Estimating Markov Chain Mixing Times: Convergence Rate Towards Equilibrium of a Stochastic Process Traffic Assignment Model.

The article focuses on developing and analyzing methodologies to estimate the Markov chain mixing time (MCMT), which measures the convergence time of transport systems modeled by Markov chains to their stationary distributions. It addresses limitations of traditional network equilibrium models, particularly the absence or multiplicity of stable equilibria and uncertainty about convergence speed after system shocks. The study introduces aggregation techniques, notably level 1 and level 2 aggregations, to reduce computational complexity in large-scale transport systems and employs coupling methods to estimate upper bounds of MCMTs efficiently. Through analytical and numerical analyses of various transport scenarios—including constant travel costs, congested systems, positive user interactions, and multidimensional departure time choices—the research finds that MCMTs are generally inversely proportional to the probability of users revising their choices and often exhibit a log-linear or faster-than-logarithmic relationship with the number of users, depending on system complexity. The findings suggest that despite complex user interactions, transport systems tend to reach steady states within reasonable time frames, and the proposed methods enable practical assessment of convergence times in large-scale, possibly unstable, transport models.