Modeling and solving the dynamic user equilibrium route and departure time choice problem in network with queues

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Abstract

This paper considers a simultaneous route and departure (SRD) time choice equilibrium assignment problem in network with queues. The problem is modeled on discrete-time basis and formulated as an equivalent “zero-extreme value” minimization problem, in which the first-in-first-out (FIFO) behavior at intersection is guaranteed by proper formulation of the dynamic link travel times. A heuristic solution algorithm is proposed, which simulates a normal day-to-day dynamic system by a route/time-swapping process, thereby reaching to an extreme point of the minimization problem. The existence of discrete-time dynamic user-equilibrium (UE) solutions is investigated. The iteration-to-iteration stability of the proposed algorithm is discussed, together with numerical results on two example networks.

Introduction

Dynamic traffic assignment (DTA) models have important applications to the rapidly developed advanced traveler information systems (ATIS) and advanced traffic management systems (ATMS). In general, these DTA models can be classified into two categories: the reactive assignment model and the predictive assignment model. The reactive assignment model assumes that each traveler chooses the shortest route to his destination according to present instantaneous traffic condition. As a result of the time-varying traffic condition, travelers between the same origin–destination (OD) pair departing at the same time may arrive at the destination differently if different routes have been chosen. The models proposed by Wie et al., 1990, Boyce et al., 1993, Lam and Huang, 1995, Ran et al., 1993 and Kuwahara and Akamatsu (1997) can be classified as this category. In contrast, the predictive assignment model considers the impact of future traffic condition on route choice behavior, i.e., the shortest route is determined based on the actually experienced travel time or cost by a traveler leaving from a particular location at certain time. The models proposed by Bernstein et al., 1993, Friesz et al., 1993, Kuwahara and Akamatsu, 1993, Smith, 1993, Heydecker and Addison, 1996, Smith and Wisten, 1995, Wie et al., 1995, Ran and Boyce, 1996 and Chen and Hsueh (1998) can be put under this category.

Most of the predictive assignment models aim to satisfy the dynamic user-equilibrium (UE) condition which requires that at equilibrium, the actual travel time or cost experienced by travelers between the same OD pair departing at the same time is equal and minimal. In other words, if such an equilibrium state has arisen today, then there is no incentive for any route-inflow to change tomorrow. This definition about equilibrium can also be used in departure time choice, as well as simultaneous route and departure (SRD) time choices (Bernstein et al., 1993, Friesz et al., 1993, Wie et al., 1995). However, the predictive assignment problem is much more difficult than that of the reactive assignment due to the burdensome computational requirements and vague property of the actual route travel time (or cost) in general networks. Modeling and solving the dynamic SRD equilibrium problems become challenging from both the theoretical and practical viewpoints.

Janson and Robles (1993) formulated a link-based bi-level program for dynamic UE assignment in which departure times are affected by arrival time costs. Lam et al. (1999) also proposed a time-dependent assignment model for logit-based departure time and UE route choices. However, for various reasons these models adopted much longer time intervals, for instance, 10 or 15 min in Janson and Robles, 1993, Janson and Robles, 1995 and 1 h in Lam et al. (1999), than other discrete-time dynamic assignment models that are generally regarded as their continuous-time counterparts. So, these models may be more applicable for long-term transportation planning, rather than instantaneous traffic analysis.

The approach of variational inequalities has been demonstrated to be quite useful for modeling the dynamic SRD equilibrium problem. Friesz et al. (1993) were the first to offer an infinite-dimensional variational inequality (VI) formulation for this type of problem. Wie et al. (1995) extended the model to a discrete-time dimension and presented a heuristic algorithm for obtaining an approximate solution. They employed a non-linear link exit function to describe the physical phenomenon of traffic congestion. This yields some difficulties in computing precisely the actual link travel times which is necessary for preserving the first-in-first-out (FIFO) queue discipline and modeling correctly the vehicle dynamics (or flow propagation).

This paper investigates the dynamic UE route and departure time choice problem in network with queues. A discrete-time predictive assignment approach is adopted. We assume that the time needed to pass through a capacity-constrained bottleneck can be modeled as a deterministic queuing process. The link queue would develop linearly when the inflow rate exceeds the capacity. The actual travel times on links and routes, under this assumption, can be expressed with explicit functions and be calculated accurately.

