CALCULATION OF DYNAMIC TRAFFIC EQUILIBRIUM ASSIGNMENTS ABSTRACT

CALCULATION OF DYNAMIC TRAFFIC EQUILIBRIUM ASSIGNMENTS Benjamin Heydecker Neville Verlander Centre for Transport Studies University College London AB...
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CALCULATION OF DYNAMIC TRAFFIC EQUILIBRIUM ASSIGNMENTS Benjamin Heydecker Neville Verlander Centre for Transport Studies University College London

ABSTRACT The ce ulation of dynamic equilibrium assignments of traffic is important in the analysis of congested road networks. The calculations involved are substantially more intricate than are the corresponding ones for static equilibria. Few authors have published details of dynamic equilibrium assignments; where details are given, some fail to achieve,a good approximation to equilibrium whilst others generate assignments that are discontinuous in time. This paper addresses the issue of formulating dynamic tr&ic assignment in a way that is readily solvable and which leads to solutions that are of good quality and hence are plausible. Two aspects of the formulation have been found to be crucial in achieving this: these are the way in which time varying costs and flows are associated with each other in the mathematical condition that is solved for equilibrium, and the assumptions that are made implicitly about the continuity of the assigned flows. We consider simple test examples that use small networks and simple demand profiles so that dynamic equilibrium assignments can be calculated directly. We show that use of the best general formulation will lead to good quality solutions that are close approximations to the known one. However, we show that relatively innocuous variations in the formulation, such as inappropriate association of flows with costs or assumption of continuous assignments, can lead to unsatisfactory and noisy solutions that are implausible. This work provides a possible explanation and remedy for the implausible character of many published dynamic assignments. 1. INTRODUCTION The calculation and use of dynamic traffic assignments is now widely recognised as playing an important part in the analysis and management of congested road networks. ,Although a considerable literature exists on the formulation and solution of dynamic traffic assignment (see, for example, Ran and Boyce, 1996), relatively little attention has been paid to the quality of the solutions. In particular, few authors have published details of dynamic equilibrium assignments. Amongst those who have, some fail to achieve a good approximation to equilibrium (for example, Wie, Tobin, Friesz and Bemstein, 1995) whilst others have intervals throughout which the assignments that are calculated are discontinuous (for example, Papageorgiou, 1990; Jayakrishnan, Tsai and Chan, 1995; and Lam and Huang, 1995). These results, then, are either not good solutions to the problem in hand, or through their lack of stability are not credible. In either case, the solutions cannot be considefed to be of good quality.

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In the present paper, we establish a formulation of dynamic traffic assignment according to the user equilibrium principle that when developed with due care, leads to estimates of flows that are stable and for the main part smooth, whilst achieving the dynamic equilibrium that is required of them. We believe that good quality solutions of this kind are important in establishing repeatable and credible model estimates. We conclude from this that good quality models are now available for dynamic traffic assignment and should be adopted in preference to others. Their use in the development and analysis of traffic management plans will be informative, and will lend transparency and credibility to the resulting evaluation. It also offers the prospect of leading to an improved understanding of the processes involved in the operation of congested road networks, and of emergent behaviour of traffic using them.

2. FORMULATIONS OF DYNAMIC ASSIGNMENT

2.1 Introduction The model formulations for dynamic analysis of road networks have substantially more stringent requirements on them than do their counterparts for static assignment. This additional stringency arise from considerations such as the propagation through the network of temporal variations in flow rates, and the relationship between costs and flows. In static formulations, these are either relatively unimportant or are accommodated implicitly. In this section, we specify clearly the details of the formulation that is analysed here. Although these details are shown to be of crucial importance to the successful calculation of dynamic assignments, the analysis presented here shows that straightforward formulations can be established which perform well. Against this, we also show that relatively innocuous variations in this formulation can lead to unsatisfactory results. The topic of this paper is the calculation of dynamic equilibrium assignments. A central issue in this is the specification in mathematical form of a sufficient condition for an assignment to be in equilibrium. Here, we adopt the widely used dynamic extension of Wardrop's (1952) equilibrium principle that the travel costs incurred by traffic on all routes that are used between each origin-destination pair at an instant are equal and less than those that would be incurred on any unused route at that instant. This can be stated after Beckmann (1956) in the form of complementary inequalities that are satisfied by the costs and flows which we extend here to apply at each instant of continuous time: > o 3 C,(s)= CId(S) v p E Pod v s e p ( 4 = o 3 c,(s)~c~d(s)

