Convergence of Time Discretization Schemes forContinuous-Time Dynamic Network Loading Models

Rui MaDepartment of Civil and Environmental Engineering

Rensselaer Polytechnic Institute (RPI)110 Eighth Street, Room JEC 5107 Troy, NY 12180-3590

Xuegang (Jeff) Ban (Corresponding Author)Department of Civil and Environmental Engineering

Rensselaer Polytechnic Institute (RPI)110 Eighth Street, Room JEC 4034 Troy, NY 12180-3590

Phone: (518) 276-8043 Fax: (518) 276-4833Email: banx@rpi.edu

Jong-Shi PangDepartment of Industrial and Enterprise Systems Engineering,

University of Illinois at Urbana-ChampaignUrbana, Illinois 61801, U.S.A.

Henry X. LiuDepartment of Civil Engineering,

University of MinnesotaMinneapolis, MN 55455, U.S.A.

6,000 words + 6 figures + 1 table = 7,750

ABSTRACTThis paper discusses the importance of obtaining convergence and choosing discretization schemesto numerically solve continuous-time dynamic network loading (DNL) models, using anα point-queue model as an example. The authors prove consistency, stability and convergence of thediscretization schemes for solving the recently developedα point queue model. Also discussed isthe implicit and explicit discretization schemes and theirimplications to DNL solution algorithms.

Key Words:Discretization scheme; dynamic traffic assignment; dynamic network loading; point-queue model;ordinary differential equation; convergence analysis

INTRODUCTION AND MOTIVATIONDynamic Traffic Assignment (DTA) has been extensively studied for decades; see [14, 25, 28] forcomprehensive reviews on this topic. Different from its static counterpart, DTA is challenging dueto not only conceptual complexity such as the dynamic evolution of traffic states over a network(i.e., flow propagation), but also numerical difficulties: it is natural to formulate DTA in continuoustime, which however usually needs to be discretized in orderto obtain numerical solutions. Thedynamic network loading (DNL) process is one of the essential components of DTA. DNL aimsto find the time-varying status of a road network for a given inflow profile [34]. Given such aninflow profile, network status such as exit flow, queue length,travel times can be calculated bythe DNL process. The family of DNL models includes whole linkmodels [9, 23], point-queuemodels [23], and continuum traffic flow models such as the LWR hydrodynamic model [19, 29].A whole-link model uses a delay function to describe the travel time on a link as a function of thenumber of vehicles on the link and/or other variables, such as inflow and exit flow rates. A point-queue model assumes the physical length of any vehicle is zero and the queue is not built up untilvehicles are at the exit node of the link. Vehicle speed is always free flow speed before they jointhe queue. Continuum flow model describes the temporal-spatial evolution of macroscopic flowquantities, with two fundamental relations, conservationof vehicles and flow-concentration-speedrelation [35]. In the past, extensive research has been conducted for DNL models such as how toensure the first-in-first-out (FIFO) property. Nie and Zhang[23] provided an extensive review ofdifferent types of DNL models and compared their performance.

Most DNL models is described in continuous time because timeis not discrete but continu-ous by nature. This generally results in a continuous-time ordinary differential equation (ODE) orpartial differential equation (PDE). Ideally there could be analytical solutions to the ODE or PDEif the model is analytically solvable. This is, however, notthe case for a general DNL model withvariable inflow patterns. Thus, numerical solution methodsusually have to be applied to solvecontinuous-time DNL models. Two critical issues then arise. The first one is how to discretize acontinuous-time model into its discrete-time counterpart, i.e. what kind of discretization schemeone should choose to generate and solve the corresponding discrete-time model. The second issueis how to make sure that the solution obtained from the numerical methods (i.e., the solution fromthe discrete-time model) is convergent to the solution of the continuous-time model. These twoissues are the major focus of this paper.

There are two commonly used schemes to discretize a continuous-time model: an implicitscheme versus an explicit scheme [31]. Implicit and explicit schemes perform differently in keymodel properties such the convergence as we will show later in this paper; they also require differ-ent computational efforts. The former calculates the unknowns at the current time step by solvinga discrete problem defined by such unknowns and involves inputs from the known quantities ob-tained from the previous time steps. One such example is the Euler backward difference which isan implicit scheme for numerically solving ODEs or PDEs. Thelatter calculates the unknowns atthe current time step from the calculated iterates of previous time steps. Euler forward differenceis an example of an explicit scheme. See next section for a mathematical description of these twoschemes. Convergence of a discretization scheme means thatthe solution of the resulting discrete-time model approximates a solution of the continuous-time model; more precisely, convergence

2

here means if the discrete time step goes to zero, the discrete-time solution should converge, insome sense, to a solution of the continuous-time model. Thisis a major priority when numericallysolving continuous-time models, such as DNL models. In other words, if convergence cannot beguaranteed, the discretization scheme would fail to solve the original continuous-time problem.For numerical schemes of solving ODEs and PDEs, two other concepts are also closely relatedto convergence: consistency and stability. The former means the truncation error between thecontinuous-time and discrete-time models vanishes as the discrete time step goes to zero; the lattermeans the discrete solution is bounded if the continuous-time solution is bounded.

