## SUE and Junction Modelling - the Extension of Stochastic User Equilibrium Assignment to Model Realistic Junction Delays

### Authors

HUGHES P C and MAILER M, Napier University, UK

### Description

A practicable and reliable method for Stochastic User Equilibrium (SUE) assignment is an important goal for research in traffic assignment (Van Vuren (1994)). If both congestion and differences in driver perception are modelled, the assignment should be a

### Abstract

A practicable and reliable method for Stochastic User Equilibrium (SUE) assignment is an important goal for research in traffic assignment (Van Vuren (1994)). If both congestion and differences in driver perception are modelled, the assignment should be able to give realistic routeing patterns in most networks. However, particularly in urban networks, the delay at a junction is often at least as important as the delay incurred in simply travelling along the approaching link. The complex interactions at priority and signallsed junctions mean that these delays need to be modelled properly, rather than with a simple cost-flow function.

SUE methods can be classified based on the underlying stochastic method. Two main approaches to stochastic assignment have been adopted, leading to logit models (Dial(1971)), which use a logistic splitting function, and probit models, which assume Normal link cost distributions. Probit models can either be numerical (as in the SAM model, described in Maher & Hughes (1995, 1996, 1997), or based on random sampling (as in Burrell (1968)). In the paper, both a logit and a probit (SAM) based method will be extended to model junction delays, and test results from several networks shown.

In the cases where junction delay is to be modelled properly, the delay formulae will in general depend on all the flows entering the junction (with the exception of fixed-time signals, as will be explained below). Also, given that a steady state situation is being modelled, the capacity of each junction needs to be respected; the flow into a junction may be over-capacity for short periods during the peak times of day, but that situation cannot be sustained indefinitely.

In the case of signalised junctions, the Webster delay formula (1961) gives a formula for delay at a particular approach, which, for fixed signal settings, depends only on the flow on that approach. It also forces the model to respect the strict (fixed) capacity of that approach. As the ratio of flow to capacity (the degree of saturation) approaches unity, the delay approaches infinity. This is realistic for modelling a steady state, which is the aim of this paper; a flow over capacity is not sustainable indefinitely.

For priority junctions, on the other hand, the delay formula will necessarily depend on the major road flow. The capacity of the minor road is determined by a formula which gives a value, which is decreased as the major road flow increases. Given this capacity, the delay is then calculated by a simple queuing formula which, as in the case of signals, approaches infinity as the flow approaches capacity.

Previous work has been done on solving for User Equilibrium (LIE) and System Optimum (SO) with junction delay formulae. The conditions for the minimisation program for LIE (due to Beckmarm et aL (1956)), and the corresponding one for SO, are in general violated when link interactions are modelled. In the case where the Jacobian of the travel time functions ta is symmetric and positive definite with respect to the link flows xa, the result still holds (see for example Sheffi (1985)). The intuitive meaning of this condition is that for two links a and b, t~ depends as much on xb as ta does on xb; also, the cost on any link depends "mostly" on the flow on that link. Except in cases where two-way streets (rather than priority junctions) are being modelled, this eondition is unlikely to be satisfied. However, a "diagonalisation" method has been used by Maber et al (1993), using different formulae for the junction modelling, but reaching similar findings, and although the method has no theoretical guarantees, it is found generally to work in practice. Koutsopoulos and Habbal (1994) test the accuracy of four levels of junction delay modelling. They found that the modelling of all turning movements was not in itself as important as the inclusion of link interactions, which gave a large benefit in terms of accuracy.

An efficient SUE algorithm has been developed by the authors, and described in previous papers; it uses a minimisation program given by Sbeffi and Powell (1982). This program, as in the LIE case, is rendered strictly invalid by the introduction of junction delay formulae; in a similar way, as this paper will show, a diagonalisation method can be used with good results. The next section will give a brief outline of the basic SUE algorithm, after which section 3 will discuss the main modifications which are necessary when junction delay formulae are introduced. Sections 4 and 5 give the results of some tests, after which there is a brief conclusion.

#### Publisher

Association for European Transport