Stochastic Models of Systems

In this monograph stochastic models of systems analysis are discussed. It covers many aspects and different stages from the construction of mathematical models of real systems, through mathematical analysis of models based on simplification methods, to the interpretation of real stochastic systems.
The stochastic models described here share the property that their evolutionary aspects develop under the influence of random factors. It has been assumed that the evolution takes place in a random medium, i.e. unilateral interaction between the system and the medium. As only Markovian models of random medium are considered in this book, the stochastic models described here are determined by two processes, a switching process describing the evolution of the systems and a switching process describing the changes of the random medium.
Audience: This book will be of interest to postgraduate students and researchers whose work involves probability theory, stochastic processes, mathematical systems theory, ordinary differential equations, operator theory, or mathematical modelling and industrial mathematics.

Inhalt
1 Introduction.- 1.1 Classification and properties of stochastic systems.- 1.2 Renewal processes.- 2 Markov renewal processes.- 2.1 Definition of Markov renewal process.- 2.2 Semi-Markov processes.- 2.3 Ergodicity and stationary distribution.- 3 Phase merging algorithms.- 3.1 Reducible-invertible operators.- 3.2 Perturbation of reducible-invertible operators.- 3.3 Martingale characterization of Markov processes.- 3.4 Pattern limit theorem.- 3.5 Ergodic phase merging.- 3.6 Splitting phase merging.- 3.7 Heuristic phase merging principles.- 4 Evolutional stochastic system in a random medium.- 4.1 Stochastic additive functionals.- 4.2 Storage Processes.- 4.3 Random evolution.- 4.4 Ergodic average and diffusion approximation of random evolutions.- 4.5 Splitting average and diffusion approximation of random evolution.- 4.6 Application of average and diffusion approximation algorithms.- 4.7 Counting processes.- 4.8 Proofs of limit theorems.- 5 Diffusion approximation of Markov queueing systems and networks.- 5.1 Algorithms of diffusion approximation.- 5.2 Markov queueing processes.- 5.3 Average and diffusion approximation.- 5.4 Stationary distribution.- 5.5 Markovian queueing systems.- 5.6 Markovian queueing networks.- References.