) (t,x) diagram of the exact solution for example A; (b) numerical surface (N,t,x) using Dt = 1 sec. (c) k-map of numerical solution with Dt = 2 sec; (d) with Dt = .5 sec; (e) numerical N-curves along f for different mesh-sizes; (f) numerical N-curves at 5 locations.
    3.4  (a) (t,x) density map of numerical solution for example B with Dt = .5 sec; (b) numerical Nf .
    3.5  A realistic example of a heavy truck with v(0)=5 mph under the conditions of example A.
    4.1  (a) Lane-changes in the t-x plane; (b) discretized freeway representation.
    4.2  Schematic representations of the IT principle.
    4.3  (a) A 2x1 lane-drop; (b) exact solution in the FD; (c) numerical N-curves; (d) numerical k-map with Dt= 0.2 sec.
    4.4  (a) Error propagation showing stability of \@mathcal M; (b) total number of lane-changes as a function of Dt.
    4.5  k-maps of the lane-drop example for Dt= 2 sec (top,) Dt= 1 sec (middle) and Dt= 0.5 sec (bottom.)
    5.1  (a) (t,x) density map of a sample simulation; (b) approximate trajectory for analytical solution.
    5.2  Capacity plots showing agreement between simulation and approximation.
    5.3  Comparison of HCM values and eq. (5.2) for two typical cases.
    5.4  (a) Recoverable flow gap; (b) disruptive lane-changing.
    5.5  k -maps for a moving obstruction traveling at (a) v = 1 mph and (b) v= 20 mph.
    5.6  Dimensionless bottleneck discharge rate, r, as a function of the slow vehicle speed v for n {2,3,4} lanes.
    5.7  Location of the lane-changing maneuvers relative to the truck (simulation.)
    A-1  Example of § 3.5 solved with the multi-lane method.
    A-2  Crossing between two moving bottlenecks.
    B-1  (t,x)-diagrams for (a) possible trajectories; (b) approximate trajectory.
    B-2  (a)(t,x) diagrams of a single truck on the uphill; (b) mean of the simplified process.

List of Tables

    2.1  Truck types assumed for analysis
    3.1  Traffic states for two examples: A and B
There is no doubt in my mind that the person who deserves all the credits is my wife Viviana Montenegro. Not only she provided me with the greatest love and support during these demanding years, but she also has been the greatest mom for our daughter Sofía.
I am also very grateful to Carlos Daganzo for everything he taught me during these years. Apart from giving me invaluable academic advice, inspiration and passion for discovery, his unconditional support, care and friendship made me feel at home even though we are thousands of miles away. I will spend my career trying to match the quality of his research, teaching and more importantly, his personal values.
I greatly benefited from the exceptional group of students I had the chance to meet at Berkeley, whose friendship and encouragement will be forever appreciated. I am particularly grateful to Alejandro Lago, Juan Carlos Muñoz, Souyong Ahn, Yu-Wei Li, Anne Goodchild, Yanfeng Ouyang and Monica Menéndez for their inspiration, interesting discussions and feedback they always provided.
Finally, I would like to thank my parents Jorge and Gloria, and my sisters Tota, Anita, Cacao and Mane for their permanent love, care and support, which made us always feel near "home".

Chapter 1
Introduction

This dissertation studies the dynamics of traffic streams composed by fast and slow vehicles by extending the kinematic wave theory of moving bottlenecks (KW-MB) [[1,[2,[3,[4]. For decades we have known that heavy vehicles [[5,[6,[7], roadway geometry [[8,[9,[10,[11,[12] or lane-changing [[13,[14,[15] can significantly affect traffic performance, but understanding the mechanisms of cause and effect is still an open question.
On the one hand, existing extensions of the kinematic wave (KW) theory [[16,[17] that include multiple user classes [[18,[19,[20,[21,[22,[23,[24,[25,[26,[27,[28] treat slow vehicles as a fluid rather than a moving obstruction and do not include realistic vehicle acceleration capabilities. The latter is surprising since "free-motion" models for the kinematics of an isolated vehicle are accurate and well understood [[29,[30,[31,[32,[33,[34,[35]. Additionally, most of these extensions are "single-pipe" models (ie, no lane distinction is made). This severely reduces their comprehensiveness.
On the other hand, microscopic models -used by the vast majority of practitioners worldwide- have not been successfully validated as they are highly sensitive to the input parameters. Moreover, since the number of parameters can be quite large and many of them are not observable, we are left with untractable optimization problems that may admit multiple solutions. In light of the complexity of the task, the Federal Highway Administration recently launched the NGSIM initiative (next generation simulation), to provide a standard framework for developing microscopic models and a test-bed for their validation.
It is the author's opinion that the main challenge faced by microsimulation is a consistent modeling of discretional lane-changing (ie, lane changes to increase travel speed) and other non-car-following interactions imposed by multi-lane traffic. These interactions tend to be unstable at the microscopic level, in the sense that individual lane-changing decisions may change dramatically with small variation on the initial conditions. At a more aggregate level, however, recent observation shows that lane-changing rates caused by a merge bottleneck are reproducible [[36]. This implies that the phenomenon can be explained with simpler models.
In this context, this dissertation develops Hybrid Models that combine the robustness and stability properties of the KW model together with accurate microscopic free-motion models using simple physical principles. To this end, chapter 2 presents a brief background on non-hybrid models: (a) the KW and KW-MB models including numerical solution methods and (b) the free-motion models used in the numerical demonstrations.
The formulation of the single-pipe hybrid model is developed in chapter 3 where simple numerical examples are included to show the convergence of the model. Although a number of problems can be addressed by this model (eg, trucks on an upgrade or buses on city streets) the limitations imposed by the single-pipe framework preclude a parsimonious modeling of lane-changing.
In chapter 4 we show that using a multi-lane KW model allows for modeling lane-changing as a continuum with only one additional parameter, as the attributes that trigger lane-changing become endogenous variables (ie, the speeds on different lanes.) This model is an improved formulation and discretization the model of Munjal and Pipes [[37,[38]. A numerical demonstration of convergence is provided.
Chapter 5 includes two example applications, which summarize the main practical findings of this thesis:
  1. The upgrade bottleneck application shows the effects of heavy vehicles using the single-pipe method. This example also includes the derivation of a simple expression for the capacity of the road segment. For the case of two vehicle classes, the formula accurately replicates simulation under all conditions. We found that these results differ considerably from the predictions of the Highway Capacity Manual (HCM) [[39] based on microsimulation.
  2. Lane-changing is modeled as moving bottlenecks using a discrete approximation of the lane-changing flows generated in chapter 4. The example of lane-changing caused by a slow vehicle reveals an exceptional agreement with empirical data. This experiment also suggests that the "capacity drop" often observed on merge bottlenecks could be explained by the same physical principle.
Finally, the main conclusions and extensions are included in chapter 6.

Chapter 2
Background

This chapter presents some background on KW and KW-MB models together with the finite-difference schemes used for their numerical solution. Current extensions of the KW model, plus a review on lane-changing and free-motion models are also included.

1  The KW model

The KW model is a scalar conservation law for the density, k(t,x), of vehicles on a road at time t and location x, supplemented with a fundamental diagram (FD) that gives the flow, q(k,t,x), as a function of the local density k(t,x). The time and space indexes will be omitted as much as possible. For a road without entrances or exists the conservation law becomes:
k

t
+ q(k)

x
=0.
(1)
System (2.1) belongs to the family of first order nonlinear hyperbolic partial differential equations (PDEs). When q(·) is concave, as often assumed in traffic flow, the solution of (2.1) is unique and stable only if an entropy condition [[40] is imposed; ie, choosing the solution that maximizes the flow [[41].
To illustrate the predictions of the model, consider initial values given by
k(0,x) =



ku
if x < 0,
kd
otherwise.
(2)
System (2.1)-(2.2) is called a "Riemann problem" in the theory of PDEs and constitutes the building-block for developing current numerical solution methods for the KW model. In general, Riemann problems are not trivial, but in the case of the KW model the solution is simple. It can be shown [[41] that the flow at x=0, [(q)\tilde](ku,kd), is given by the cell transmission (CT) rule [[42]: 1
~
q
 
(ku,kd)= min
{l(ku),m(kd), Q},
(3)
where l(k,t,x) and m(k,t,x) are two monotonic functions that uniquely define the FD (see the solid lines in Fig. 2.1a,) and Q is the capacity of the road at x=0. As a result of the entropy condition the solution depends on ku and kd as follows:
Notice that unlike deceleration waves, rarefaction fans have not been observed in the field.

