Dual form of Markov renewal equations and an application to asymptotic analysis of a single-server queue

We propose dual form of Markov renewal equations. While the dual form is theoretically equivalent to the classical standard form of Markov renewal equation, it is shown to be a useful tool for analysis of recent stochastic models. To demonstrate the power of the dual form of Markov renewal equations, we apply one to tail asymptotic analysis of the stationary workload distribution for a single-server queue, where its arrival process is governed by a countable-state Markov chain. This application extends the existing result for the case with a finite-state Markov chain. We find that our approach with the dual form of Markov renewal equation gives a more straightforward proof than those in the previous works and makes the extension simple.