{

I

is the rate at which traffic enters route p at time s , Pod is the set of all routes from origin o to destination d , C,(s) is the cost incurred on route p by traffic entering it at time s , and C*,d(s) is the minimum cost of travel from o to d starting at times. where

ep(s)

There are several reasons why this formulation as it stands is insufficiently detailed for practical use in calculating dynamic traffic equilibria. These stem in part from the fact that practical solution procedures are based on the calculation of flows and costs jn

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discrete time intervals rather than at each instant over continuous time. In the remainder of this section, we develop a detailed interpretation of the dynamic equilibrium condition (1). We then proceed to show that it is adequate for use in the calculation of good quality dynamic traffic equilibria in simple networks, and contrast the results of this with those arising from the solution to minor variants of the formulation.

2.2 Traffic flows We suppose that the demand for travel varies over time and is specified exogenously. In our discrete time formulation, we adopt a time index s, that is used to indicate the whole of the interval [s,, sn+l)where s,,~ = s, + As for some increment A s . We suppose initially that the flows are constant throughout each such time increment, and vary freely between successive ones. We therefore interpret the flow e p (sn ) as the constant rate at which flow enters route p throughout the time interval [s,, order to satisfy the exogenous demand for travel, we require that De,(S")

&+I)

. In

= To&,)

(2) 2 0 tJPEp,, where T,,(s,,) is the mean rate at which traffic departs from origin o for destination d during the time interval s, . PEP,

e,(s,)

2.3 Costs of travel We suppose that the costs of travel that would be incurred by entering a route at each instant are furnished by an appropriate traffic model. In order to respect causal determinism, we require that these costs depend only on the traffic conditions that are encountered on the trajectory of a vehicle: this prevents the rates of flow entering a route from influencing the costs and trajectory of a vehicle that has entered that route at an earlier time. Although this requirement is entirely reasonable from a trufJic modelling point of view, not all of the cost functions that have been used in formulations of dynamic assignment respect it: for discussion of this, see, for example Hurdle (1986), Daganzo (1995), Astarita (1996), and Heydecker and Addison (1998). Considerations of conservation of traffic in conjunction with variations in speed mean that the rate of flow can vary along the trajectory of a vehicle (see, for example, Astarita, 1996; Heydecker and Addison, 1996). We suppose that the first-in first-out W O ) traffic discipline is observed strictly, so that the cost of travel at instant s, cannot be affected by traffic entering during the as the cost that would be interval [sn,s,+J . We therefore interpret the cost c,(s,) incurred by a vehicle that enters route p at the instant s,+l , ie immediately after the interval [s,, s,+l) . This convention of using the same index for flows during a time interval and costs at the end of that interval then respects the influence of flows on costs.

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2.4 A mathematical programming formulation The complementary inequality formulation (1) of traffic equilibrium is not in itself a good basis for an approach to the calculation of dynamic assignments. However, Smith's (1979) transformation to variational inequality form does provide a suitable basis. Thus an assignment at time s,,, expressed in the form of a column vector of route inflows e(s,) ,is an equilibrium if and only if

-[f

- e(s,

'I)

.

o

V f E D(s,) (3) where D(s,) is the set of route inflow vectors that satisfy the travel demand constraints (2). Because the costs used in (3) are those that arise immediately after the flows during the time interval with which they are associated, this is a predictive formulation.

I

Use of the assignment f = e(s,) in the left-hand side of (3) gives the value 0, so that an equilibrium assignment is achieved when the maximum value with respect to choice off is exactly zero. Thus the equilibrium assignment e*(s,)E D(sn) for time interval s,, solves the programming problem z(sn,e) Min (4) e q % ) where the objective function of this optimisation is specified as z(s,,e) = MX -[f-e]'.c(s,). fq%) According to (3), the value of the objective (5) at the solution to (4)is exactly 0 . Thus the dynamic equilibrium assignment e' is the solution to the optimisation problem Min zz(sn,e) e " (6) Subject to e(s,)e D(s,) V n . 2.5 A solution approach Because of the influence of assigned flows e(sJ on future costs, the optimisation problem (6) is a dynamic programming formulation. Such problems are often of limited practical value because of their formidable difficulty of solution: the backward dynamic programming procedures that have general applicability for this class of problem have substantial computational demands because they calculate many partial solutions, the vast majority of which do not form part of the whole solution. However, the special property of the present formulation that the optimal value of the objective Z(sn,e ) is exactly 0 at each time step n can be used to good effect. In view of this property, the contribution of all future optimal elements of the summand in ( 6 ) is known to be exactly 0, even though the assignments that achieve this need not be known. Accordingly, the dynamic programme (6) can be solved in the natural increasing time order by solving separately in sequence the individual steps (4) from low to high values of n : the solutions to earlier steps will influence those to later ones through the cost

