A dynamic traffic equilibrium assignment paradox
Introduction
Static traffic assignment theory has been the basic framework not only for estimating traffic demands but also for theoretically considering the problems of transportation demand management policies such as congestion tolls. However, when we consider these problems in the framework of dynamic assignment, taking into account the effect of traffic queues, some theoretical conclusions can differ significantly from those derived from the conventional framework of static assignment.
Consider the two symmetrical networks shown in Fig. 1. The network in Fig. 1(a) has a single origin and multiple destinations. We regard it as an “evening-rush-hour” on a network of a city with a single CBD, and shall refer to it as an “E-net” hereafter. The network in Fig. 1(b) is the “reverse”, with a single destination and multiple origins. We can obtain it by reversing the direction of all the links and origin/destinations of the E-net. Moreover, we may regard it as a “morning-rush-hour” on the same network and shall refer to it as an “M-net”. The standard static traffic assignment produces exactly the same flow patterns for both networks.1 This is because each path in an M-net has the same cost function as that of the reverse path in the corresponding E-net when the standard assumptions of static traffic assignment are employed. Does this simple conclusion also hold for dynamic assignment? The answer is “no”. Indeed, there seem to be essential differences between the two flow patterns for E-nets and M-nets in the dynamic assignment case.
The purpose of this paper is to explore the differences between the two dynamic flow patterns for E-nets and M-nets. Specifically, we first derive explicit solutions of the dynamic equilibrium assignment on “saturated networks” (the rigorous definition is explained later) with one-to-many origin–destination (OD) patterns. We then identify the differences in the properties of the two dynamic flow patterns by comparing the structure with that of the reverse network. Furthermore, we discuss a particular type of capacity-increasing paradox (a dynamic version of Braess’s paradox) as an example that demonstrates the essential differences between the two flow patterns.
In Section 2, we briefly review the dynamic user equilibrium assignment, restricting ourselves to the minimum knowledge required for considering our problem. Section 3 compares the structure of the dynamic equilibrium assignment problems for E-nets and M-nets. Section 4 considers the dynamic version of Braess’s paradox, and Section 5 summarizes the results and remarks on further research topics.
Section snippets
Networks and notation
Our model is defined on a transportation network G[N, L, W] consisting of the set L of directed links with L elements, the set N of nodes with N elements, and the set W of OD node pairs. The origins and the destinations are subsets of N, and we denote them by R and S, respectively. In this paper, we deal only with networks with a one-to-many OD (i.e., the element of R is unique) or those with a many-to-one OD (i.e., the element of S is unique). Sequential integer numbers from 1 to N are
Equilibrium flow patterns on saturated networks
In general, it is impossible to obtain an analytical solution of the DUE assignment formulated in Section 2. Therefore, exploring the properties of the DUE assignment under general settings is not appropriate for our purpose. Instead, proceed by assuming “saturated networks”, which enable us to obtain an analytical solution. “Saturated networks” are networks that satisfy the following two conditions:
(a) there are inflows on all links over the network, i.e., ,
(b) there are
Paradoxes
This section presents a discussion of a capacity-increasing paradox as an example that demonstrates the essential difference between the two dynamic flow patterns for E-nets and M-nets. The paradox is a situation where improving the capacity of a certain link on a network worsens the total travel cost over the network. This is a dynamic version of Braess’s paradox (see Murchland (1970)), which is well-known in the analysis of static assignment.2
Concluding remarks
This paper showed that, unlike the static assignment framework, the direction of flow plays a significant role in the dynamic assignment case. Specifically, we first derived the closed form solutions of the dynamic equilibrium assignment for a network with a one-to-many OD pattern (E-net) and the reversed network (M-net). We then compared the structure of the solutions and theoretically clarified the essential differences between the two dynamic flow patterns, such as the indeterminacy
Acknowledgements
I gratefully acknowledge helpful discussions with Nozomu Takamatsu and Naoshi Oishi on several points in the paper. I would like to thank Benjamin Heydecker for reading the draft and making a number of insightful comments and suggestions. Thanks are also due to an associate editor and two anonymous referees for their helpful comments.
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