1.1  Fundamental Diagrams

It is widely accepted that traffic states can be grouped in two phases or regimes: (i) free-flow regime, given by the increasing portion of the FD, where vehicles do not interact very much; (ii) congested or queued regime, given by the decreasing portion of the FD, where drivers are not able to travel at their desired speed because of vehicles ahead.
The appearance of FD's obtained in the field depends on the aggregation interval and on the occurrence regime changes, as shown in [[43,[44]. The dots in Fig. 2.1a depict a typical scatter plot of loop-detector flow-occupancy data for all lanes on consecutive 30-sec intervals. If 3-min intervals are used, the plot becomes less scattered as shown by the dots on part (b) of the figure. Note how well a triangular FD fits the data, and that points inside it are regime transitions [[43,[44].
F2-1.gif
Figure 1: (a) Demand and supply functions; (b) triangular FD. Dots depict empirical data for (a) 30-sec and (b) 3-min intervals for loop-detector # 7 on I-80 @ University Av. (Berkeley, CA) on 09/28/02 from 6 am to 9 pm, westbound.
A recent interpretation for traffic states was provided by Daganzo [[42] in the context of numerical methods. He points out that in free-flow, l(k) represents the ability of a segment to send flow to the segment downstream, or the drivers desire to advance; in congestion, m(k) gives the ability of a segment to receive flow from the upstream segment. Later, Lebacque [[45] argues that the sending ability represents the demand for advancing to the downstream segment while the receiving ability is a simile for the supply for serving upstream demand.
In the continuum, the FD can be expressed as:
q(k)= min
{l(k),m(k), Q},
(5)
where Q(t,x) is the capacity of the road at (t,x).
On all the numerical examples given in this dissertation we will use triangular FD's, which are defined with three observable parameters: free-flow speed u, congested wave speed -w and jam density k; see Fig. 2.1b. They are the simplest ones consistent with empirical data, which indicates that acceleration and deceleration waves travel at a nearly constant speed and without spreading [[46,[47,[48,[49,[50,[44,[51,[52,[53,[54,[55,[56,[57,[58,[59]. Notice that the KW model predicts spreading acceleration waves, unless one uses a triangular FD.
It will be convenient to define the equilibrium speed-density relation V(k)=.(k)/k.

1.2  Godunov's method

During the last decade much has been learned about discretizing the KW model with finite difference methods [[42,[60,[61,[62,[41,[63,[64]. In these methods, the highway is partitioned into small sections (of length Dx) and time into discrete time steps (of duration Dt.) The resulting numerical grid (tj=.jDt, xi=.iDx) is shown by the dots in Fig. 2.2.
            F2-2.gif
Figure 2: (a) Numerical domain of dependence and stencil for Godunov's method with triangular FD; (b) exact solution of a Riemann problem when ki+1 ki are congested.
Let kij be the numerical approximation of k(jDt, iDx). For a method to converge to the right solution one must be able to express it in conservation form [[40]:
ki: = ki - Dt

Dx
[F(ki-p,ki-p+1,,ki+q) - F(ki-p-1,ki-p,,ki+q-1)]
(6)
for some flux function F of p+q+1 arguments. We use the update symbol ":=" to indicate that the terms on the RHS are evaluated on the previous time-interval w.r.t terms in the LFS. Otherwise, (2.6) would read: kj+1i = kij - .
When q/k changes sign (such as in traffic flow,) Godunov's method [[65] is the best first order numerical scheme. This method provides the entropy solution by solving the Riemann problem between neighboring cells forward in time. Godunov's method can be written as
ki: = ki - Dt

Dx
[
~
q
 
(ki, ki+1)-
~
q
 
(ki-1,ki)],
(7)
ie, it is a conservative method with p=0, q=1 and flux function (2.3).
The stencil of the method is the set of points in the discrete (t,x) plane that are involved in the RHS of (2.6); see arrows in Fig. 2.2a for the Godunov stencil with triangular FD. The numerical domain of dependence of the scheme at (j,i) is the set of grid-points that are connected to (j,i) through a network of stencils; see bold dots in the figure. The domain of dependence of the PDE at (j,i), namely D(j,i), is the (t,x) region that could affect the value of kij according to the PDE; see shaded area in the figure.
The Courant-Friedrich-Levy (CFL) stability condition states that the numerical domain of dependence must be a subset of D(j,i). This is satisfied by choosing grids such that
Dx

Dt

max
k 
| q(k)

k
|,
(8)
which means that vehicles must spend at least one time-step inside a cell.

2  N-curves

Newell's cumulative count curves, N(t,x), give the cumulative number of vehicles that have crossed location x by time t. The bivariate function N(t,x) is (i) continuous, (ii) non-decreasing in time, (iii) non-increasing in space, and (iv) constant along a vehicle trajectory. If we let sub-indices represent partial derivatives then q = Nt and k = - Nx, and the KW model expresses the existence of a fundamental diagram F(·) such that
Nt=F(-Nx,t,x).
(9)
In this context, the conservation equation becomes the identity Nxt = Ntx. Homogeneous problems with triangular FD can be easily solved exactly using Newell's minimum principle [[66]; inhomogeneous problems with concave FD's is best solved with Daganzo's variational method [[67] with an error that is uniformly bounded. In essence, these methods assert that the value of N at any given (t,x) point is given by the minimum of the N values obtained from every point in the domain of dependence of (2.9).

3  Theory of Moving Bottlenecks

From an historical perspective, Gazis and Herman [[68] were the first to propose a model to predict the impacts of a slow vehicle. They did so for a slow truck on a two-lane road, but their analysis is partial. The KW-MB theory was introduced by Newell [[66,[2] for describing the effects of a long slow moving convoy on traffic streams. Lebacque et al [[3] analyze the problem in a moving coordinate system and proposed the first numerical method for its solution. Muñoz and Daganzo [[4] improve the formulation of the model based on empirical data and show the suitableness of the point-bottleneck approximation.
In the KW-MB theory, any vehicle that moves slower than the traffic stream can become an active moving bottleneck, causing a queue upstream and a free-flow state downstream, given by points U and D in Fig. 2.3a. The slope of the line connecting these points is the speed of the moving bottleneck, denoted v, and the speed of cars queueing behind the slow vehicle is vU. The flow of state D, QD, is called the capacity of the moving bottleneck.
Notice that this theory assumes that moving bottleneck trajectories and capacities are defined exogenously; ie, that one knows v(t) and QD(t) beforehand. Since moving bottlenecks in this thesis are part of the traffic stream, one must check that the problem remains well-posed; ie, that the values of v(t) and QD are consistent with traffic conditions in contact with the slow vehicle at all times. As noted in [[4], only the bold segments in Fig. 2.3a constitute physically possible states. Other states imply some sort of collision: (i) states below E imply v V(k); (ii) states above [`(DU)] imply that some proportion of the traffic stream will collide with the slow vehicle.
However, issue (ii) above is automatically handled by the theory, as states above [`(DU)] will immediately transition to either D or U. Fig. 2.3b shows the solution on the time-space plane of a problem with a moving bottleneck that enters a freeway flowing at capacity, travels at a constant speed v and leaves T time units later.
F2-3.gif
Figure 3: (a) Representation of a moving bottleneck in a triangular FD; (b) solution of a simple example in the time-space plane.
Flow R in Fig. 2.3a represents the maximum passing rate at which vehicles pass the moving bottleneck; ie, it gives the flow measured by an observer moving with the moving bottleneck when it is active. Flows R and QD are related by a linear transformation, which for triangular FD's is
R=QD(1- v

u
).
(10)
It was shown experimentally [[4] that for constant v, there exists a reproducible relation R=R(v). We can also express this dependency in terms of the capacity QD, ie
QD = QD(v),                 with dQD

d v
> 0.
(11)
This experimental evidence has not been explained yet. A psychological interpretation is that drivers are more "afraid" to pass slower moving bottlenecks, which is not very intuitive. Up to now, the only alternative is to determine QD(v) empirically. This implies a tremendous data-collection effort because one needs at least one experiment for each combination of number of lanes, type of geometry and speeds of the moving bottleneck.
In § 5.2 we provided a possible explanation based on the bounded accelerations of vehicles passing the moving bottleneck. Because vehicle acceleration is not instantaneous the passing vehicle can cause a temporary obstruction during the maneuver. The slower the moving bottleneck the more severe the obstruction. Since the obstruction occurs on the passing lane only, we require a multi-lane version of the KW model, which is developed in chapter  4.