Most existing DNL discretization applied an explicit scheme, since in general, implicitschemes require a higher computational cost. However, implicit schemes also have advantages inensuring convergence as will be demonstrated later in this paper. The selection of a discretizationscheme thus has both theoretical and computational implications on solving continuous-time mod-els. So far in the literature, the issue of selecting a properdiscretization scheme for DNL modelshas not been fully discussed. To the best of the authors’ knowledge, the only two references thatfocused on these two schemes are [4, 11]. In [11], it was shownthat the implicit scheme can alwaysguarantee First-In-First-Out (FIFO) for whole link models, which is not always true if an explicitscheme is used. In [4], both the explicit and implicit schemes were used to solve point-queue mod-els, and their impacts to FIFO and convergence were discussed. The importance of consistency,stability, and convergence of a numerical scheme for solving continuum traffic flow models, whichare usually formulated as PDEs, has been well recognized andinvestigated; see [35] for a com-prehensive review. Interestingly, this seems not the case for whole link models and point-queuemodels, for which the convergence issue (as well as consistency and stability) of a numerical solu-tion scheme has not been well studied or understood; see nextsection for a more detailed review.Many mentioned about convergence, which however were generally not rigorously investigated.The only known exception is [4] which provided direct proofsof the convergence of some implicitschemes when discretizing and solving ODE-type DNL models.Therefore in this paper we il-lustrate the concepts of convergence (also consistency andstability) using a modified point-queuemodel recently proposed in [4]. Notice that a whole link model is usually a regular continuous-timefunction, while a point-queue model is often formulated as an ODE. This means that both modelsare conceptually and computationally easier than the PDE-type continuum flow models. We findthat by focusing on those simpler-form continuous-time DNLmodels, the implications of differentdiscretization schemes and the importance of convergence/stability can be more readily illustrated.

In this paper, after reviewing the current literature of thewhole link and point-queue mod-els, we focus on the discretization schemes of the continuous-time DNL models, and the keyconcepts of consistency, stability, and convergence. To illustrate the analysis, we select a modifiedpoint-queue model, namely theα-model, as recently proposed in [4] for several considerations.First, the resulting ODE of theα-model is absolute continuous for a “Lipschitz continuous”right-hand side, i.e., there is no additional difficulty due to discontinuities and shocks. This will allowus to apply traditional ODE convergence analysis techniques, such as the Lax’s Equivalence The-orem [32], to analyse the convergence and stability issues of the discretization schemes. Althoughthe convergence conditions of an implicit scheme to theα-model has been established in [4] us-ing a direct approach, we will provide convergence results for both the explicit scheme and the

3

implicit scheme, extending the results in [4]. Secondly, the α-model approximates the LinearComplementarity System (LCS) model proposed in [4] which corrects the deficiency of the orig-inal point-queue model (thus a more preferred point-queue model to use for DNL). Furthermore,as a parameter of theα-model approaches infinity, theα-model coincides with the LCS model. Inthis sense, the LCS model may be used as a benchmark to comparedifferent schemes to solve theα-model. We show in this paper that both the explicit and implicit schemes of theα-model areconsistent. The implicit scheme is always stable, and thus convergent. For the explicit scheme,however, the stability in only ensured under certain conditions. As a result, the convergence of theexplicit scheme can only be achieved under similar conditions. Numerical experiments are thencarried out to compare the stability and convergence properties of the two discretization schemes.

TIME DISCRETIZATIONWith a few exceptions such as the cell transmission model (CTM), most DNL models start withcontinuous-time descriptions. Time discretization is introduced later either to further develop adiscrete-time model or to solve the continuous-time model.Two discretization schemes, implicitand explicit, may perform differently on some key aspects ofsolving the continuous-time model.Through the literature review, we found that little attention has been paid in the past to the impactsof different discretization schemes for solving DNL models. As we show later in this paper for theα-model, an implicit scheme can perform very differently from an explicit scheme with regard toconvergence and other important characteristics (such as to guarantee the non-negativity of queuelength).

Discretization SchemesContinuous-time DNL models are often formulated as ODEs (orPDEs). Here we focus on theODE-type DNL models such as the point-queue models. Consider an ODEx(t) = f(t, x(t), u(t)),wherex(t) andu(t) are the state and input to the system respectively. In general, there are twobasic discretization schemes [30], implicit and explicit.In both schemes, a time steph > 0 is givenand the time derivativex(t) is approximated by the finite-difference quotient:

x(t) ≈ h−1 ( x(t + h) − x(t) ) .

The explicit scheme computes the unknown state variable at the current time step using the knownstate from previous time steps. Specifically, at a given timeinstanceth,i with xh,i known, such ascheme computes the next discrete-time iteratexh,i+1 corresponding to the time instanceth,i+1 ,

th,i + h by the simple substitution:

xh,i+1 = xh,i + h f(th,i, xh,i, uh,i) (1)

Since the right-hand side of the above equation is known at timeth,i+1, the calculation of the nextiteratexh,i+1 is trivial.

Presumably, compared to an explicit scheme, which has the advantage of easy implemen-tation, an implicit scheme can achieve a more stable solution. In what follows, a general iterationof an implicit scheme is described. Specifically, at a given time instanceth,i with xh,i known, sucha scheme computes the next iteratexh,i+1 corresponding to the time instanceth,i+1 by solving for

4

xh,i+1 in the (generally non-linear) equation:

xh,i+1 = xh,i + h f(th,i+1, xh,i+1, uh,i+1) (2)

Since the right-hand side of equation (2) contains the unknownxh,i+1, an equation solver is neededto obtain the next iterationxh,i+1.