This paper adopts a route-based modeling approach in which all paths between OD pairs are enumerated. Obviously, this approach requires greater computational effort and data storage for the model run than the link-based one. However, the route-based modeling approach is capable for generalizing the dynamic assignment features so that vehicle flows can be dynamically tracked through the network either on the link-by-link or node-by-node basis. Computational advances, such as parallel and high-speed computational processing, are expected to make it possible to run the route-based SRD models for large-size networks.

Like most DTA models in the literature, this paper treats departure time choice as another dimension of route choice, simply regarding that the decision-making is based on the trade-off between travel costs (including travel times) and arrival costs (either early or late). This leads to more extreme results in which some time intervals have no departures when the arrival costs exceed the travel cost advantage despite queuing. The logit-based departure time choice model can prevent from generating extreme results. Furthermore, many other factors affecting departure time choice, such as the utilities of performing at-home activities and/or out-of-home activities on the way of travel (or commuting), can be incorporated into the proposed model formulation. In fact, this extension has been done in Lam and Huang (2000).

This paper does not consider the physical queue length effects (or spillback effects) on link capacities. Modeling the spillback queuing effects on upstream links, in analytical approach, is a well-known extremely difficult topic in transportation science (Daganzo, 1998). An experimental method of adjusting upstream link capacities for spillback queues was designed by Janson and Robles (1995) and can be extended to the model proposed in this paper.

In the next section, an equivalent “zero-extreme value” minimization condition to the discrete-time SRD equilibrium problem is firstly formulated. Then, the FIFO behavior at intersection is examined, based on appropriate formulation of the dynamic link travel time functions. In Section 3, we investigate the existence of the discrete-time SRD equilibrium solutions in deterministic queuing networks, and examine the monotonicity of the path cost functions. In Section 4, we propose a solution algorithm based on route/time-swapping process and discuss its convergence. Numerical examples, both of a simple two-route network and a more complicated grid network, are presented in Section 5, where we can demonstrate the degree of reaching equilibrium by the proposed algorithm. Section 6 concludes the paper.

Section snippets

The setting

We consider a network G(N,L) composed of a finite set of nodes, N, and a finite set of directed links, L. The set of origins is represented by R and the set of destinations by S,R⊂N,S⊂N. Let a denote a link, and let p denote a path (i.e. route, used without difference in this paper) which is simply an acyclic ordered set of m links, {a1,a2,…,am}, that connects an origin r(r∈R) and a destination s(s∈S). Let Prs denote the set of all feasible paths between an OD pair (r,s). The time period T of

Existence of equilibria

Smith (1979) proved the equivalence between the variational inequality model of the steady-state equilibrium assignment and its fixed point problem. He further pointed out that in the dynamic case the proof is precisely the same provided we apply it to the set of route inflows (Smith, 1993). The set of route inflows, Ω, must be a compact convex set, which is naturally respected in this paper. Hence, the variational inequality model (4) is equivalent to such a fixed point problem: For any ρ>0,

Algorithm

In this section, we present an iterative algorithm to solve the equivalent “zero-extreme value” minimization problem (7). This iterative algorithm is analogous to the “day-to-day” route-swapping process suggested by Smith and Wisten (1995), or the route choice adjustment process based on projected dynamical systems introduced by Nagurney and Zhang, 1997a, Nagurney and Zhang, 1997b. Cybis (1995) has done some foundation works for our algorithm. These authors, however, did not consider the

Numerical examples

The proposed solution algorithm was coded in FORTRAN and run on a personal computer (200 MHz) for solving the two exemplary SRD equilibrium problems. The results are given and discussed below.

The first example is a two-route network drawn from Arnott et al., 1990. All input data are: s1=8000 (veh/h), s2=3000 (veh/h), t10=0.2 h, t20=0.3 h, F (thetripdemand)=22000 veh, α=6.4($/h), β=3.9($/h), γ=15.21($/h), Δ=0.25h, k*=9.0h. In order to use our model and algorithm to solve this example, we set T

Conclusions

In this paper, an SRD dynamic equilibrium assignment problem in network with queues is formulated as a discrete-time path-based VI model and then an equivalent “zero-extreme value” minimization problem. The established link travel time functions ensure the FIFO discipline to be respected in flow propagation. The existence of dynamic UE solutions is proved by verifying the continuity of the nested path travel cost functions that consist of actual path travel times and schedule delay penalties. A

Acknowledgements

This research was supported by the grants from the Research Grants Council of the Hong Kong Special Administrative Region (project ref. no. HKPU-5029/98E) and from the National Natural Science Foundation of China through two projects (79770006, 79825001). The authors would like to thank two anonymous referees for their helpful comments and suggestions on an earlier draft of the paper.

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