3.1  Discretization of the KW-MB model

The discretization of the KW-MB model has a short history. To the author's knowledge, the first attempt was suggested verbally in Lebacque et al [[3]. It corresponds to an extension of Godunov scheme where the flow at the cell containing the (point) bottleneck is redefined. This method is of limited importance because it can only handle homogenous freeways and moving bottleneck trajectories that pass through lattice points.
Giorgi et al [[69] uses the ideas in [[3] but redefines the cells near the bottleneck so that the CFL stability condition is satisfied. It is not clear to the author that this method would work for arbitrary trajectories and crossings among moving bottlenecks. Additionally, inhomogeneous roads are not handled naturally by the method and requires a special procedure that imposes additional computations.
Later, Daganzo and Laval [[70] replace the continuous trajectory of a moving bottleneck, f(t), by an approximate step-trajectory, fa(t), restricted to the numerical grid; see Fig. 2.4a. In terms of vehicle number, the error introduced by this approximation is uniformly bounded. This bound tends to zero as the lattice spacing is reduced. This method allows the use of "off-the-shelf" software for solving general inhomogeneous problems involving crossings.
The only disadvantage of this method is shown in part (b) of the figure: the approximate flows, densities and speeds do not converge to the exact ones, even as vehicle counts do; these quantities "flip-flop" between states C and D, the latter defined as the queued state at flow QD. Thus, to provide estimates one needs to average their values over multiple cells, introducing a small first-order error.
F2-4.gif
Figure 4: (a) Approximate trajectories; (b) appearance of the solution.
To overcome this disadvantage, Daganzo and Laval [[71] developed another method that produces smooth flows. This is done by solving "composite Riemann problems" -ie, two neighboring Riemann problems solved simultaneously- using an extension of Newell's minimum principle [[66].
Finally, a variational method has recently been developed [[67] which outperforms all others. In this method any traffic problem (which could include moving bottlenecks) can be solved with unprecedented accuracy and very quickly using dynamic programming. In the case of triangular FD's the method is exact. In other cases the error is uniformly bounded. The method is not used in this dissertation because it was developed concurrently.

4  Extensions to the KW model

Over the past several decades, a steady stream of extensions of the KW model have been proposed. They were motivated by the alleged inability of the model to explain the fluctuations from equilibrium values observed in the field, eg, capacity drop [[72,[73,[74,[75,[76,[48,[49,[77,[78], hysteresis [[79], capacity of moving bottlenecks [[4], propagation of oscillations [[80,[81,[51,[82,[83,[84,[85], clusters [[86,[87,[88,[89], persistent traffic jams [[48,[90,[91] and fast waves [[48].
Recently, it has been shown that KW can indeed explain most of them [[44,[82,[53,[58,[43], while capacity drop, fast waves and the capacity of moving bottlenecks remain unexplained. However, Daganzo's multi-lane models [[21,[22] suggest that lane-changing could be at the root of capacity drop and fast waves, revealing that multi-lane models are fundamental for an adequate traffic representation.
Next, a small subset of relevant extensions to the KW model are described.

4.1  Second-order models

Lately, under the presumption that first-order models do not provide "interesting non-equilibrium results", abundant higher-order models [[92,[93,[94,[95,[96,[97] have been proposed since the seminal works of Payne and Whitham [[98,[99]. These models use (2.1) together with another PDE for describing the acceleration of vehicles. Second-order models have been heavily criticized in the literature [[100,[101,[102]. The formulation of proper numerical solution methods and the modeling of boundary conditions [[101,[103] is still an open question. Furthermore, they ignore the most likely cause of traffic instabilities, ie lane-changing.

4.2  Single-pipe multi-class models

Current single-pipe multi-class models [[23,[24,[20,[25,[27,[26] can be expressed as the conservation equation for each vehicle class, m, together with a separate FD for each class, Fm: k_m+ q_m=0,
q_m=_m F_m(k_m), where the am's are variables that should capture traffic interactions. The different models published in the literature differ in the assumption to compute the am's. Assuming equal spacing among vehicles leads to the models in [[23,[27]; equal gap leads to [[20]; gap proportional to vehicle length gives the free-flow model in [[25]; equal speed gives the models for congested traffic in [[104,[25]; finally, user optimum leads to [[26].
The problem with these models is that they cannot address the interactions among different user classes with enough physical coherence, since (i) slow vehicles are treated as a fluid rather than moving obstructions, and thus lane-changing cannot be included explicitly; and (ii) the arrangement of vehicles across lanes is ignored (it is very different to have cars and trucks mixed or segregated in different lanes). In many cases FD's are chosen for numerical rather than physical reasons, which has led to numerous claims based on spurious foundations. 2

4.3  Multi-lane single-class models

The continuum multi-lane extension of the KW model was already devised in 1971 by Munjal and Pipes [[37]. For an n-identical-lane road the model reads:
kl

t
+ ql

x
=Fl,                             l = 1n
(12)
where kl and ql give the density and flow on lane l. Lane-changing rates onto lane l, Fl, are assumed given by a simplified form of Gazis et al's model [[80]:
Fl=a(kl-kl),                         "l,l contiguous,
(13)
where a is a constant in units of time-1. It is clear from (2.14) that lane-changing traffic has priority over the through traffic on the target lane because they are computed ignoring all possible competing flows.
Although this "priority problem" is somewhat unrealistic, it allows to simplify the formulation to a single equation that can be solved with stream-lined procedures when no regime changes are allowed [[105]. To include regime changes, an unrealistic parabolic FD on each lane is assumed in that reference, leading to "numerical insights" on the steady-state conditions of the system that may be unrealistic. Munjal and Pipes' model was first extended in [[106], which added a source terms to include ramp flows explicitly, defined a as a linear function of (kl-kl) and incorporated the complete Gazis et al's model including a time-lag. Notably, the first discretization of the multi-lane model is presented, but the numerical scheme used at that time (Lax-Wendroff) is unable to treat properly boundary conditions in the KW model. This reference also develops a second-order and a two-dimensional formulation, but the results favor the simpler first-order model.
No other first-order macroscopic multi-lane models have been developed maybe because a multi-lane model alone cannot unveil new insights about traffic flow. Other approaches have been used too. The stochastic models in [[107,[28] aim at describing the steady-state conditions rather than traffic dynamics. Gas-kinetic and second-order models are also available [[108,[109] but suffer from the drawbacks mentioned in § 2.4.1 and require several additional parameters.
The following section is devoted to Daganzo's models [[18,[19,[21,[22] that suggests that lane-specific formulations are valuable when different user classes are considered and/or driver psychology is taken into account.

4.4  Multi-lane multi-class models

The only publications under this category are Daganzo's models [[18,[19,[21,[22], devoted to solve two specific problems. For tractability, only two user types and two lane types are included, and traffic rules are very simple but physically grounded. In this simplified setting, the Riemann problems can be solved graphically.
Note that in the `'rabbits and slugs" model the effects of lane-changing are underestimated since it is assumed that rabbits can accelerate infinitely fast to the speed on the target lane. Chapter 4 explores a more parsimonious multi-lane model where lane-changing vehicles can be treated as temporary moving bottlenecks.

4.5  Bounded accelerations

Lebacque [[110] incorporates gravity into the KW model by including a single upper bound for vehicle acceleration. This model assumes only one vehicle type that share the same acceleration bound independent of the geometry. An undesirable feature of this formulation is that the scale-invariance of the numerical scheme is lost; ie, the results depend on the mesh size. Note that [[69] successfully combined this approach with moving bottlenecks for the case of simple bottleneck trajectories.
Although this work does not provide a framework for modeling moving bottlenecks, it is the only publication aiming at including realistic accelerations into the KW model.

5  Lane changing

When Lighthill and Whitham formulated the KW model, they did so for a single lane where lane-changing is not an issue. To extend the model to two-lane roads they proposed to double the flow for a given density per lane. This assumption is only reasonable if lane-changing does not cause an interruption on the target lane. It is also an optimistic assumption when comparing its predictions with the weaving-section formulas in the HCM.
The following are some recent examples that manifest the importance that lane-changing can have on traffic:
Unfortunately, there are no continuum models that capture the reductions in capacity caused by the bounded accelerations of lane-changing vehicles. Rather, the literature is generous on microscopic models [[113,[114,[115,[13,[116,[117], models based on neuronal networks [[118,[119], steady-state stochastic models [[107,[120], gas-kinetic models [[108] and discrete choice models [[121].

6  Free motion models

Free-motion models for the kinematics of an isolated vehicle are well understood [[29,[30,[31,[32,[33,[34,[35]. They are based on Newtonian mechanics and capture the physical limitations imposed by roadway geometry on the engine for typical driver behavior.
The free motion model incorporated in TWOPAS [[33] is used in this dissertation but any other can be used. It gives the maximum possible (or "desired") acceleration of a vehicle, a(·), as a function of its current speed v(t), and the following parameters:
  1. vehicle characteristics: weight-to-power ratio, W, and frontal area-to-power ratio, A,
  2. grade at every location, G(x), and altitude, h(x).
The acceleration for trucks, a, is given by:
a = bap

b+ 1.5 S(ap)(ap - ac)
                        (in ft/s2,)
(14)
where S(·) returns the sign of its argument; the rest of the parameters are defined as follows: = { ll 0.4v if v < 10,
10 otherwise, .
a_p = a_c + 15,368 C_p / W v,
a_c = -.2445 -.0004 v -.021 C_d v ^2/ A - 222.6 C_p / W v - g G,
C_p = 1 -.00004 h,
C_d = (1 -.00000688 h) ^ 4.255, where h is expressed in ft, v in ft/s, W in lb/HP, A in ft2/HP and G as a decimal; the acceleration of gravity is g=32.17 ft/s2.
For the numerical analysis in this dissertation we chose the two truck types on the following Table:
Table 1: Truck types assumed for analysis
Veh Type W (lb/hp) A (lb/ft2)
1. Heavy trucks 228 682
2. Light trucks 140 312
Fig. 2.5a depicts the accelerations predicted by (2.15-2.16) for light and heavy trucks and G=2 %.
F2-5.gif
Figure 5: (a) Accelerations predicted by (2.15-2.16); (b) crawl speed predicted by (2.18).
For cars, the free motion model takes a linear form that only depends on the car's maximum speed, vmax, and zero-speed acceleration, a0:
a = a0 (1 - v / vmax) - g G.
(15)
The car type used in the numerical demonstrations of § 5.2 is a high performance car, defined with vmax=142.7 ft/s and a0=14.1 ft/s2.