The selection of an explicit or implicit scheme can be problem-specific in order to properlysolve the continuous-time model. In general, an implicit scheme requires more modelling andcomputational efforts but the results tend to be more stable[4]. An example in the DNL literaturethat could be used to illustrate the benefit of the implicit scheme is the “predictive” dynamic userequilibrium (DUE) model proposed by Heydecker and Verland [16]. This model associates thediscrete-time link travel time at a given time instant with inflow to the link at the next time instant.It has been found [2, 15, 16] that the predictive DUE solutionyields less variations compared withits “reactive" counterpart that computes the link travel time at a given time instant with inflow to thelink at exactly the same time instant. The predictive DUE only partially implements the implicitdiscretization idea to the DNL model because it only does so when calculating link travel times.However, it still provides some evidence demonstrating thebenefit of an implicit schemes (i.e.,the solution process is more stable). Later it will be shown that for theα-model, implicit schemesprovide unconditional stability and convergence.

Consistency, Stability and ConvergenceConsistency, stability and convergence are well-known concepts for designing proper discretiza-tion schemes to develop finite difference equations (i.e., the discrete-time model) when solvinga continuous-time model; see [17]. Here we only give their conceptual definitions and omit thedetail mathematical derivations.Consistency:

A discretization scheme is consistent with the continuous-time model if the difference be-tween the discrete-time model and the continuous-time model, i.e. the truncation error, vanishesas discrete time step goes to zero.Stability:

A discretization scheme is stable if the discrete solution is bounded when the solution ofcontinuous model is bounded; it is unstable if the discrete solution is unbounded when the solutionof continuous model is bounded.Convergence:

A discretization scheme is convergent if the discrete-timesolution approaches to the solu-tion of the continuous-time model as the size of the discretetime step goes to zero.

Detailed mathematical formulations of these concepts can be found in [6, 17]. Anotherrelated concept for continuous-time model is the well-posedness which is given as below:Well-posedness::

Well-poseness for ODE/PDE indicates the solution should depend on the initial data ina continuous way. It is an important property for the time evolution of a well-behaved physicalprocess; see [32] for more formal definitions.

With these concepts in mind, we have conducted a literature review on existing research ofDNL models, as shown in Table 1. Here we only focus on the wholelink and point queue models

5

as aforementioned (the only exception is the CTM in [12]). For continuum flow DNL models, theconvergence (as well as stability and consistency) has already been considered as a critical topic insolving the PDE formulations for continuum flow models, see [35]. One would expect convergenceissues to be simpler for point-queue and whole-link models,since they are only one dimensionalproblems while continuum flow DNL models have one more spatial dimension. Nevertheless,these issues have not been well studied so far for point-queue and whole link models, which areimportant for solving those models and thus the focus of thispaper.

From the table, we can see that Carey and his colleagues [7, 8,9, 10, 11] have conductedthe most comprehensive investigations on the whole link model including mathematical propertiesof the model, and solution algorithms. In particular, Careyand Ge [10] studied and comparedboth the explicit and implicit discretization schemes of the continuous-time whole link model;in [11], they concluded that the explicit scheme may violateFIFO but the implicit scheme willalways guarantee FIFO. Nie and Zhang [24] proposed a solution algorithm based on cumulativedeparture curves for the whole link model. They proved that the algorithm can always proceedas long as the travel time function satisfies FIFO and the length of the time interval is smallerthan the minimum link travel time. It was also shown that, forlinear travel time function, discretesolution of their algorithm converges to the solution of thecontinuous model when the length ofsimulation interval approaches zero, which is however not always true for general form whole linkmodels. In [18, 20], the explicit discretization scheme of the point-queue model was discussed andused for solving instantaneous dynamic user equilbirum (DUE). Both schemes were used in [4]to solve point-queue models, with conditions established for the convergence of implicit schemes.Comparisons of results by the explicit scheme of the whole link, point queue, and CTM DNLmodels were conducted in [24].

Several observations from Table 1 are worth mentioning. First, most DNL studies did nottry to link the discrete-time solutions to the continuous-time solutions. In addition, the discretiza-tion schemes applied so far for the DNL models are mostly the explicit scheme, which is intuitiveand easy to implement. To the best of the authors’ knowledge,[10, 11] are the only publishedstudies so far that discussed the implicit scheme, in addition to the recent study in [4]. Thirdly,rigorous investigations of whether the discrete-time solution converges to the solution of the orig-inal continuous-time model are also very sparse. Many mentioned that such convergence can beachieved under certain conditions, e.g., if the time step issmall enough [18], but no formal proofwas provided. Nie and Zhang [23] give a lemma on the convergence under certain conditions basedon [34]; the conditions however may not hold in general. Fourth, FIFO is the primary considera-tion for most studies when developing discretization schemes for DNL models. FIFO certainly isone of the most important requirements for both discrete-time and continuous-time DNL models.However, if a continuous-time DNL is formulated and solved,convergence is equally important toensure that the discretization scheme can really lead to the“true” solution of the continuous-timemodel. Last but not least, no study, to the best of the authors’ knowledge, has explicitly consideredconsistency and stability of the discretization scheme that is applied to solve a continuous-timeDNL model. The only study which briefly mentions consistencyis the CTM model in [12]. Thesetwo properties are thus not listed in Table 1.

From Table 1 and the above observations, it is clear that there is a gap for investigating

6

DNLModelType

DiscretizationScheme

Convergent FIFO

Carey and Ge(2003)

whole link Implicit Newton’s method. Conver-gent if the initial value is suf-ficiently close to solution; noproof is provided.

N/A

Carey and Ge(2007)

whole link Explicit N/A Conditionallyyes

Carey and Ge(2007)

whole link Implicit N/A Yes

Kawahara etal. (1997)

PQ Explicit If time step is smaller, dis-crete solutions become closerto the continuous solutions;no formal proof.

