Convergence rates in matrix analytic models

The authors consider a class of Markov chains , which includes Markov chains of the type and the type. Here is the level process and the modulating process. This paper gives conditions under which is an upper bound for the convergence rate to stationarity for suitable functions . Conditions for the convergence rate of to be for some are also given. Similar results are obtained for the workload in the queue. In addition, the paper extends various standard results on stopped sums from the i.i.d. to a Markov-modulated setting.