6.1  Crawl speed

The crawl speed vc is the steady state speed of a vehicle on an infinite upgrade, where the speed drops to a point where the engine lacks power to accelerate. For a particular vehicle type, it is solely a function of G.
To compute the crawl speed one can combine (2.16d) with (2.16c), set the latter equal to zero and solve for v. We found that the following polynomials give an accurate estimate of the crawl speed (in mph) for the relevant truck types for .01 G .09:
vc(G) =



-.09 G 3 + 2G 2 - 18 G + 71
, for heavy trucks,
.44 G 2 - 9.4 G + 71
, for light trucks.
(16)
Fig. 2.5b depicts (2.18) for these truck types.
For the case of cars vc can be obtained analytically setting (2.17) equal to zero end solving for v. One gets
vc=vmax(1- g

a0
G)                           , for cars.
(17)
To get an idea, take as a rough approximation g/a0 2 and G=4 % to see that on a typical freeway uphill, cars lose only 10 % of their speed.

Chapter 3
Single-Pipe Formulation

1  Introduction

This chapter presents the formulation of the single-pipe hybrid model, which extends the KW-MB by computing moving bottleneck trajectories endogenously and realistically. The causes that force a subset of vehicles to drive slower, eg gravitational forces imposed by roadway or safety factors may be explicitly incorporated.
In this chapter we assume that roadway geometry affects only a subset of vehicles (trucks,) whose "desired speed", [^(v)], can be obtained from vehicle and road characteristics by means of a free-motion model. Fig. 3.1 shows a flow chart of the hybrid model. The model is Markovian since the state of each user class in the future can be obtained with conditions in the present. The state variables are the densities of the KW stream and the speeds and positions of slow vehicles. The constrainted free-motion (CFM) rule takes [^(v)] and the speed of cars immediately downstream, v*, and gives the minimum of the two, which will be the actual speed of trucks, v, during [t,t+Dt). Knowing v allows for (i) updating the state of moving bottlenecks and (ii) applying the KW-MB theory to update the state of cars, provided we know the capacity of the slow vehicle at that time, QD(t).
                           F3-1.gif
Figure 1: Schematic representation of the hybrid model.
We are confident that a model based on the figure will make robust predictions because its core component, the KW-MB theory, is macroscopic and stable in nature; and the CFM rule involves a minimum operation. Consequently, the hybrid system benefits from the practical advantages of stable macroscopic models: (i) input parameters are readily measurable in the field; (ii) rough input estimates yield reasonable predictions; and (iii) the model output will be valid if individual components have been validated.
Next, we formulate the method for computing trajectories endogenously. The convergence of the method is demonstrated by means of two numerical examples.

2  The CFM rule

As mentioned in § 2.3, one must check that traffic states in contact with the moving bottleneck remain on the bold segments in Fig. 2.3a. Traffic states on the remainder of the FD imply collisions: (i) states above line [`(DU)] imply car-truck collisions; and (ii) states below E imply truck-car collisions. Item (i) is automatically handled by the KW-MB theory, as states above [`(DU)] will immediately transition to either D or U. The purpose of the constrained free-motion (CFM) rule is to handle (ii) by accounting for the influence that traffic stream may have on slow vehicles.
As shown in § 2.6 the free motion model gives the desired acceleration of a vehicle, a(·), as a function of its current speed v(t), its weight-to-power ratio, W, its frontal area-to-power ratio, A, and the grade at the current location, G(f(t)). 3 The desired speed of the vehicle, [^(v)](t), can therefore be defined in discrete time as
^
v
 
:=v+aDt,
(1)
The CFM rule simply takes the minimum between the desired speed predicted by the free motion model and the speed of cars immediately downstream of f, denoted v*(t); ie, v={v^*,},
:=+v t.
Note that (3.2a) implies instant deceleration capabilities. While this is unrealistic, it induces small errors since deceleration forces are 4 or 5 times greater than acceleration forces for a typical vehicle [[110]. Additionally, deceleration maneuvers cannot affect the discharge rate of a bottleneck, as this maneuver takes place inside its queue.

3  Implementation

The only additional parameter needed for numerical implementation is the moving bottleneck capacity, QD(t). In keeping with simplicity, we take QD(t) as the capacity of the unblocked lanes, ie
QD .
=
 
Q n- 1

n
,
(2)
where n is the number of lanes in the freeway. Although [[4] observed that QD increases with v, (3.3) can be used as a rough approximation.
From the available alternatives to carry out the KW-MB procedure [[70,[71,[67,[69] we chose the method in [[70] because of its simplicity and software availability. In this method we assume that f(t) and QD(t) are constant during the interval [t,t+Dt). Thus, one simply defines a static bottleneck of capacity QD(t) in a cell, i, near to the moving bottleneck, and we use
~
q
 
(ki,ki+1)= min
{l(ki),m(ki+1), QD},
(3)
instead of (2.3)- but only in cell i. The density update (2.7) remains unchanged.
Recall that the KW-MB method in [[70] does not produce smooth flows (see Fig. 2.4b) and therefore one has to average several cells in order to have a reliable estimate for v*(t). Formally, we use
v*=V( 1

4
i+4

m=i 
km),
(4)
where V(k) is the speed-density relation consistent with the FD in use.
The next section shows that the method converges to the KW-MB solution.

4  Numerical demonstration

Consider a level n-lane freeway segment of length 0.4 mi flowing at capacity state C in Fig. 3.2 when, at (t,x) = (0,0) a slow vehicle enters the segment. At the same time but at x=0.4 an incident occurs for 30 sec imposing a capacity QJ. Each lane obeys a triangular FD as in Fig. 3.2 with u = 60 mph, w = -u mph and k = 150 vpmpl. We use w=-u because our KW numerical solver is then exact, and therefore the results isolate the errors introduced by the approximation in [[70]. It is assumed that [^(v)] is constant (=30 mph) in order to obtain the exact solution by hand.
                                    F3-2.gif
Figure 2: (k,q)-diagram showing Cases A and B.
To see how (3.2a) provides the desired KW solution, we examine the N-curves along a truck trajectory, denoted Nf(t)=.N(t,f(t)). As opposed to car trajectories -where the vehicle number N is constant- Nf increases whenever cars pass the truck; ie, when the moving bottleneck travels slower than traffic. The numerical estimation of Nf was carried out with the moving-observer formula [[122]:
Nf:=Nf+ki(v* - v) Dt,
(5)
where it is clear that the passing rate, [(N)\dot]f=.Nf/t, increases with (v*-v).
In order to test the method on all possible conditions, we consider the following two cases; see Fig. 3.2:

4.1  Case A: truck is slowed by a queue

To illustrate this case we use an example with n=2 lanes and vJ=12 mph. The exact solution is shown in Fig. 3.3a. The moving bottleneck trajectory is the bold line on the figure while the resulting state variables are shown in the following table.
Table 1: Traffic states for two examples: A and B
Example AExample B
Stateqkv qkv
C9,0001506027,00045060
J3,0002501219,89556835
D4,500756022,50037560
D4,5002252022,50052543
U7,50017542.8525,50067537.8
R2,2250-11,2500-
Part (a) of the figure shows the exact (t,x) solution obtained by hand. The bold line is the truck trajectory f. Part (b) presents the (N,t,x) surface using Dt = 1 sec. The bold line is the Nf curve.
Parts (c-d) present the numerical density maps for two values of Dt. It is apparent how the solution converges to the exact solution of part (a) as the mesh is refined. Notice how the truck queue "flip-flops" between C and D, as expected from Fig. 2.4b. The reader can verify from part (c) that inside this queue there are two cells in state C for every cell in state D; thus the average flow is (2Q+QD)/3=7,500 vph as in the exact solution.
Part (e) of the figure shows the numerical estimate of Nf for different mesh-sizes, where convergence is clear. Note that the three solutions only differ while the truck is inside the queue, but are practically identical elsewhere. As expected, the passing rate is zero inside the queue and maximum, QR = 2,250 vph, when the moving bottleneck is active.
To complete the description, part (f) of the figure shows the N-curves Ns(t) measured at the five evenly-spaced locations xs, s=1 5, shown as doted lines on part (d). The vertical lines connecting the two figures conveniently show the changes in state for N2.
F3-3.gif
Figure 3: (a) (t,x) diagram of the exact solution for example A; (b) numerical surface (N,t,x) using Dt = 1 sec. (c) k-map of numerical solution with Dt = 2 sec; (d) with Dt = .5 sec; (e) numerical N-curves along f for different mesh-sizes; (f) numerical N-curves at 5 locations.

