The stability of stochastic equilibrium in a two-link transportation network

Most research and applications of network equilibrium models are based on the assumption that traffic volumes on roadways are virtually certain to be at or near their equilibrium values if the equilibrium volumes exist and are unique. However, it has long been known that this assumption can be violated in deterministic models. This paper presents an investigation of the stability of stochastic equilibrium in a two-link network. The stability of deterministic equilibrium also is discussed briefly. Equilibrium is defined to be stable if it is unique and the link volumes converge over time to their equilibrium values regardless of the initial conditions. Three models of route choice decision-making over time are formulated, and the stability of equilibrium is investigated for each. It is shown that even when equilibrium is unique, link volumes may converge to their equilibrium values, oscillate about equilibrium perpetually, or converge to values that may be considerably different from the equilibrium ones, depending on the details of the route choice decision-making process. Moreover, even when convergence of link volumes to equilibrium is assured, the convergence may be too slow to justify the standard assumption that these volumes are usually at or near their equilibrium values. When link volumes converge to non-equilibrium values, the levels at which the volumes stabilize typically depend on the initial link volumes or perceptions of travel costs. Conditions sufficient to assure convergence to equilibrium in two of the three models of route choice decision-making are presented, and these conditions are interpreted in terms of the route choice decision-making process.