An analysis of instability in a departure time choice problem

Journal of Advanced Transportation, Fall, 2008 by Takamasa Iryo

This study performs a theoretical analysis of instability in a departure time choice problem. Stability of equilibrium is an important factor for reliability of travel time. If equilibrium is not stable, travel time changes over a period of days even if demand and network performance are stable. This study examines the stability of a dynamic user equilibrium problem by using the departure time choice problem. The mechanism of day-to-day changes in a traveller's behaviour is determined first, and then a function that indicates dissimilarity to equilibrium is defined. The day-to-day changes in the dissimilarity function are mathematically examined using approximations. A numerical test is also carried out to verify the result. Results of these analyses show that there can be a case where the system does not converge to equilibrium. It is also indicated that this instability should be caused by the non-monotonicity of the schedule cost.

1. Introduction

There can be several factors that deteriorate the reliability of travel time in congested road networks. One of the most important factors is the change in the demand for travelling because congestion is severer for an increase in the number of people wanting to drive on the road. Further, network capacity can change over a period of days due to external factors such as the weather. It is natural that travel time is subject to such changes caused by 'exogenous' factors.

One may consider that the network congestion would be stable if conditions such as those mentioned above were stable over several days. Strictly speaking, it can be assumed that the travel time of every link must be unchanged if the same travellers make their trips in the network with an unchanging performance. However, such a consideration may not be true because travellers can choose their behaviour depending on their experiences in the past. This concept is often employed in behavioural studies, for example, see Arentze and Timmermans (2005). Although there can be sources of travel time information via various media, including roadside information boards or intelligent transport systems, they do not provide exact travel time information in the future. Therefore, travellers must collect travel time information on the road network 'after' they finish their trip, and this implies that their behaviour in the networks will depend on the travel time in the past. Such behaviour will lead to the day-to-day evolution of traffic congestion, which will alter travellers' behaviour and congestions over a period of days and cause the system to become unstable. This instability may deteriorate the reliability of travel time. It can be considered as an 'endogenous' factor for the unreliability of travel time.

Among the existing studies on the stability of equilibrium in congested road networks (see Watling (1999) for a comprehensive review of this field), there is a series of studies that discusses stability in deterministic day-to-day processes. A deterministic process is considered as an extension of Wardrop's equilibrium, where each traveller aims to choose the fastest or cheapest option for making his/her trip. The system is in equilibrium if it reaches a stable point as a result of the day-to-day evolution mechanism. Smith (1984a) showed that Wardrop's equilibrium is asymptotically stable for a deterministic day-to-day process if route cost functions are weakly monotonic. This conclusion implies that the monotonicity of route cost functions should be important (but does not indicate whether it is necessary or not) to guarantee the stability of the day-to-day evolution mechanism, which can lead to the reliability of travel time.

However, the monotonicity of route cost functions is generally not satisfied in many realistic traffic loading models, especially within day dynamic traffic loading models such as the bottleneck model. It is known that the link cost function of the bottleneck model is weakly monotonic if each route has only one bottleneck, no blocking due to spillbacks of queues, and no other cost such as the schedule cost added to travellers at destinations (Smith and Ghali, 1990). However, it is known that the monotonicity of the 'route' cost is not guaranteed if a route has two or more bottlenecks (Kuwahara, 1990; Mounce, 2001). So far, the stability of day-to-day evolution has been proved only in the case where travellers can choose only the routes with a single bottleneck and have no option to alter the departure time from the origins (Mounce, 2006). Stabilities in more general models are still under discussion. Szeto and Lo (2006) showed a numerical example of a day-to-day evolution where the physical queue scheme is adopted. Although more theoretical analyses are required for understanding this result, it provides important information that there can be many cases where equilibrium is not stable.

This study focuses on the case where travellers have schedule constraints at destinations and can choose the departure time from the origins. This model is often referred to as the departure time choice problem, and was introduced by Vickrey (1969). Characteristics of equilibrium solutions have been investigated by many studies, and the existence and uniqueness of the solution have been proved for different assumptions. Smith (1984b) and Daganzo (1985) proved the existence and uniqueness, respectively, of the solution by assuming homogeneous schedule cost functions and a single bottleneck network. Lindsey (2004) showed both by considering heterogeneous schedule cost functions. Iryo and Yoshii (2007) showed the characteristics of the solution which are closely related to the existence and uniqueness in more general networks by constructing an optimisation problem which is equivalent to the equilibrium condition. On the other hand, it appears that theoretical studies on the stability of equilibrium in departure time choice problems have not been satisfactorily performed. This study shows the case where the day-to-day dynamics are not stable in a departure time choice problem. This study assumes a simple network with one OD pair, one bottleneck and homogeneous travellers, and it shows that the system is not stable by adopting a day-to-day evolution mechanism, which is similar to the one proposed by Mounce (2006).