4.2  Case B: moving bottleneck joins a fast queue without slowing

An illustration in this case uses n=6 lanes and vJ=35 mph. The resulting state variables are also shown on Table 3.1. The k-map of the numerical solution with Dt = .5 sec are shown in Fig. 3.4a. Given that [^(v)] vJ, in the exact solution f is a straight line, as illustrated by the solid line in the figure. It is apparent how the truck does not affect the jam, as predicted by the KW-MB theory. As expected, now the passing rate [(N)\dot]f is positive inside the jam. According to the moving observer formula, [(N)\dot]f inside the jam should be

N
 

f 
=qJ-kJv = 19,895-568·30 = 2,842 vph
as in part (b) of the figure.
F3-4.gif
Figure 4: (a) (t,x) density map of numerical solution for example B with Dt = .5 sec; (b) numerical Nf.

4.3  Realistic accelerations

In real applications (such as the upgrade bottleneck in § 5.1,) one would like to have v* given by a free-motion model. Fig. 3.5 shows such an example for a heavy truck with v(0)=5 mph under the traffic conditions of example A. Notice that the area of the initial downstream state D is larger than the one after exiting the queue. This happens because the truck exits the queue at vJ=12 mph, which is grater than v(0). This effect -only captured by hybrid models- will turn out to be important.
F3-5.gif
Figure 5: A realistic example of a heavy truck with v(0)=5 mph under the conditions of example A.

5  Discussion

Single-pipe hybrid models can be applied to any problem where the single-pipe assumption is reasonable; eg, trucks on an uphill or buses on a rural road. However, empirical evidence on congested off-ramps or diverge bottlenecks with n > 1 indicate that a multi-pipe approach is more realistic [[59]. The multi-pipe approach for example can capture the following:
  1. in real life not all lanes may be affected by the moving bottleneck. It is the author's experience that in freeways of 4 or 5 lanes a moving bottleneck rarely affects more than 3 lanes. This may be because trucks usually use the right lane and vehicles exiting in the next off-ramp would rather queue behind the moving bottleneck than passing it,
  2. discretional lane-changing, where the driver tries to travel on the fastest neighboring lane. In this case, one needs to know traffic conditions on different lanes, which are not endogenous variables in the KW model.
The above motivates the multi-pipe formulation in the next chapter.

Chapter 4
Multi-Pipe Formulation

1  Introduction

This chapter presents a multi-lane KW model that enables representing multi-pipe traffic streams with hybrid models. Existing formulations of the multi-lane model [[37,[38,[106] do not take advantage of the latest numerical methods [[42,[41] for solving the continuum model, inducing unrealistic priorities among competing flows and inadequate treatment of the propagation of discontinuities.
To overcome these difficulties §§ 4.2 and 4.3 present an extension of the CT rule [[60] where competing flows are a result of supply-demand interactions. The proposed formulation requires only one additional parameter -the capacity for changing lanes- which is readily observable in the field. This formulation exhibits desirable convergence properties for the case of a lane-drop, as shown in § 4.4. The final section shows how moving bottlenecks can be incorporated, and simple examples that include interactions are presented in appendix A. Notice that the full potential of the resulting model is presented in chapter 5, where each individual lane-changing maneuver is treated as a moving obstruction.

2  The continuum formulation

Consider a freeway with n identical lanes where identical vehicles flow. 4 Let k=.[k1,kn] be a vector composed of the density on each individual lane kl(t,x), l = 1,2n, and let ql(t,x) be the flow on lane l. We saw in § 2.4.3 that the conservation equation for a single lane can be expressed as [[37]:
kl

t
+ ql

x
=Fl,                             l = 1n,
(1)
where Fl (k,t,x) is the net lane-changing rate onto lane l, in units of vehicles per unit time per unit distance (the word "rate" will be used to indicate such units). Flow transfers occur between adjacent lanes only; ie, the set of movements that a driver faces at any time has, at most, three components: all movements ll such that l {l-1,l,l+1}, if all adjacent lanes exist. 5

2.1  Solution method

Let Fll (k,t,x),l l be the lane-changing rate from lane l to lane l, so that
Fl=

l l 
Fll - Fll.
(2)
To understand these quantities physically consider the time-space region A in Fig. 4.1a. A lane-changing is a trajectory that crosses only one boundary of A (5 in the figure). The number of lane-changes onto lane l (black dots in Fig. 4.1a) is given by


l l 



A 
Flldtdx,
and corresponds to the number of trajectories that only cross the upper or right boundary of A (2 in the figure); similarly, the lane-changes from lane l only cross the lower or left boundary of A (3 in the figure).
To solve (4.1)-(4.2) we assume that a "cell-transmission" rule balances people's desires for changing lanes with the actual possibilities for doing so. Towards this end, let Lll(k, t, x),l l be the desired lane-changing rate on ll (ie, demand for lane-changing), Tl(kl, t, x ) the desired through flows on l, and ml(kl, t,x ) the receiving function on l. The competition for the available supply among the conflicting movement demands determines the actual lane-changing rates Fll and through flows ql . This competition can be carried out for each lane independently since conflicting movements belong to adjacent lanes only. This procedure can be characterized by a vector-valued function of four arguments and three outputs, ie:
F(Ll-1,l, Tl, Ll+1,l,ml) (Fl-1,l, ql,Fl+1,l)
(3)
For stability, the function F should satisfy an entropy-like condition that maximizes the flow. It should also reflect sensible priority rules, which depend upon the nature of the lane-changing maneuver (discretional or mandatory). For the case of discretional lane-changing we chose to extend the incremental transfer (IT) principle [[19] as explained next. 6
F4-1.gif
Figure 1: (a) Lane-changes in the t-x plane; (b) discretized freeway representation.

3  The discrete formulation

As in most discrete traffic flow models, we use the finite-difference method, where the highway is partitioned into small sections of length Dx and time into discrete time steps of duration Dt, related by
Dx .
=
 
-
u
 
Dt,
(4) for stability, where [`(u)] is the fastest characteristic in (4.1). Additionally, we partition the freeway in cells (i,l) as shown in Fig. 4.1b, where i is the section along the roadway and l is the lane index. The resulting numerical grid is denoted (tj=.jDt, xi=. iDx,l).
For discrete variables we will use subscript i and superscript j to denote the value at the discrete point (tj,xi); eg, kilj is the discrete approximation of kl(tj, xi). These indexes will be omitted as much as possible. In the discrete, the conservation equation becomes
kilj+1- kilj

Dt
+ qilj-qi-1,lj

Dx
=

l l 
Fi-1,llj - Fillj ,                         "l.
(5)
This equation it is ready for stepping through time, as there is only one term with time index j+1 . At each iteration one defines L, T and m for every cell using {kj}, then computes the lane-changing rates and through flows with the IT principle [[19]. In the IT principle the total supply ml is viewed as a reservoir being filled gradually with pipes coming from upstream. This is shown in Fig. 4.2, where each cell has one reservoir per demand type. Small reservoirs give the lane-changing demand while big ones, the through demand. The IT principle states that each reservoir will send fluid downstream until upstream reservoirs are empty or the receiving reservoir is full. Appendix A1 describes the algorithm.
                F4-2.gif
Figure 2: Schematic representations of the IT principle.
Up to this point the formulation has been as general as possible, but now we further specify functional forms. Lane-changing demand rates are assumed to be Markovian as they can be obtained from the current state of the freeway; ie,
Lll = pll kl,                 "l, "l l,
(6)
where pll=.p(kl,kl) is the fraction of drivers that would like to change from lane l to lane l in the next unit of time. The Markovian assumption ensures that the probability for changing lanes in the next Dt time units is pllDt, as Dt 0. It follows that the probability of staying in the same lane in the next Dt time units is 1- l lpllDt. Therefore, the demand for through flow is given by
Tl = l(kl)( 1-

l l 
pllDt )        "l,
(7)
where l(·) is the sending function for lane l. 7 The term pll Dt can be interpreted as a choice function in discrete-choice modeling [[123], provided that Dt 0 . For simplicity, it is assumed proportional to the (positive) speed difference between lanes, Dvll , ie
pllDt .
=
 
Dvll

u
Dt

t
,                 "l, "l l.
(8)
The parameter t has units of time and should be comparable to the time to complete a lane-changing maneuver. Notice that t-1 is an upper bound to pll ; as such, it can be interpreted as the capacity at (x,l) for changing lanes, and should be readily observable in the field.

4  Convergence

F4-3.gif
Figure 3: (a) A 2x1 lane-drop; (b) exact solution in the FD; (c) numerical N-curves; (d) numerical k-map with Dt= 0.2 sec.
In this section we examine the example of a lane-drop to show the convergence of the model. As it is customary in this dissertation, the fundamental diagram on each lane is assumed triangular with sending function l(·) and receiving functions m(·), ie: (k_) =uk_,
(k_) = w(-k_). As we did in chapter 3, the parameter values used in this section are: u=-w = 60 mph, k = 150 vpmpl and t = 3 sec.