Yes

Li et.al.(2000)

ModifiedPQ

Explicit Mentioned, no proof. Yes

Ban et.al.(2011)

PQ Explicit N/A. Yes

Ban et.al.(2011)

PQ Implicit Yes, via direct proof Yes

Daganzo(1994)

CTM Explicit If the time step goes toinfinitely small, CTM con-verges to the solution of LWR(even discontinuities exist);no formal proof is provided.Show that the discrete-timemodel is consistent

Yes

Nie and Zhang(2005b)

whole link Explicit Convergent under certainconditions

N/A

Nie and Zhang(2005a)

whole link Explicit N/A N/A

Nie and Zhang(2005a)

PQ Explicit N/A N/A

TABLE 1 Review of Existing DNL Models

the consistency, stability, and convergence of a discretization scheme when applied to solving anODE-type continuous-time DNL model. We notice further thatmany existing DNL especiallyDTA models are discrete-time - they may start from a continuous-time description but quicklyintroduce the discretization and develop specific discrete-time models. Examples of this includethe discrete-time link-node DTA formulation in [2] and the link-based DTA model in [33]. For

7

those DNL or DTA models, the convergence issue is less critical because the main focus of thosemodels is to solving the discrete-time problems; no connection was ever made trying to link thesolution of the discrete-time model to that of the continuous-time model. In fact, it is possiblethat such connection can hardly be done because significant changes are usually made duringthe discretization (e.g., how to approximate the flow propagation as shown in [33]); as a result,the discrete-time models may be significantly different from the original continuous-time models.This partially explains why the convergence (also consistency and stability) issues have not beenwidely studied so far.

One may ask if there are relations among these three concepts. According to Lax’s equiv-alence theorem [32], the convergence of a finite difference method is related to the consistencyand stability of the discretization schemes. Convergence can be ensured when one can prove con-sistency and stability for a well-posed linear system. It isusually more difficult to directly proveconvergence than consistency and stability (exceptions doexist, e.g., the direct convergence proofin [4]). For linear systems, therefore, one of the most efficient ways to prove the convergence ofa discretization scheme is via Lax’s equivalence theorem: once consistency and stability are ob-tained, convergence is guaranteed. On the other hand, for a non-linear problem, convergence isusually difficult to prove in general.

We next illustrate the consistency, stability, and convergence of the explicit and implicitschemes for theα-model developed in [4].

DISCRETIZATION AND CONVERGENCE OF THE α-MODELThe α–modelIn [4], two modified point-queue models were developed to correct some of the issues the originalpoint-queue model has. The first one is the so-called LCS model that guarantees that the queuelength is always non-negative, a desirable properties thatthe original point-queue model cannotensure especially in discrete form. The second model is theα-model, an approximation to theLCS model using a parameter, which can be expressed as follows: for some constantα > 0, thedynamics of queue lengthq(t) on a link can be expressed as:

q(t) =

0 if t ∈ ( 0, τ 0 )

max(

p(t − τ 0) − C, −α q(t))

t > τ0.(3)

With q(t) determined, the exit flow ratev(t) and the actual travel timeτ(t) are given by:

v(t) , p(t − t0) − q(t), for t ∈ (t0, T ]

τ(t) , t0 + ( C )−1q(t + t0), for t ∈ [0, T − t0].

Heret is the continuous time,q(t), p(t), v(t) are the time-dependent queue length, inflowrate, and exit flow rate to the link respectively;C is the link exit flow capacity andτ 0 is thefree flow travel time of the link. It is shown in [4] that (i) theα-model coincides with the LCSmodel when the parameterα goes to infinity; (ii) the queue length and exit flow are alwaysnon-negative in continuous time; (iii) FIFO always holds for both the continuous-time and discrete-time formulations; (iv) the implicit scheme is convergent;and (v) queue lengthq(t) is absolutelycontinuous for a “Lipschitz continuous” right-hand side.

8

Before discussing the discretization schemes, we first present an assumption applied in thispaper regarding the inflow profile, which is assumed to be given for DNL models.Assumption (A): The inflow rate to a linkp(t) is non-negative for the time period[0, T0], afterwhich p(t) = 0. Furthermore, the total number of vehicles entering during[0, T0] to the link isfinite and bounded, i.e., we have (Ω > 0 is the bound):

0 ≤∫ T0

0

p(t)dt ≤ Ω.

Note that Assumption (A) only ensures that the total number of vehicles entering the link isbounded which is true for real world applications; it does not impose any restriction on the inflowrate itself (such as boundedness).

We then consider time discretization as follows: Discretize theα-model using a given time

step sizeh > 0 starting from timeτ 0 for q(t). Denote r = h−1

∫ (r+1)h

rh

p(t)dt ≥ 0 for an integer

r = 0, 1, . . ., and thus r for each time stepr is also finite and bounded. From Assumption (A) r = 0 whenr > N0 = T0

h. As shown in [4], the explicit scheme and the implicit schemeto

compute the queue length at time stepr + 1 can be expressed as follows:

Explicit Scheme qr+1 = qr + h[

max(

r+1 − C, −α qr) ]

, (4)

Implicit Scheme qr+1 = max

1

1 + h αqr, qr + h

(

r+1 − C)

, (5)

Consistency, stability and convergenceIn this subsection the consistency, stability and convergence of both explicit and implicit discretiza-tion schemes of theα-model will be discussed.

First, the two discretization schemes are straightforwardEuler forward and backward dif-ferences on a Lipschitz continuous system. Since Euler methods has second order truncation errorson a Lipschitz continuous function, consistency is guaranteed [32]. Therefore we focus on the sta-bility and convergence of the two schemes hereafter in this section.

