## Stochastic User Equilibrium and Link Capacity Constraints: Formulation and Theoretical Evidence

### Authors

BIFULCO G N and CRISALLI U, Universita degli Studi di Roma "Tor Vergata", Italy

### Description

This paper presents a model for a stochastic assignment model with fixed demand. The presented assume that the system can be considered in steady state (equilibrium assignment) and that link capacity constraints are explicitly taken into account. This las

### Abstract

This paper presents a model for a stochastic assignment model with fixed demand. The presented assume that the system can be considered in steady state (equilibrium assignment) and that link capacity constraints are explicitly taken into account. This last assumption plays is relevant when demand flows are large, so that capacity constraints could bind. In general, the exiting flow from any link can not exceed the link capacity, thus downstream flows can be effected by upstream bottlenecks. Flow that exceeds capacity gives rise to link queue, and flow conservation should be accordingly reformulated. If the assignment model fails in considering queue formation, the resulting flow pattern could be quite unrealistic and, typically overestimated. Such an effect is more evident in city centres, where surrounding bottlenecks could "protect" the centre from excessive traffic flows. It could be easily shown that the most proper way to deal with link capacity constraints is to use within-day-dynamic assignment models (Cascetta and Cantarella, 1991). However, such dynam!c models are quite complex and data needing, moreover their use is olden not justified for strategic planning.

Some major attention has been done in the last years to "capacitated" steady-state assignment models (Larsson & Patriksson, 1995; Bell, 1995; Ferrari, 1997; Kheifits & Yehuda, 1997). Most contributions (except Bell) assume deterministic route choice behaviour, based on quite unrealistic hypotheses, formulated by the well known Wardrop's first principle (Wardrop, 1952). The general approach (except by Kheifits & Yehuda) is to reformulate the variational inequality (Smith, 1979) or the linearly constrained convex mathematical programming problem (Beckmann et al., 1956) that mathematically represent the equilibrium. In any case all the approaches "reduce" the equilibrium problem to an equivalent convex programming problem with linear constraints. In the case of the Bell's stochastic approach the Wardrop's principle, as well as its formulation as convex programming problem, is extended in order to take into account a probabilistic logit-type route choice model. Linear constraints represent conditions such as non-negativity of link flows, demand conservation and, in the case of capacitated assignments, capacity constraints. Lagrange multipliers are introduced for demand feasibility and capacity constrains and Khun-Tucker optimality conditions are applied to the resulting problem. The interpretation of the optimal multipliers is different from Bell and Patriksson on one side and Ferrari on the other side. The first two authors mainly suggest to interpreter optimal multipliers as the queuing delays due to binding capacity constraints, while Ferrari suggests to interpreter to use them as pricing tolls to be applied to oversatured links in order to ensure the network equilibrium that, otherwise, does not exist. In the case of the Bell and Patriksson's interpretation, one could observe that the obtained queuing delays are not necessarily consistent with the values that could be observed (and/or modelled) for oversatured links. On the other hand, Ferrari's interpretation seems to implicitly suggest that the only way a capacitated assignment model could be used is to allow (commit) the decision-maker to impose pricing measures on some links.

The approach used by Kheifits and Yehuda is quite different. They propose to account for queue formation. Therefore, for this aim, they propose a "queued" flow propagation model, framed within a deterministic assignment approach. The authors mainly investigate algorithmic issues, unfortunately the underlying model is not formalised and the related theoretical properties are not investigated. In this article a steady-state Stochastic User Equilibrium assignment model is presented, which deals with capacity constraints by explicitly accounting for queue formation. The model, however, is not able, in the presented form, to deal with spill-back phenomena, which occur when the queue lengths (if any) exceed link lengths. With respect to the previously described approaches the presented model shows on one hand a rigorous and consistent mathematical formulation and on the other hand is the only model based to a fixed-point approach. The route choice model is a general (not only logit) probabilistic model. In section 2 the assignment problem will be formalised, mainly investigating the effect of a capacity-dependent flow propagation model on the whole assignment model. Theoretical properties of existence and uniqueness will be investigated, as well as some resolving approaches will be briefly introduced. In section 3 an application to a real size network is briefly presented in order to show the suitability of the model on real-size systems. A comparison with the results of a classic Stochastic User Equilibrium assignment model is also given. Section 4 lists some further research directions.

#### Publisher

Association for European Transport