4.1  The numerical experiment

Consider a 0.4-mile, 2-lane freeway segment that has a lane-drop on lane 2 at x=0.4 mi; see Fig. 4.3a. At t=0 all lanes have optimal density k0=75 vpmpl and the input demand is also k0 for t > 0.
The numerical N-curves predicted by our method (Dt= 0.2 sec) for all lanes combined are shown in part (c), which are measured at the 5 evenly-spaced locations shown in part (a). This figure shows that this solution consists of an instantaneous transition between states C and B; see part (b) of the figure. This solution coincides with the (single-pipe) KW solution and indicates that the multi-lane model does not affect the discharge rate of the bottleneck. 8
When we look at individual lanes separately, we observe reasonable patterns. Consider part (d) of the figure, which shows the k-maps of the numerical solution for each lane and for both lanes combined. Note how the density on lane 2 near the bottleneck is approx 140 vpm and decreases gradually upstream. The opposite happens on lane 1 with a density of k0 at the bottleneck and increases gradually upstream. This behavior is reasonable, as common experience indicates that vehicles on the ending lane tend to drive slower the closer to the bottleneck they are.
To show the convergence of the model one must show that it is stable and consistent [[40]. To this end, let M be the discrete model operator, and K the continuum model operator presented in this chapter.

4.2  Stability

Let N0(x) be the initial N-profile used in the previous example; ie, the N-values along the road at t=0, which in this case are given by N0(x)=150x. Also let a perturbed profile [(N)\tilde]0(x)=152x. If Mj is the operator obtained after j iterations of the numerical scheme (4.4-4.7), stability means that for all j:
||N0 -
~
N
 

0 
|| e ||MjN0 - Mj
~
N
 

0 
|| e.
(9)
for some norm ||·||. Condition (4.9) states that errors in the input data do not grow with time.
Fig. 4.4a shows the values of the errors ||MjN0 - Mj[(N)\tilde]0|| for the L-norm, Dt= 0.5 sec and tj 30 sec. It is clear that (4.9) is satisfied since the errors decay as time passes.
F4-4.gif
Figure 4: (a) Error propagation showing stability of M; (b) total number of lane-changes as a function of Dt.

4.3  Consistency

Let MDt be the operator obtained after one iteration of M with time-step Dt, and let KDt be the exact continuum operator up to time Dt . The model is consistent if
||MDt N0 - KDt N0 || 0                 as Dt 0.
(10)
Condition (4.10) indicates that the solution of M tends to the continuum solution as the grid is refined.
Fig. 4.5 shows the k-maps of the lane-drop example for different mesh sizes. It becomes clear that the solution tends to Fig. 4.3d, obtained with Dt= 0.2 sec. Fig. 4.4b shows the total number of lane-changes (on the entire freeway during the whole simulation period) as a function of Dt , which also converges to approx. 17.5 as Dt 0.
F4-5.gif
Figure 5: k-maps of the lane-drop example for Dt= 2 sec (top,) Dt= 1 sec (middle) and Dt= 0.5 sec (bottom.)

5  Moving bottlenecks in multi-lane models

This section combines the single-pipe ideas for treating moving bottlenecks of chapter 3 with the multi-lane model presented in this chapter. It turns out that including moving bottlenecks in the multi-lane framework is more parsimonious because the physical problem is better represented. For example, we know that the flow-gap in front of the moving bottleneck originates only on the lane l occupied by the moving bottleneck. Therefore, the problem reduces to a moving bottleneck on lane l only. The capacity of the moving bottleneck is thus implicitly defined as that of the remaining lanes.
As we did on chapter 3, we approximate f by a sequence of static bottlenecks; ie, if cell (i+1,l) contains the moving bottleneck we use
Fill=0, qil=0,                 "l
(11)
in lieu of (4.3). The CFM rule of § 3.2 applies unchanged, with the caveat that downstream traffic speed is now lane-specific.
For illustration purposes, Appendix A2 shows two applications of this method. The first shows a moving bottleneck joining a queue, and the second a moving bottleneck passing another one. From these examples it can be seen that "pencil and paper" solutions to apparently simple problems can be extremely difficult to obtain, as lane-changing flows create different shock waves in different lanes and shocks tend to be curves rather than straight lines.

Chapter 5
Applications

This chapter includes two example applications that show the possibilities of the models developed in chapters 3 and 4. The upgrade bottleneck application shows the effects of heavy vehicles using the single-pipe method. We found that our results differ considerably from the predictions of the HCM based on microsimulation.
In the second application lane-changing is modeled as moving bottlenecks using a discrete approximation of the lane-changing flows generated in chapter 4. The example of lane-changing caused by a slow vehicle reveals an exceptional agreement with empirical data. This experiment also suggests that the "capacity drop" often observed on merge bottlenecks could be explained by the same physical principle.

1  Upgrade bottleneck

In this section we use the single-pipe method to estimate the steady state flow on a of freeway with n identical lanes with an uphill in the middle section. Each lane obeys a triangular fundamental diagram with u = 60 mph, w = -15 mph and k = 150 vpmpl so that Q=n[(w u)/(w + u)]k is the capacity of the flat segments. The uphill starts at xo=.5 mi for L mi and has G percent grade. There are two user types, cars and trucks, in proportions 1-r and r << 1, respectively. We assume that cars and trucks share the same free-flow speed u on the flat segments. If not impeded by a queue, cars are able to maintain free-flow speed u on the uphill while trucks are forced to slow down due to gravity.
As we did on chapter 3, the capacity of the moving bottlenecks is taken as the capacity of the remaining unblocked lanes, ie
QD .
=
 
Q n- 1

n
,
(1)
which implies that trucks only use one lane. This is a reasonable assumption for the case of a single truck type and allows the derivation of closed-form steady-state expressions.

1.1  Simulation

We applied the hybrid model of chapter 3 for all the combinations of n={1,2,3} lanes, L={.1,.4,.6,.9,1.3} mi, G={2,4,6} % grade, r=1,25 % and the two truck types defined in §2.6. The capacity was obtained by dividing the number of vehicles crossing some location by the simulation period of two hours.
Fig. 5.1a shows the k-map of an extract of a sample simulation. Truck arrivals to the uphill are depicted as white circles in the figure. As expected, trucks change speed as they climb the uphill and reach the flat road again; notice how the second truck climbs slower than the first one because it reached the bottom of the hill at a slower speed; the third truck remains inactive inside the queue but activates when it becomes the head of the queue. Capacity losses are due to the traffic state downstream of the moving bottlenecks propagating forward and causing the voids in flow indicated in the figure.
F51-1.gif
Figure 1: (a) (t,x) density map of a sample simulation; (b) approximate trajectory for analytical solution.

1.2  Analytical solution

The approximate solution presented here assumes that all truck trajectories are identical and equal to the "approximate" trajectory in Fig. 5.1b. This trajectory has only two slopes, the free-flow speed u and the truck's crawl speed, vc(G). Appendix B shows that given L and G, the dimensionless capacity of the upgrade r(n,r,L,G) can be approximated by =_ ,
T Lw+v_c ;
Q_U Q_D + wv_c ; where T is the duration of the queueing episode caused by a truck, QU is the flow queueing behind a truck moving at vc and rmin=.QU/Q.
Consideration of (5.2) shows that it has all the desired properties in the limit cases. In particular, as r increases r stabilizes at rmin, which can also be expressed as
rmin = 1- w(u-vc)

nu(w+vc)
.
(2)
Note that rmin > 0.7 for the worst case scenario of heavy trucks G=6.5% and n=1 lanes.
It turns out that (5.2) matches simulation results very accurately under all conditions studied here; see for example bottom row in Fig. 5.2. In the entire data set the average absolute difference between the r values obtained with simulation and analytically is .01; see the top row in the figure. As expected, the bigger differences arose for short upgrade, but remain within a 5 %; see the middle row in the figure.
    F51-2.gif
Figure 2: Capacity plots showing agreement between simulation and approximation.

1.3  Comparison with HCM

The HCM 2000 [[39] capacity formulas for upgrade sections are based on the microsimulation model FRESIM, developed by the Federal Highway Administration. Using the notation of this paper and neglecting recreational vehicles, the HCM capacity expression reads
r .
=
 
1

1+r(ET-1)
,
(3)
where ET=ET(L,G,r) is the equivalents of trucks, ie, the number of cars that produce the same capacity reduction than a truck. The values for ET are tabulated for different values of r, L and G in exhibit 23-9 of the HCM for the average truck, which has a weight-to-power ratio "between 125 and 150 lb/hp".
It was not possible to reconcile HCM values with those presented here. HCM recommends r curves that tend to zero as r 1 instead of stabilizing as in (5.3). Moreover, there are cases were r is increasing in some range of r. This is shown in Fig. 5.3 for two typical cases, where the HCM curve is compared to (5.2) for heavy and average trucks. There may be several reason for the disagreement. First, the validation of microsimulation models is a tremendous task and nothing guaranties that, if conditions change, the output will be reasonable; second, the number of lanes is not considered in (5.4); finally, the free motion models in FRESIM may need validation (as suggested in [[124].)
        F51-3.gif
Figure 3: Comparison of HCM values and eq. (5.2) for two typical cases.