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functions c(.), but unusually for a dynamic programme, the solutions to later steps w ill not influence those to earlier ones. This solution approach is one of forward dynamic programming. It can be applied to the present formulation because of two special properties. First, the flows assigned in the future have no influence on current costs: this is a reasonable requirement of costs functions which obtains in simple networks provided that appropriate traffic models are used to generate them. Second, the optimal value of the objective function is known to be 0 at each step without any need to calculate it.

3. EXAMPLE CALCULATIONS

3.1 Introduction We now compare various approaches to the calculation of dynamic equilibrium assignments. We introduce a simple test example, and then identify by direct analysis the exact form of the equilibrium assignment. We proceed to show how the formulation developed in Section 2 can be applied successfully to calculate good quality dynamic equilibrium assignments. We then consider the consequences of varying certain details of the formulation for the quality of the assignments calculated. In particular, we consider the effect of including in the objective function (5) the cost at the start of the time interval [s, s,+1) rather than that at the end of it. Last, we consider the effect of assuming that the equilibrium assignment solution is continuous in time rather than calculating the assignment during each time increment according to the cost functions but otherwise independently of the values of earlier assignments.

3.2 Specification of a test example The test example that we consider is chosen deliberately to be simple. It uses a network of two separate routes between a single origin-destination pair. The traffk model that we adopt is the deterministic aueue, so that *

(L(t0)= 0, e(t0 - cp)- Q (otherwise)

- (P)5 Q )

where L(t) is the number of vehicles in the queue at time t , (P is the free-flow travel time, and Q is the capacity of the queue. The delay incurred by a vehicle that reaches the queue at time t is L(t) / Q , so that the time of anival ~ ( s )at the destination associated with departure at time s is given by ~ ( s ) = s + cp + L(s+cp)/Q . (8) In this case, we take travel time to be the sole component of cost of travel, so that c(s) = (P + L(s+cp)/Q. (9) It is clear from this specification that the cost of travel at any time s can depend on flows at earlier times though traffic accumulated in the queue that is encountered, but that this cost does not depend on any future inflows.

The free-flow travel times and capacities for the two routes in the present example network are specified in Table 1. According to this, route 1 has a lower free-flow travel time than does route 2, so that it will be used by all traffk initially. Only when the queue has been overloaded for sufficiently long that the delay in the queue reaches 10 seconds will route 2 come into use: this will happen when the number of vehicles in the queue reaches 5. Route

Free-flow travel time cp (seconds) 40 50 ~

1 2

Capacity Q (vehicles I second) 0.5 0.3

Table 1: specification of routes in the example network.

The specification of the test example is completed by the exogenous inflow profile. We adopt a temporal profile that varies over time, starting at 0, increasing linearly to 1 vehicle / second at time 20 seconds, remaining constant at that rate until time 40 seconds, decreasing linearly to 0.6 vehicles per second at time 60 seconds and then remaining at that level indefinitely. This demand profile is chosen to be continuous over time, to have a maximum level that exceeds the capacity of the network which is 0.8 vehicles / second, and to remain indefinitely at a level that exceeds the capacity of the route that is fastest under free-flow conditions. During each interval of the discretetime formulation, we use the constant inflow rate that will achieve the same total inflow as the specified demand profile, corresponding to the mean value over that interval.

