Event coupling and performance sensitivity analysis of generalized semi-Markov processes

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.