2  Disruptive lane-changing (DLC)

In chapter 4 we presented a theory for predicting lane-changing flows on multi-lane traffic. Since lane-changing flows are continuous variables and accelerations are not bounded in this model, it fails to reflect what we believe is the major impact of lane-changing: the temporary blocking on a target lane by a slow-moving lane-changer.
When a lane is interrupted by a lane-changing vehicle coming from a slower lane, it creates a gap in flow in front of it that propagates forward in space on that lane only. This gap has the potential for propagating indefinitely, inducing a permanent loss in capacity. To see this, Fig. 5.4 shows a time-space diagram of a single lane flowing at a speed vd and interrupted by lane-changing vehicle with initial speed va. Two cases need to be considered: if vd is less than the free-flow speed u then the moving bottleneck will be able to catch up with traffic and the flow gap will eventually be recovered, see part (a) of the figure; however, if vd = u it is clear from part (b) that the loss in flow is permanent. We call this a disruptive lane-change (DLC).
One could argue that these flow gaps may be filled with vehicles from neighboring lanes. However, when all the lanes are flowing at capacity, the gap cannot be filled without creating a similar gap on other lanes; ie, the loss in flow cannot be recovered. Therefore, the location of the maneuver should play an important role in determining the amount of disruption. We expect maneuvers "near" the bottleneck to be more likely to result in a DLC.
                F52-1.gif
Figure 4: (a) Recoverable flow gap; (b) disruptive lane-changing.
Here we combine the hybrid theory of chapter 3 with the multi-lane model of chapter 4 into a model that can predict the effects of DLC without introducing any new parameters. This theory can be used with bottlenecks of any nature, eg a lane-drop, a slow truck or an on-ramp. This implies that our proposed physical principle has the potential for explaining the discharge rate of several bottleneck types.
After the formulation of the method in the following section, we present an example that accurately predicts the capacity (and other effects) of a moving bottleneck caused by a slow truck.

2.1  Formulation

In this section we "quantize" the continuum lane-changing process of chapter 4. To model each jump discretely let hxll(t) be the number of lane-changes occurred on the freeway up to time t from lane l to lane l at location x , ie,
hxll(t)=
t

0 
Fll(s,x)ds.
(4)
where Fll(t,x) is the lane-changing rate of movement ll and hxll(t) is real. There are several ways to quantize (5.5) with integer jumps but the most attractive one is using an inhomogeneous Poisson process with rate Fll given by the multi-lane model of chapter 4. 9 In this case (5.5) is the expected number of lane-changes, but the actual number simulated is always discrete. The advantages of the Poisson approach are: (i) the lane-changing process is Markovian since drivers react to traffic conditions in the present, while the past seems irrelevant; (ii) there is no need to define additional parameters; and (iii) it is simple to implement. 10
The implementation of this idea is as follows. Each time that a lane-changing rate is positive, we perform a Bernoulli trial with probability FllDtDx. If the outcome is a success, we introduced a moving bottleneck on the target lane with initial speed vl, and a vehicle type possibly drawn from typical vehicle mixes. The following numerical experiments assume that the lane-changing vehicles are equally fast cars.

2.2  Flow downstream of a moving obstruction

Consider again a long and flat homogeneous n-lane facility flowing at capacity. The four parameters that define the model are u = 60 mph, w = -15 mph, k = 150 vpmpl and t=6 sec. At (t,x) = (0,0) a slow truck enters the segment and travels at a constant speed v u . Fig. 5.5 shows the k-maps for one simulation, two bottleneck speeds, v = 1 and 20 mph. Note from the white regions on the figure how passing maneuvers reduce the flow downstream of the moving obstruction.
F52-2.gif
Figure 5: k -maps for a moving obstruction traveling at (a) v = 1 mph and (b) v= 20 mph.
It can be seen from Fig. 5.5a that the locations of the lane-changing maneuver (indicated as bold dots in the figure for all maneuvers) plays a key role. The closer to the bottleneck the bigger the flow gap. This is in agreement with the intuition in the opening section, as lane-changes near the bottleneck tend to become DLCs. There is a critical distance from the bottleneck where lane-changing no longer has an effect. This is not observed in part (b) of the figure because most of the maneuvers take place very close to the bottleneck.
                    F52-3.gif
Figure 6: Dimensionless bottleneck discharge rate, r, as a function of the slow vehicle speed v for n {2,3,4} lanes.
We repeated this numerical experiment for all v {0, 5, 10, 20, 50 } and n {2,3,4} , and determined a normalized bottleneck discharge rate, r, for each one of these cases and constructed the chart in Fig. 5.6. The normalization constant is the capacity of the unblocked lanes. Previous theories would predict r = 1, which neglected the dependency between capacity and truck speed. The empirical data collected in [[4] and [[39] are indicated in Fig. 5.6 as circles and a square, respectively, while simulation results are shown as curves. It can be seen that our model agrees with available observations quite accurately in the range v > 30 mph.
                    F52-4.gif
Figure 7: Location of the lane-changing maneuvers relative to the truck (simulation.)
For 0 < v < 30 mph our predictions are somewhat intriguing as the system capacity slightly increases when the speed of the truck decreases. One would expect DLCs to be more severe as v 0 since v is also the initial speed of the lane-changing maneuvers close to the truck; let's call this the "initial speed effect". However, careful consideration shows that the maneuver's spatial distribution is more extended as v 0 as noted in Fig. 5.5. In order to compare both parts of this figure with the same scale, Fig. 5.7 shows the location of the lane-changing maneuvers relative to the truck. It is clear how lane-changing maneuvers tend to occur farther from the truck as it travels slower. This implies that a smaller proportion of the maneuvers will be DLCs. We can conclude that this effect overcompensates the "initial speed effect", allowing for higher discharge rates as v 0 . Of course, this curious result is a direct consequence of the lane-choice model considered here (4.7) and field validation needs to be performed, but it clearly highlights the importance of lane-changing location relative to the bottleneck.
The empirical data for v=0 (square in the figure) corresponds to an incident that blocks one lane on a two-lane facility as reported in [[39]. As commented in [[4] this low value could be caused by the "rubbernecking" of drivers passing through the incident. Since this effect is not included in our model it is natural to observe discrepancies. Interestingly, this discrepancy shows that the nature of the bottleneck (incident or lane-drop in this case) plays an important role because they induce different driver behavior.

Chapter 6
Conclusions and extensions

This dissertation developed the formulation of hybrid models, a class of traffic flow models that combine macroscopic and microscopic models to represent heterogeneous traffic streams. These models merge together a continuum KW stream with models describing the acceleration capabilities of each individual slow vehicle. These models are stable and can be validated by parts, which enables fast and effective calibration/validation processes compared to the requirements of microscopic models.
Arguably, the major contribution is a theory based on disruptive lane-changing (DLC) that has the potential for explaining the discharge rate of the most common bottleneck types. The validity of this theory was demonstrated with the multi-lane hybrid model of § 4 that requires only one additional parameter. We found that the loss in capacity imposed by the bounded accelerations of lane-changing vehicles is able to explain the relationship between the discharge rate of a moving bottleneck and its speed, as observed empirically in [[4]. Furthermore, we found that the spatial distribution of lane-changes plays a key role, particularly for low moving bottleneck speeds.
Another contribution is the derivation of a closed-form expression (5.2) for the steady-state capacity of upgrade bottlenecks, as a function of the length and grade of the uphill and the percentage of trucks. These formulas match simulation results with surprising accuracy. We found an important disagreement with the HCM formulas, especially under large truck proportion. A possible explanation may be that the HCM formulas are based on microsimulation, suggesting a revision of the manual's methodology. Truck management schemes near bottlenecks caused by geometry can now be systematically devised; eg, truck-only lanes, truck metering or truck bunching. More generally, formula (5.2) -with suitable parameter values- applies to any moving bottleneck problem provided that they constitute a time-independent proportion of the traffic stream. Examples include freeway segments with general geometric profiles and the effects of buses, turning vehicles or bicycles on city streets.
It is natural to conjecture that DLC theory may explain the "capacity drop" phenomenon observed on merge bottlenecks; ie, as soon as a queue forms on the freeway a lower bottleneck discharge rate is observed. This would imply that effective ITS solutions can be designed with the ability of maximizing system capacity by controlling the variables that induce the drop in capacity. Research is needed to design strategies able to (i) control the spatial and temporal occurrence of lane-changing maneuvers (eg, variable message signs upstream of the merge,) and (ii) control the arrival pattern of heavy vehicles to the bottlenecks.
The frameworks proposed in this dissertation can be extended in a number of directions. Of particular interest is the modeling of congested urban networks since the boundary conditions imposed by urban networks (eg, roundabouts and is signalized/unsignalized intersections) can be handled very naturally in the multi-lane model, and moving bottlenecks can be used to represent buses, trucks, bicycles, lane-changing vehicles and turning vehicles.
A major practical feature that remains to be explored is using the "variational principle" [[67] as the exact numerical solution method for the models presented here. The main challenge is extending the variational principle to the multi-lane case, in a way consistent with the IT principle. This would make possible the efficient and realistic simulation of large transportation networks for the first time, opening the door for real-time simulation for assisting system-wide control strategies.

