Discrete-event systems driven by Poisson processes

In a generalized semi-Markov process (GSMP) model of a discrete-event systems, “clock samples” of events drive the system (they are the timing inputs). Glasserman and Yao have identified a condition, condition M, that implies event epochs are monotone functions of clock samples. When clock samples are exponentially distributed, an alternative construction exists where system inputs are Poisson processes. The author shows that there exists a natural partial order on the latter input space and that condition M ensures monotonicity of event epochs with respect to this partial order as well. Moreover, the author shows that the probability measure defined on this input space is an associated measure. This implies that running two systems with common Poisson streams leads to guaranteed variance reduction-compared to using independent inputs-for some performance measures