Monotonicity in Generalized Semi-Markov Processes

Abstract

We establish stochastic monotonicity of the event epoch sequences of generalized semi-Markov processes through the structure of the generalized semi-Markov schemes on which they are based. Our main condition states, roughly, that the occurrence of more events in the short run never leads to the activation of less events in the long run. We consider monotonicity with respect to two types of inputs: clock times (which translate to, e.g., service and interarrival times in queueing systems), and structural parameters (which translate to, e.g., buffer size, number of servers, and job population). For the second type of comparison, we replace a structural change with an equivalent change in clock times to reduce the comparison to one of the first type. When applied to queueing systems, our results yield new comparisons and also unify several existing results previously established using special properties of individual systems.