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Appendix 1
Complements for the multi-lane model

A  The IT principle

This appendix formalizes the extension of the IT principle described in § 4.2, which was characterized by a vector-valued function of four arguments and three outputs, ie:
F(Ll-1,l, Tl, Ll+1,l,ml) (Fl-1,l, ql,Fl+1,l).
(A-1)
In order to simplify the exposition we redefine the inputs as lll given by
lll =



LllDx
if l l
Tl
if l = l
,   "l,l.
(A-2)
In this way, all the inputs have the same units (flow). It is also convenient to define
all= lll



n 
lnl
,
(A-3)
as the proportion of the total demand coming from movement ll. In this way, the problem is to split the total supply ml=.m(kl) to satisfy upstream demands. As mentioned in § 4.2, the procedure must satisfy two conditions: (i) be a reasonable approximation of traffic behavior, and (ii) maximize total flow at every location. In the case of the IT principle, this problem can be carried out with the following computational procedure.

Computational procedure

Given are ll-1,l, lll, ll+1,l and ml. For each l do:
1. Compute auxiliary flows fll=min{lll,allml,Q}, "l,
2. Set ml:=ml-l fll. If ml=0 stop, else
3. Set lll:=lll-fll, "l. If "l:lll=0, stop, else reset all with (A-3) and go to Step 1.
At each iteration, m, this algorithm allocates as much supply as possible to each movement subject to the maximum allocation given by the demand proportion. However, if ml > nlnl then there will always be a movement such that
lll allml,                 for some movement ll,
(A-4)
and some supply will be lost if another movement could have used it. This would violate the flow-maximizing character of the entropy condition, possibly yielding an unstable model. To overcome this limitation, the algorithm will iterate until the remaining supply is split between the other movements, as sought. Notice that this algorithm converges in at most m=3 iterations. Finally, the outputs are obtained using q_ =_m f_^m,
_' =_m f_'^m/ x,         .

B  Examples of moving bottlenecks in the multi-lane model

This appendix shows two applications of moving bottlenecks in the multi-lane model. The first shows a moving bottleneck joining a queue, and the second shows one moving bottleneck passing another one.

B.1  A single moving bottleneck join a queue

Fig. A-1 shows the example on § 3.5 -a heavy truck with v(0)=5 mph under the conditions of Case A on that section. Although the figure is similar to the one-pipe solution of Fig. 3.5, there are some discrepancies on the back-of-the-queue (BOQ) of the jam J. This is expected since the jam propagates differently on each lane. In fact, the average across lanes could never display a sharp shock (as in Fig. 3.5) unless both lanes behave equally; ie, when the flow downstream of the moving bottleneck is the same on both lanes. Consequently, the truck trajectory is slightly different from the single-pipe one as the truck joins the BOQ at a different time.
F4-6.gif
Figure A-1: Example of § 3.5 solved with the multi-lane method.

B.2  Passing

It is expected that when slow vehicles are close enough interactions may occur. Although complex interactions could be incorporated within the proposed framework, this is not our goal because the stability of the overall model could be jeopardized. Instead, we present the simplest case of a slow vehicle that changes lanes to overtake an even slower one.
Crossings among point moving bottlenecks are naturally handled by the multi-lane model. All one needs to do is to make sure that the two moving bottlenecks are on different lanes while the crossing takes place. For simplicity, we assume that when a moving bottleneck wants to change lanes, it can do so immediately, so that the problem reduces to using the appropriate "l" index in (4.11).
As an example, consider the same 2-lane freeway with initial and boundary data kl= 60 vpmpl, which is lower than the optimal density of 75 vpmpl. We include two active moving bottlenecks on lane 1, MBr, r=1,2 with constant desired speed v*r and infinite acceleration capabilities. Initially, MB2 is downstream of MB1. The latter travels at speed v1 < v2*, which could be lower than v1* if it is queueing behind MB2; see top-left diagram in Fig. A-2. We assume that MB1 will start the lane-changing as soon as the gap between the two bottlenecks reaches some critical value, d, ie, when
|f2-f1|=d.
(A-5)
At this point MB1 will move to lane 2 and begin passing, which is denoted by the thin dotted line in the figure. After the passing is completed, MB1 may (or may not, as in the figure) return to the original lane. The remaining parts of Fig. A-2 present the numerical k-maps obtained with such a simple crossing model when v1*=40 mph, v2*=5 mph, d=0.05 mi and Dt= 0.1 sec. Several interesting properties of the solution can observed:
Finally, this example show that "pencil and paper" solutions to apparently simple problems can be extremely difficult to obtain, mainly because of the lane-changing flow, which create different shock waves in different lanes and shocks tend to be curves rather than straight lines.
F4-7.gif
Figure A-2: Crossing between two moving bottlenecks.

Appendix 2
Analytical Solution of the Upgrade Bottleneck

This appendix presents the analytical solution for the capacity of upgrade bottlenecks under the presence of only two vehicle classes, cars and trucks as described in § 5.1. The complexity of the problem resides in the infinite number of possible truck trajectories that can arise on a given segment, even though a single truck type is assumed. This is because the speed of trucks at the bottom of the upgrade determines the actual trajectory, and can vary from free flow-speed u to the crawl speed of trucks, vc(G). It follows that the set of truck trajectories is bounded by the free-flow and crawling trajectories as shown in Fig. B-1a.
FA-1.gif
Figure B-1: (t,x)-diagrams for (a) possible trajectories; (b) approximate trajectory.
Truck trajectories are supposed identical and equal to an "approximate" trajectory, ie, one in between the free-flow and crawling trajectories. Fig. B-1b superimposes these two curves and depicts the approximate trajectory, which is defined with the parameters L, u and vc(G). In this setting, one can apply renewal theory because truck trajectories are independent of one another and so are truck inter-arrival times, h.
As follows from Fig. B-2, the mean of the process at xo, m(h), is given by rhQU,h T and r[TQU+(h-T)Q],h T, where
T .
=
 
L w+vc

wvc
,
(B-1)
is the duration of the disturbance caused by a single truck. It is easy to verify that for triangular fundamental diagrams we have
QU=QD+ w vc

w+vc
k.
(B-2)
Let the random variable h have mean [(h)\tilde] and probability density function f(h). It follows that the time until the next truck arrival is f(h)=m(h)e-m(h), ie
f(h) =



rQUe-rQUh
when h T,
e-Tr(Q-QU)rQe-rQh
when h > T.
(B-3)
Noting that xbe-bxdx,b > 0, equals [1/(b)](1- e-bT)-Te-bT on [0,T] and [1/(b)]e-bT+Te-bT on [T,), and taking the expectation of (B-3) gives
\Tildeh=[ e-rQUT

Q
+ (1-e-rQUT)

QU
]/r.
(B-4)
Since the mean number of vehicles in a truck-led platoon is 1/r, the capacity of the upgrade can be expressed as 1/(r\Tildeh) and its dimensionless capacity as r=.1/(r\TildehQ). Therefore,
r = rmin

1-e-rQUT (1-rmin)
,
(B-5)
where rmin=.QU/Q is the minimum capacity; ie, when the queue is permanent. Notice how (B-5) exhibits of the correct properties in the limit cases: (i) r 1 as r 0 or T 0, and (ii) r rmin as r 1 or T .
                  FA-2.gif
Figure B-2: (a)(t,x) diagrams of a single truck on the uphill; (b) mean of the simplified process.

Footnotes:

1The CT rule applies even for non-concave FD's.
2For example, non-concave and different "congested branch" for each user class is used in [[23] whereas empirical evidence supports one concave user-independent branch.
3The altitude, h(x), is held constant at 100 ft for simplicity.
4In this thesis driving on the right is assumed.
5It is understood that any variable with sub-index ll gives a quantity measured in one direction; ie, from lane l to lane l.
6 Mandatory lane-changing is not analyzed in this dissertation as it requires the introduction of additional parameters; eg, how far away from an off-ramp will drivers be lined-up on the exit lane.
7The sending function for a lane should be roughly equal to the sending function of the whole segment divided by the number of lanes.
8This coincidence is a property of triangular FD's and should not be true in general.
9Notice that lane-changing flow transfers occur continuously, just as in chapter 4, not in discrete jumps of one vehicle as it would appear more natural. However, implementing the latter procedure is technically more complicated and requires additional parameters. Additionally, it is not clear that the results would be significantly different.
10Notice that a deterministic method is less realistic because it is not memoryless.


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