For the explicit scheme in (4), we first look at the non-negativity issue of the queue length,as shown in the following two corollaries.

Corollary 1. Under the explicit discretization scheme for theα-model, non-negative queue valueat current time step leads to non-negative queue value at next time step. That is, for anyqr ≥ 0,qr+1 ≥ 0 if and only if αh ≤ 1. Hereh is the time step.

Proof. First proveαh ≤ 1 is sufficient.Givenqr ≥ 0 andαh ≤ 1.Whenqr ≥ −α−1(r+1 − C), qr+1 = qr +h(r+1−C). If r+1 ≥ C, qr+1 ≥ 0+h(r+1−

C) ≥ 0. If r+1 ≤ C, qr+1 ≥ −α−1(r+1 − C) + h(r+1 − C) = (C − r+1)α−1(1 − αh) ≥ 0.Whenqr ≤ −α−1(r+1 − C), qr+1 = qr−αhqr = qr(1−αh) ≥ 0. Soαh ≤ 1 is sufficient

for qr+1 ≥ 0, whenqr ≥ 0.

9

Then proveαh ≤ 1 is necessary, i.e., for∀qr ≥ 0 and∀r+1 ≥ 0, if qr+1 ≥ 0, then wemust haveαh ≤ 1.

It is equivalent to show∃qr ≥ 0 ∃r+1 ≥ 0, if αh > 1, thenqr+1 < 0.Indeed suchqr andr+1 exists when0 < qr ≤ α−1(C − r+1). In this case,qr+1 =

qr − αhqr = qr(1 − αh) < 0 if αh > 1. Soαh ≤ 1 is necessary forqr+1 ≥ 0, whenqr ≥ 0. Thiscompletes the proof.

Corollary 2. The explicit discretization scheme for theα-model conditionally generates non-negative queue with positive time step and parameterα, given a non-negative queue as the initialvalue.

Proof. Given the initial queue as a non-negative value, from Corollary 1 it shows the queues arenon-negative step by step, if and only ifαh ≤ 1.

Non-negativity is usually referred to as a realistic property that needs to hold when actu-ally solving the continuous-time DNL models in practice. For theα-model, the continuous-timesolution is always non-negative [4]. One may thus think thatthe discretization scheme should alsomaintain this property. However, for theα-model, this is not true as indicated by the above twocorollaries. In fact, as will be shown in the next two corollaries, the non-negativity condition isstricter than the stability condition. This is because non-negativity requires the queue dischargingprocess smooth and never cross zero, while stability tolerates negative queue values at some timesteps before the process becomes stable (and non-negative)at later time steps. Before stability isproved, a corollary for bounded queue at time stepN0 is given first.

Corollary 3. The explicit discretization scheme for theα model has finite queue value at any finitetime step given finite initial queue and Assumption (A).

Proof. According to the explicit scheme expression (4),qr+1 = q0 + h∑r

i=0 max(i − C,−αqi).Sinceq0 , inflow i and number of termsr + 1 are all finite,qr+1 must also be a finite value.

Corollary 4. The explicit discretization scheme for theα-model has conditional stability with apositive time step and parameterα.

Proof. The total number of time steps is assumed to be infinite in order to prove the stability. Basedon Corollary 3, the queue length at time stepN0 is finite. For∀r ≥ N0, inflow r = 0 according toAssumption (A). Queue can be calculated asqr+1 = qr + hmax(−C,−αqr). There are two casesfor queue at time stepr. Case 1 isqr ≥ C

α. For this caseqr+1 = qr − hC. Case 2 isqr < C

α. For

this caseqr+1 = (1 − αh) qr. Clearly, in case 1, the queue decreases in a constant rateC. Weshow next that the explicit scheme will eventually reach case 2 in finite steps.

At time stepN0, if case 1 happens, i.e., queueqN0 ≥ Cα

, then given sufficient amount of timesteps,q would decrease into case 2 eventually. In fact, the number oftime steps needed for queueat time stepN0 to decrease to case 2, denoted asM , can be calculated as:qN0 −(M−N0)hC < C

α.

This leads toM > N0 + qN0

hC− 1

αh. Here we can setM = N0 + qN0

hC− 1

αh+ 1, which is clearly a

finite number. In other words, if case 1 happens at time stepN0, then after an finiteM steps, case2 will happen. On the other hand, if queueqN0 < C

α, then queue at time stepN0 is already in case

2.

10

Since case 1 would eventually reduces to case 2, the stability condition is only determinedby the stability condition for the queue in case 2. Ifqr < C

α, then

qr+1 = (1 − αh) qr.

Notice that hereqr+1 could be negative.The stability condition of the explicitα-model is:

|1 − αh| ≤ 1 or αh ≤ 2. (6)

which is a necessary condition of non-negative condition∀h > 0, α > 0.

From Corollary 2 and Corollary 4, it can be easily seen that the non-negativity conditionis stricter than the stability condition. Since convergence only requires stability and consistency,negative queue values can occur if only convergence is considered. However, this is not realis-tic in practice. This imposes certain limitations when using the explicit scheme for solving thecontinuous-time DNL models as will be discussed later afterTheorem 6.

For an implicit scheme, it can be seen from (5) thatqr+1 ≥ 0 if qr ≥ 0 which indicates thatthe discrete-time queue length is always non-negative. Based on this, the stability property of theimplicit scheme is given in the following corollary.

Corollary 5. The implicit discretization scheme for theα-model has unconditional stability witha positive time step and parameterα.

