The analysis of cyclic stochastic fluid flows with time-varying transition rates

We consider a stochastic fluid model (SFM) {(Xˆ(t), J(t)), t ≥ 0} driven by a continuous-time Markov chain {J(t), t ≥ 0} with a time-varying generator T(t) and cycle of length 1 such that T(t) = T(t + 1) for all t ≥ 0. We derive theoretical expressions for the key periodic measures for the analysis of the model, and develop efficient methods for their numerical computation. We illustrate the theory with numerical examples. This work is an extension of the results in Bean et al. (Stoch. Models 21(1):149–184, 2005) for a standard SFM with time-homogeneous generator, and suggests a possible alternative approach to that developed by Yunan and Whitt (Queueing Syst. 71(4):405–444, 2012).