3.3 Analytic solution

In view of the simplicity of the network, the dynamic equilibrium assignment can be calculated directly using the techniques presented by Heydecker and Addison (1996). In particular, the principle can be used that in equilibrium, the rate of inflow to each route of minimum cost at the time of departure is proportional to the outflow at the time of arrival. By applying this principle, we solve the equilibrium condition (1) exactly at each instant in continuous time. In the present case, the assignment is initially all to route 1 because it has least travel time under free-flow conditions. Until time 10 seconds, the assignment is within capacity, so that the queue that is encountered on that route remains empty. However, between times 10 seconds and 20 seconds, the inflow exceeds the capacity by a total of 2.5 vehicles, and between times 20 seconds and 40 seconds the demand exceeds the capacity of route 1 by the constant value of 0.5 vehicles / second. Thus by the time corresponding to departure at time 25 seconds, the length of the queue that will be encountered on route 1 has reaches the critical value of 5 vehicles, and route 2 comes into use. The total rate of inflow at that time exceeds the sum of the capacities of the two routes (0.8 vehicles / second) so that the queue on each of them will increase after that time. Because of this, the outflows from the routes will each be equal to their 84

respective capacities, so that in order to retain equilibrium, the inflows will be proportional to them. Thus the proportion of traffic that is assigned to route 1 decreases instantaneously to 0.625 (ie 518) whilst that assigned to route 2 increases instantaneously from 0 to 0.325. The assignment will continue in these proportions until the time of departure that corresponds to the next encounter of an empty queue on route 2, which occurs at the time 75 seconds. By this time, the total inflow has fallen to the constant value of 0.6 vehicles I second which is within the capacity of the network but exceeds the capacity of route 1 alone. After this time, the outflow from route 1 is equal to its capacity of 0.5, whilst that from route 2 is equal to its inflow at the time of departure, so that the equilibrium assignment proportions change instantaneously to 0.833 (ie 5/6) for route 1 and to 0.167 for route 2. This analysis shows that the dynamic equilibrium assignment to the present example network is continuous at almost al.! times, but has jump discontinuities at certain instants. 3.4 Predictive equilibrium assignment Because the formulation introduced in Section 2 associates flows during each time interval with the costs at the end of it, we refer to it as a predictive formulation. The solution to the mathematical programming problem (6) can be calculated readily for the present example by proceeding step by step from the start to the end of the study period. This entails solving for each step n in sequence the programming problem specified by (4) and (5) for the flow s, during that interval: the cost &) associated with that step is calculated according to the earlier assignments and the incremental assignment during that interval. The effect of this is that the assignments are calculated for each time increment to achieve equality of costs of those routes in use at the end of the time increment. The amount of work involved in this corresponds broadly to solving as many static assignments as there are time steps during the study period. The solution to this formulation for the present simple example network using a time increment of 1 second is shown diagrammatically in Figures 1 and 2. Figure 1 shows the proportion of the demand that is assigned to each of the two routes at each time, whilst Figure 2 shows the same information in the form of rate of flow assigned to each route together with the total demand. The solution can be described conveniently in terms of assignment proportions as shown in Figure 1. Up to time s = 25 seconds, the assignment is all to route 1. As discussed in section 3.3, at that time the cost of using route 1 reaches the free-flow cost of route 2, which therefore comes into use. The assignment proportions calculated for the time increments starting from that at 25 seconds are then proportional to the capacities of the respective routes. The proportions remain so until the time of entry that corresponds to the next arrival at an empty queue on route 2, which occurs at the time increment starting at s = 75 seconds. After that time, the assignment proportions are such that the inflow to route 1 is exactly equal to its capacity so as to maintain the travel time equal to the free-flow travel time on route 2. These assignments are shown in terms of inflow in Figure 2, in which the variations in inflows seen during the interval [25,75) seconds arise because of the constant assignment proportions applied to the varying total demand.

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The dynamic equilibrium assi,ment calculated according to this approach achieves an excellent approximation to the analytical solution described in Section 3.3. Despite the freedom of the inflows calculated to vary between successive time increments, they remain similar except at the two instants of discontinuity discussed above. These solutions seem to be of good quality and can be interpreted readily: the results calculated according to this formulation and shown in Figures 1 and 2 differ imperceptibly from the exact analytical ones. Beyond that, these results are largely unremarkable.

3.5 Reactive equilibrium assignment We now consider the effect of varying the association between assigned flows and costs. We associate the flows assigned during each time interval with the costs that obtain at the start of the interval rather than with those that arise at the end of it. Because the flows are assigned after the time at which these costs obtain, the flows have no influence on the costs. We refer to this as a reactive formulation because the assignments are calculated in response to rather than in anticipation of costs. In this formulation, the equilibrium assignment e * ( s n ) E D(s,) for time interval s, solves the programming problem (4)where the objective function in this case is specified as

z(s,,e)

=

MX

f @(%)

-[f -e]'

(10)

The sole difference between this objective and that specified by (5) is in the time at which the cost function is evaluated here it is specified as c(s,-,) rather than c(s-). However, the effect of this is profound because the flows calculated in each step do not affect these costs, this is a linear programming problem. The solution can be seen directly to be achieved by the all-or-nothing assignment of the demand to the fastest routes.

