Uniform acceleration expansions for Markov chains with time-varying rates

Abstract

We study uniform acceleration (UA) expansions of finite-state continuous-time Markov chains with time-varying transition rates. The UA expansions can be used to justify, evaluate and refine the pointwise stationary approximation, which is the steady-state distribution associated with the time-dependent generator at the time of interest. We obtain UA approximations from these UA asymptotic expansions. We derive a time-varying analog to the uniformization representation of transition probabilities for chains with constant transition rates, and apply it to establish asymptotic results related to the UA asymptotic expansion. These asymptotic results can serve as appropriate time-varying analogs to the notions of stationary distributions and limiting distributions. We illustrate the UA approximations by doing a numerical example for the time-varying Erlang loss model.