Proof. Similar to the proof in Corollary 4, the total number of timesteps is assumed to be infiniteas well in order to prove the stability. Based on Corollary 3,the queue length at time stepN0 isfinite. For∀r ≥ N0, inflow r = 0 according to Assumption (A). Queue can be calculated asqr+1 = max( 1

1+αhqr, qr − hC). There are two cases for queue at time stepr. Case 1 isqr ≥

Cα

+ hC. For this caseqr+1 = qr − hC. Case 2 isqr < Cα

+ h C. For this caseqr+1 = 11+αh

qr. In asimilar fashion as the proof of Corollary 4, we can see that case 2, if ever happens, will reduce tocase 1 eventually in finite steps. Therefore, the stability analysis will only focus on case 2.

If qr < Cα

+ hC, then

qr+1 =1

1 + αhqr.

The stability condition of implicit modified point-queue model is:

| 1

1 + αh| ≤ 1, (7)

which is always true for∀h > 0, α > 0.

Based on the Lax’s theorem, the above Corollary 4 and Corollary 5 can readily lead to theconvergence results of the explicit and implicit schemes, as summarized in the following theorem.

Theorem 6. The explicit discretization scheme for theα-model is convergent with the conditionαh ≤ 2, ∀h > 0, α > 0; the implicit discretization scheme for theα-model is convergent for∀h > 0, α > 0.

11

Theorem 6 clearly show the advantage of the implicit scheme compared to the explicitscheme, i.e., the unconditional convergence. The implicitscheme is much “safer” to use to en-sure convergence than the explicit scheme, no matter how themodel parameters and the discretetime step are selected. This is particularly true for theα-model for which theα parameter ispreferred to be large to better approximate the LCS model (orthe original point queue model;see [4]). As a result, the discrete time steph has to be selected as very small to ensure stabilityand convergence in the explicit scheme. This will obviouslyrequires more computational effort.For the implicit scheme, a relatively largeh can still be selected even for very largeα, thus re-ducing the computational time to generate a convergent discrete-time solution. On other otherhand, given0 ≤ αh ≤

√2, which satisfies the stability condition of the explicit scheme, we have

|(1 − αh)qr+1| ≤ |( 11+αh

)qr+1|. This indicates that if the convergence speed is considered, the ex-plicit scheme performs better than the implicit scheme in this particular case. This will be shownnumerically in next section.

NUMERICAL RESULTSNumerical results of theα-model under the explicit and implicit schemes are shown in this section.Here we focus on a link with free flow travel time0.1 hour and the exit capacity 1000 vehicles perhour (vph). The inflow rate is constantly 2000 vph during the first hour. After that, the inflowdrops to zero immediately. Numerical results of the LCS model in [4] is used as a benchmark forcomparison purposes for these two schemes. Figure 1 depictsthe inflow profile, as well as queuesand exit flows for both LCS model and an explicit scheme of theα-model. It shows that for mosttime steps LCS and this explicit scheme have quite close results to each other, except during thetime steps when queue is near zero (i.e., at about 2.1 hour in Figure 1). Our focus in this sectionis mainly on these time steps when different schemes differ significantly. Furthermore, since thequeue dynamics is the key component in point-queue models, exit flows (which can be deriveddirectly from inflow and queue ) will not be shown hereafter inthis paper; see [4] for more resultson exit flows.

Figure 2 illustrates that for the explicit scheme, stability and convergence do not necessarilylead to the non-negativity of queue length. The figure shows aspecial case, for which the explicitscheme is stable but contains negative queue length. Here1 ≤ αh = 1.5 ≤ 2, which satisfiesthe stability condition but violates the non-negative condition of the explicit scheme. While atsome time steps the queue lengths calculated by the explicitscheme are negative, the distancebetween solutions of LCS and the explicit schemes get smaller step by step, which means thisscheme is stable. The region1 ≤ αh ≤ 2 is thus unique because it guarantees stability but notnon-negativity. This negative but stable region is an important property an explicit scheme mayhave. This implies that besides convergence, there are alsoother important requirements, suchas non-negativity of queue length or FIFO, which need to be considered and captured separatelywhen designing discretization schemes for solving a continuous-time DNL model. We notice thatthe variation of results is quite small in Figure 2 (at the level of 10−13), which can be neglected inpractice. However, it may not be the case for other DNL modelsfor which large variations couldhappen; reducing variation to practical levels could be as important as maintaining stability andconvergence.

Next the time step is fixed ash = 1/60 hours, or 1 minute. For three differentα’s (6, 12,

12

FIGURE 1 Inflow, Exit flow, and queue under LCS and the explicit scheme of theα-model

FIGURE 2 Explicit scheme with negative but stable results

and 240), we show the results of both the explicit and implicit schemes in Figure 3. We can see thatfor the case ofα = 6, 12, the explicit scheme converges faster to the LCS solution. This confirmsthe discussion on convergence speed of the two schemes and shows that the explicit scheme canperform better for some selections of the time step and modelparameters. In this case, the explicitscheme can achieve faster convergence and thus save computational cost.

On the other hand, when the parameterα is large (e.g., 240), the implicit scheme can stillwork well while the explicit scheme generates unstable results since the stability condition is nolonger satisfied and thus the explicit scheme is not convergent to the LCS solution. This shows thatthe implicit scheme can guarantee stability and it is “safe”regardless of different selections of thetime step and models parameters; such property does not exist in the explicit scheme.