In the case of the present example, the resulting flow assigned to route 2 and the demand profile are shown diagrammatically in Figure 3. As in the case of the corresponding predictive equilibrium assignments shown in Figure 2, the assignment before time 25 seconds is all to route 1. However, at that time, route 2 becomes infinitesimally faster, so all flow is assigned to it at the next step. The result of this is to overload route 2 whilst allowing the queue on route 1 to decrease, so that the next assignment is all to route 1. Because the demand remains at a level that exceeds the capacity of each route individually, no such all-or-nothing assignment can remain a solution to this formulation for indefinitely many successive time intervals. Thus the assignment calculated according to the reactivate formulation (10) oscillates, overloading each route in turn for one or more successive steps. Whilst the amount of traffic assigned to each route according to this formulation during a prolonged interval might correspond broadly to that according to the predictive formulation and the short-term assignments might bear interpretation in terms of the entry of small packets of traffic into routes, the exact detail of the assignment will clearly depend on arbitrary choices including the duration of the time increment As that is used. This dynamic assignment seems to be both of low quality

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and difficult to interpret: comparison of two assignments calculated in this way for different network configurations might well be difficult.

3.6 Assumed continuity The formulation presented in Section 2 considers the assignment during each time interval to be calculated independently from that during earlier intervals, except for their influence on the costs that will arise. We note in the exact analytic solution to the present example dynamic assignment problem that the equilibrium solution is continuous almost everywhere: it has the two step discontinuities noted above at times of departure 25 seconds and 75 seconds, which are required to maintain equilibrium. We now consider the effect of calculating assignments that are necessarily continuous in time. In this case, we adopt the predictive formulation (4) and (5), but consider assignments that vary linearly during each time interval and are continuous between successive ones. We achieve this by solving (4)and (5) for the instantaneous assignment i(sJ at the end of time interval s,, and then setting e(sJ = 0.5[i(sn.l)+ i(sn)] . Because the assignment to each route should be positive at all times and should never exceed the total demand, we limit the instantaneous assignments i,,(sn) to the range [0, T(sn)]. In the case of the present example, the resulting flows e&) assigned to route 2 are shown diagrammatically in Figure 4 together with the demand profile. As in the case of the corresponding predictive and reactive equilibrium assignments shown in Figures 2 and 3, the assignment before time 25 seconds is all to route 1. However, at that time, route 2 comes into use: in order to achieve equilibrium at the end of the next time increment, a dscontinuous change in assignment is required. Because this formulation does not allow such a change, a correction is required in the next time increment. As can be seen from Figure 4, the size of the subsequent corrections does not decrease until the demand falls, and even then it does not decrease towards 0. Thus by precluding the possibility of any discontinuity in the assignments, those calculated are subject to substantial oscillations in order to achieve the required mean value within each time increment. The total amount of traffic assigned to each route during a time interval according to this formulation will correspond to that according to the predictive formulation provided that the instantaneous flow required to achieve that remains within the total level of demand throughout the interval. However, that will not always be possible, so during some time intervals equilibrium will not be achieved. As in the case of reactive assignment, this dynamic assignment seems to be both of low quality and difficult to interpret so the same reservations apply.

