Dynamic and stochastic shortest path in transportation networks with two components of travel time uncertainty

Abstract

The existing dynamic and stochastic shortest path problem (DSSPP) algorithms assume that the mean and variance of link travel time (or other specific random variable such as cost) are available. When they are used with observed data from previous time periods, this assumption is reasonable. However, when they are applied using forecast data for future time periods, which happens in the context of ATIS, the travel time uncertainty needs to be taken into account. There are two components of travel time uncertainty and these are the individual travel time variance and the mean travel time forecasting error.

The objectives of this study are to examine the characteristics of two components of travel time uncertainty, to develop mathematical models for determining the mean and variance of the forecast individual travel time in future time periods in the context of ATIS, and to validate the proposed models. First, this study examines the characteristics of the two components of uncertainty of the individual travel time forecasts for future time periods and then develops mathematical models for estimating the mean and variance of individual route travel time forecasts for future time periods. The proposed models are then implemented and the results are evaluated using the travel time data from a test bed located in Houston, Texas. The results show that the proposed DSSPP algorithms can be applied for both travel time estimation and travel time forecasting.

Introduction

The dynamic and stochastic shortest path problem (DSSPP) has been the subject of extensive research in the transportation area for many years. In this problem the link travel time is assumed to be a time-dependent random variable. With the advent of Advanced Transportation Management Systems (ATMS), which are designed to improve transportation system performance, an opportunity exists for extending the DSSPP and implementing it on an actual transportation network. In ATMS, real-time travel time information is obtained directly from probe vehicles or estimated from inductive loop data. Probe vehicles are outfitted with special automatic vehicle identification (AVI) equipment or geographic positioning system (GPS) units. One of the important advantages of probe vehicles is that the travel time of the individual vehicles over each link in their route can be measured and recorded, which has been impossible with the inductive loop data. Recently there have been considerable research in using probe vehicles to estimate and/or predict travel time (Boyce et al., 1993; Eisele and Rilett, 2002; Hellinga, 2001; Park and Rilett, 1998, Park and Rilett, 1999; Sen et al., 1997; Srinivasan and Jovanis, 1996; Tarko and Rouphail, 1993; Turner and Holdner, 1995). The real-time link travel times are used as input to link travel time forecasting algorithms which provide forecast link and route travel times in future time periods.

With the advent of Advanced Traveler Information Systems (ATIS), and in particular Route Guidance Systems (RGS), the prediction of short-term link travel times has become increasingly important. Intuitively, the RGS’s route selection algorithms should use link travel times that are based on the time at which the driver is expected to arrive at a given link rather than use link travel times that are based on current conditions. Because drivers implicitly base their routes on the anticipated link travel time, the RGS should have the same capabilities (Park et al., 1999; Rilett and Park, 2001). Given this requirement, most of the existing travel time forecasting models provide forecast mean link travel time for some future time using observed link travel times (Boyce et al., 1993; Park and Rilett, 1998, Park and Rilett, 1999; Park et al., 1999, Park et al., 2002, Park et al., in press; Rilett and Park, 2001; Sen et al., 1997; Tarko and Rouphail, 1993; Turner and Holdner, 1995; Van Arem et al., 1997). These models are typically “discrete” in that the travel time data are aggregated over pre-defined intervals (e.g. 5 min) and the travel time in any interval is modeled as a random variable with a mean and variance. The variance or uncertainty of the link travel time forecast for future time periods is the forecasting error associated with “mean” link travel time rather than the variance of the “individual” drivers’ link travel times. In this sense, the overall uncertainty or variance of the individual driver’s link travel time forecasts consists of two components: (i) the mean travel time forecasting error and (ii) the individual travel time differences among vehicles (referred to as individual variance in this paper).

All of the existing DSSPP algorithms and stochastic shortest path problem (SSPP) algorithms in the transportation engineering and operations research literature assume that mean and variance of link travel time (or other specific random variable such as cost) are available (Hall, 1986; Loui, 1983; Miller-Hooks and Mahmassani, 1998, Miller-Hooks and Mahmassani, 2000; Mirchandani, 1976; Murthy and Sarkar, 1996). When these models are applied to the previous time period (i.e. observed aggregated travel time), this assumption may be true because the travel time variance corresponds to the travel time differences among vehicles within an aggregation interval. However, when they are applied using forecast travel times and the travel time uncertainty includes the randomness or possible error of the individual travel time forecast, this is not the case. The assumption is wrong because the mean link travel times for the “discrete future time periods” are forecast, which implies that only the mean travel time forecasting error will be available. In other words, the variance associated with the individual drivers is ignored.

Fu and Rilett (1998) proposed approximation models which estimates route travel time mean and variance using the mean and variance of link travel time as a function of time of day. The route travel time variance is defined with respect to individual drivers and therefore, in practice, it is appropriate for estimating individual travel time only for “previous” time periods. Strictly speaking it is not applicable for a forecasting application unless the travel time uncertainty of the forecasting model explicitly considers the mean link travel time forecasting error and individual variance.

In this sense, there is a need to reformulate the DSSPP so that it can consider both individual travel time variance and the mean travel time forecasting error. The objectives of this paper are to examine the characteristics of two components of travel time uncertainty and to develop the mathematical models for determining the mean and variance of the forecast individual travel time in future time period in the context of ATIS.

This paper first introduces the notation and subsequently discusses the assumptions and general practices in travel time estimation for previous time periods in ATIS environment. Then, the characteristics of the two components of uncertainty of the individual travel time forecasts for future time periods are illustrated and their implications from a traffic flow perspective are discussed in Section 4. Following the reformulation of the DSSPP in Section 5, mathematical models for estimating mean and variance of individual route travel time forecast for future time period are proposed in Section 6. The proposed models are then implemented and the results are evaluated using the travel time data from Houston, Texas that had been collected as part of the AVI in Section 7. Finally a concluding discussion follows in Section 8 and includes a summary of the findings and recommendations for future extensions.