13

FIGURE 3 Explicit and implicit discretization schemes for different α’s, h = 1/60 hours

Next, different time steps are chosen in the explicit schemefor a fixed parameterα =1000. Again the LCS result is used as a benchmark. Four different time steps are tested:h =0.8 × 10−3, 1.5 × 10−3, 2.4 × 10−3, and4.8 × 10−3 hours, respectively. Figure 4 shows that whenαh = 0.8, the result is stable and non-negative, and quite close to the LCS result. Asαh becomeslarger to 1.5, the non-negative condition does not hold and thus there is negative queue in the resultbut stability still holds. Whenαh is larger than 2, as seen forαh = 2.4 and4.8, the results areunstable, and thus not convergent at all. This clearly showsthat model parameters and the discretetime step need to be carefully selected to ensure convergence and other desirable properties (suchas non-negativity) when the explicit scheme is applied.

For the implicit scheme, Figure 5 depicts discrete-time solutions for differentα’s and thesame time steph = 0.01 hour. The result shows that for a fixed time step, largerα gives moreaccurate solution, compared to that from benchmark LCS model. We can also see that no matterhow to select the parameterα, the implicit scheme always give non-negative queue lengthand thesolution process is stable, and the results are thus convergent.

Figure 6 shows results from the implicit scheme under different time steps forα = 1000.According to the stability analysis of the implicit scheme,it is shown in the proof of Corollary 4that when the queue at current stepqr ≥ C

α+ hC, it drops with a constant rate; otherwise it drops

exponentially, which means decreasing slower. The figure shows that this is the case for all fourtime steps. The figure further shows that: (i) for all the timesteps, the discrete-time solutions arestable and convergent; (ii) no negative queue is ever produced by any time step, which confirms thenon-negativity of implicit schemes for theα-model; and (iii) the smaller the time step is, the closeris the discrete-time solution to that of the LCS result. Thisconfirms that for theα-model with fixedparameterα, the implicit scheme with a smaller discrete time step can give more accurate results.

14

FIGURE 4 Explicit discretization scheme for different time steps,α = 1000

FIGURE 5 Implicit discretization schemes for different parameter α, time steph = 0.01

hour

Comparing Figure 4 and Figure 6, we can see that for a givenα, the implicit scheme allows moreflexibility in selecting the discrete time steph than the explicit scheme. In fact, the implicit schemedoes not impose any restriction onh as far as convergence is concerned (it does matter in terms ofhow close the discrete-time solution is to the continuous-time solution). This however is not truefor the explicit scheme.

CONCLUSIONSThis paper studied the convergence issues for two types of discretization schemes, the explicit andimplicit schemes, for ODE-type dynamic network loading (DNL) models. By reviewing the exist-

15

FIGURE 6 Implicit discretization schemes for different time steps,α = 1000

ing literature on DNL, especially the whole link model and the point-queue model, it was foundthat (i) most discretization of DNL models applied some explicit scheme; (ii) convergence of thediscrete-time solution to the continuous-time solution was rarely studied. We then investigated thetwo discretization schemes using the recently developedα-model as an example. We showed thatthe implicit scheme always guarantees stability and convergence, in addition to the non-negativityof the discrete-time queue length. On the other hand, the explicit scheme is only stable and conver-gent under certain conditions; more importantly, it can only guarantee non-negativity under evenstricter conditions. The implicit scheme thus allows more flexibility in terms of selecting modelparameters and the discrete step size which may lead to improved computational efficiency (whencomputing an approximate, convergent discrete-time solution), while the explicit scheme may havefaster convergence speed under certain cases.

Since theα-model is (piece-wise) linear and well-posed, convergencecan be shown if con-sistency and stability are ensured. However, for other ODE-type DNL models with discontinuitiesor non-linear components, such as the original point-queuemodel or the LCS model [4], conver-gence cannot be directly achieved since the Lax’s equivalence theorem may not be applied. Inthis case, direct convergence proofs such as those in [4] maybe applied for ODE-type DNL mod-els with discontinuous right hand side and complementarityconstraints based on the differentialcomplementarity system (DCS) framework proposed in [26].

Although the discretization schemes and the convergence concepts are illustrated only us-ing ODE-type DNL models in this paper, similar schemes and concepts can also be applied to othertypes of DNL models (such as applying some implicit scheme tosolving continuum flow models)or to the continuous-time DTA or dynamic user equilibrium (DUE) models. For continuous-timeDTA/DUE models in particular, convergence is a critical issue that needs more attention. Forthem, the convergence of discretization schemes plays an essential role in numerically calculatingnot only the DNL but also the route choice. For example, theα-model studied has been applied to

16

instantaneous DUE problems (IDUE) in [3], which showed thatthe implicit scheme works well indiscretizing and solving IDUE problems. For future research, the impacts of different discretiza-tion schemes on convergence and other related essential issues to general DTA/DUE problems(such as FIFO and non-negativity) may be further investigated.

REFERENCES[1] T. AKAMATSU . An efficient algorithm for dynamic traffic equilibrium assignment with

queues.Transportation Science35 (2001) 389–404.

[2] X. BAN , H.X. L IU , M.C. FERRIS AND B. RAN. A link-node complementarity model andsolution algorithm for dynamic user equilibria with exact flow propagations.TransportationResearch, Part B42 (2008) 823–842.

[3] X. BAN , J.S. PANG, H.X. L IU AND R. MA . Modeling and Solution of Continuous-TimeInstantaneous Dynamic User Equilibria.Submitted to Transportation Research, Part B.

[4] X. BAN , J.S. PANG, H.X. L IU AND R. MA . Continuous-Time Point-Queue Models inDynamic Network Loading.Submitted to Transportation Research, Part B.

