Perturbed Semi-Markov Processes and Reduction of Phase Space

Abstract

In this chapter, we present models of regularly and singularly perturbed semi-Markov processes and formulate basic perturbation conditions and asymptotic comparability conditions based on the notion of a complete family of asymptotically comparable functions. We also describe asymptotic procedures of removing virtual transitions and one-state reducing the phase space, and an asymptotic recurrent phase space reduction algorithm for perturbed semi-Markov processes. These procedures and algorithm are provided by explicit formulas for recalculating normalisation functions, Laplace transforms for limit distributions, and limit expectations appearing in perturbation conditions for semi-Markov processes with reduced phase space. This chapter includes four sections.