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4. DISCUSSION AND CONCLUSIONS The predictive formulation of dynamic equilibrium assignment presented in Section 2 appears to have satisfactory computational properties and has solutions that are close to the exact ones for simple example problems. The solutions are plausible and clear, and hence are easy to interpret. By contrast, the solutions to the two variant formulations developed here of reactive and continuous dynamic assignment lead to low quality solutions that are implausible and would be hard to interpret. Notwithstanding the deliberate simplicity of the test example presented here, the formulation presented in Section 2 has substantial generality. It can, for example, accommodate the use of other traffic models to represent network loading and from which travel times can be calculated provided that they respect causal determinism and FlFO discipline. The formulation can be applied to networks with multiple origins and destinations and intricate topology. However, an important aspect that has not yet been explored is the treatment of flows that enter the network after others but downstream of them and hence influence their trajectories: this will cause an influence of future assignments on travel costs. The original problem of dynamic equilibrium traffic assignment that is represented by the complementary inequalities (1) is specified in continuous time. However, for all but the simplest networks, numerical solution will be required in discrete time. The original formulation applied instantaneously: what is established by the results presented here is that the way in which this continuous problem is discretised is crucial to the quality of the solutions that are calculated. The choice between predictive and reactive formulations arises only in the discrete time formulation and raises some interesting behavioural issues. Notwithstanding the matter of detailed interpretation of the assumptions implicit in the specification of predictive route choice behaviour in the model presented here, this is required in order to achieve good quality solutions that are close approximations to the exact ones. From this, we see that the exact details of the formulation of dynamic traffic assignment are crucial to its satisfactory solution. Even minor variants can lead to dramatic changes in the assignments that are calculated. Furthermore, inappropriate assumptions, such as that of continuity which obtains in the solution almost but not quite everywhere, can lead to substantial reduction in the quality of the assignments calculated. This lack of robustness in the problem formulation to variations that appear to be innocuous emphasises the need for careful analysis and model formulation, thorough testing and critical appraisal.

Acknowledgements The work reported here was funded by the UK Economic and Social Research Council under the LINK Inland Surface Transport p r o g r m e . The authors are grateful their colleagues and partners in this project, and in particular to Miles Logie and Mike Smith, for their relevant and supportive work. They are also grateful to Puff Addison and Richard Allsop for a number of stimulating discussions on the topic of this paper.

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REFERENCES Astarita, V (1996) The continuous time link based dynamic network loading problem: some theoretical considerations. Proceedings of the 24" PTRC European Transport Forum, Seminar D. PTRC, London, P404(1). Beckmann, M, Mc Guire, CB and Winsten, CB (1956) Studies in the economics of transportation. Yale University Press, New Haven. Daganzo, CF (1995) Properties of link travel time functions under dynamic loads. Transportation Research, 29B(2), 95-8. Heydecker, BG and Addison, JD (1996) An exact expression of dynamic traffic equilibrium. In: Transportation and Traffic Theory (ed J-B Lesort). Pergamon, Oxford, 359-83. Heydecker, BG and Addison, JD (1998) Analysis of traffic models for dynamic equilibrium traffic equilibrium. In: Transportation Networks: Recent Methodological Advances (ed MGH Bell). Pergamon, Oxford, 35-49. Hurdle, VF (1986) Technical note on a paper by Andre de Palma, Moshe Ben Akiva, Claude Lefevre and Nicolaos Litinas entitled "stochastic equilibrium model of peak period traffic congestion. Transportation Science, 20(4), 287-9. Jayakrishnan, R, Tsai, WK and Chen, A (1995) A dynamic traffic assignment model with traffic-flow relationships. Transportation Research, 3C( l), 51-72. Lam, WHK and Huang H-J (1995) Dynamic user optimal traffic assignment model for many to one travel demand. Transportation Research, 29B(4), 243-59. Papageorgiou, M (1990) Dynamic modelling, assignment, and route guidance in traffk networks. Transportation Research, 24B(6), 471-95. Ran, B and Boyce, D (1996) Modelling dynamic transportation networks. Springer, London. Smith, MJ (1979) The existence, uniqueness and stability of traffic equilibria. Transportation Research, 13B(4), 295-304. Wardrop, JG (1952) Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1(2), 325-62. Wie, B-W, Tobin, RL, Friesz, TL and Bemstein, D (1995) A discrete-time, nested cost operator approach to the dynamic network user equilibrium problem. Transportation Science, 29(1), 79-92.

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Figure 1: Predictive equilibrium proportions

1

-I

c .-0 ?= 0

Q

2

Q 4-

5

E S .-m

0.5

v)

2

0 0

I

50

100 Start time (s)

150

200

Figure 2: Predictive equilibrium assignments

1

Total Route 1

Route 2

0

I

0

50

100 Start time (s)

90

150

200

Figure 3: Reactive equilibrium assignments

1 h

v)

\

v)

a, 0

.c

a,

L 0.5 3

0 cc

-S

0 0

50

100 Start time (s)

150

200

Figure 4: Continuous equilibrium assignments 1

0 0

50

100 Start time (s)

91

150

200

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