[5] M. BEN-AKIVA , M. BIERLAIRE, D. BURTON, H.N. KOUTSOPOULOS ANDR. MISHA-LANI . Network State Estimation and Prediction for Real-Time Traffic Management.Net-works and Spatial Economics1 (2001) 293–318.

[6] R.L. BURDEN AND J.D. FAIRES. Numerical Analysis.Cengage Learning(2010), pp. 339-341.

[7] M. CAREY AND E. SUBRAHMANIAN . An approach to modelling time-varying flows oncongested networks.Transportation Research, Part B34 (2000) 157–193.

[8] M. CAREY. Dynamic traffic assignment with more flexible modelling with links.Networksand Spatial Economics1 (2001) 349–375.

[9] M. CAREY AND M. M CCARTNEY. Behavior of a whole-link travel time model used indynamic traffic assignment.Transportation Research, Part B36 (2002) 85-93.

[10] M. CAREY AND Y.E. GE. Comparing whole-link travel time models.Transportation Re-search, Part B37 (2003) 905-926.

[11] M. CAREY AND Y.E. GE. Retaining desirable properties in discretising a travel-time model.Transportation Research, Part B41 (2007) 540-553.

[12] C.F. DAGANZO. The cell transportation model: A dynamic representation of highway trafficconsistent with the hydrodynamic theory.Transportation Research Part B28 (1994) 269-287.

17

[13] T.L. FRIESZ, D. BERNSTEIN, T.E. SMITH , R.L. TOBIN, AND B.W. WIE. A variationalinequality formulation of the dynamic network user equilibrium problem.Operations Re-search41 (1993) 179–191.

[14] T.L. FRIESZ AND T. K IM AND C. KWON AND M.A. RIGDON. Approximate networkloading and dual-time-scale dynamic user equilibrium.Transportation Research Part B45(2010) 207–229.

[15] S. HAN. Dynamic traffic modeling and dynamic stochastic user equilibrium assignment forgeneral road networks.Transportation Research, Part B37 (2003) 225–249.

[16] B.G. HEYDECKER AND N. VERLANDER. Calculation of dynamic traffic equilibrium as-signments.Proceedings of the European Transport Conferences, Seminar F pp 79–91.

[17] J.D. HOFFMAN. Numerical methods for engineers and scientists.Marcel Dekker(2001),pp. 266–268.

[18] M. KUWAHARA , T. AKAMATSU . Decomposition of the reactive dynamic assignments withqueues for a many-to-many origin-destination pattern.Transportation Research, Part B31(1997) 1–10.

[19] M.J. LIGHTHILL AND J.B. WHITHAM . On kinematic waves. I: Flow movement in longrivers; II: A theory of traffic flow on long crowded roads.Proceeding of Royal Society A,229pp, 281-345.

[20] J. LI AND O. FUJIWARA AND S. KAWAKAMI . A reactive dynamic user equilibrium modelin networks with queues.Transportation Research, Part B34 (2000) 605-624.

[21] H.K. LO AND B. RAN AND B. HONGOLA. Multiclass dynamic traffic assignment model:formulation and computational experiences.Transportation Research Record1537 (1996)74-82.

[22] H.S. MAHMASSANI. Dynamic Network Traffic Assignment and Simulation Methodologyfor Advanced System Management Applications.Networks and Spatial Economics1 (2001)267–292.

[23] X. NIE AND H.M. ZHANG A comparative study of some macroscopic link models used indynamic traffic assignment.Network and Spatial Economics5 (2005a) 89-115.

[24] X. NIE AND H.M. ZHANG Delay-function-based link models: their properties and compu-tational issues.Transportation Research Part B39 (2005b) 729-751.

[25] S. PEETA AND A.K. Z ILIASKOPOULOS. Foundations of dynamic traffic assignment: thepast, the present and the future.Networks and Spatial Economics1(3-4) (2001) 233–265.

[26] J.S. PANG AND D.E. STEWART. Differential variational inequalities,Mathematical Pro-gramming, Series A113 (2008) 345–424.

18

[27] J.S. PANG, L. HAN , G. RAMADURAI , S. UKKUSURI. A continuous-time dynamic equilib-rium model for multi-user class single bottleneck traffic flows.Mathematical ProgrammingA, in review.

[28] B. RAN AND D.E. BOYCE. Modeling Dynamic Transportation Networks: An IntelligentTransportation Systems Oriented Approach. Springer-Verlag (New York 1996).

[29] P.I. RICHARDS. Shockwaves on the highway.Operations Research4 (1956) 42-51.

[30] L.F. SHAMPINE AND S. THOMPSON. Solving DDEs in Matlab.Applied Numerical Math-ematics37 (2001) 441–458.

[31] L.F. SHAMPINE, I. GLADWELL , AND S. THOMPSON. Solving ODEs with MATLAB.Cambridge University Press (Cambridge, England 2003).

[32] J.C. STRIKWERDA. Finite Difference Schemes and Partial Differential Equations (1st ed.).Chapman & Hall(1989), pp. 26, 222.

[33] B.W. WIE AND R.L. TOBIN AND M. CAREY. The existence, uniqueness and computationof an arc-based dynamic network user equilibrium formulation. Transportation ResearchPart B33 (1999) 341–353.

[34] Y.W. XU AND J.H. WU AND M. FLORIAN AND P. MARCOTTE AND D. ZHU. Advances inthe continuous dynamic network loading problem.Transportation Science34 (2000) 402–414.

[35] H.M. ZHANG. Continuum Flow Models, Traffic Flow Theory, A State-of-the-Art Report,Transportation Research Board. Original text by Reinhart Kuhne and Panos Michalopoulos.

19