Preface

The book presents new methods of asymptotic analysis for nonlinearly perturbed semi-Markov processes with finite phase spaces. These methods are based on special time-space screening procedures for sequential reduction of phase spaces for semi-Markov processes combined with the systematical use of operational calculus for Laurent asymptotic expansions. We compose effective recurrent algorithms for the construction of Laurent asymptotic expansions, without and with explicit upper bounds for remainders, for power moments of hitting times for nonlinearly perturbed semi-Markov processes. We also illustrate the above results by getting asymptotic expansions for stationary and conditional quasi-stationary distributions of nonlinearly perturbed semi-Markov processes, in particular for birth-death-type semi-Markov processes, which play an important role in various applications. It is worth noting that asymptotic expansions are a very effective instrument for studies of perturbed stochastic processes. The corresponding first terms in expansions give limiting values for properly normalized functionals of interest. The second terms let one estimate the sensitivity of models to small parameter perturbations. The subsequent terms in the corresponding expansions are usually neglected in standard linearization procedures used in studies of perturbed models. This, however, cannot be acceptable in the cases where values of perturbation parameter are not small enough. Asymptotic expansions let one take into account high-order terms in expansions, and in this way, to improve accuracy of the corresponding numerical procedures. Semi-Markov processes are a natural generalization of discrete and continuous time Markov chains. These jump processes possess Markov property at moments of jumps and can have arbitrary distributions concentrated on a positive half-line for inter-jump times. In fact, this combination of basic properties makes semi-Markov processes a very flexible and effective tool for the description of queuing, reliability, and some biological systems, financial and insurance processes, and many other stochastic models.