2 Nonlinearly Perturbed Markov Chains and Information Networks

This chapter is devoted to studies of perturbed Markov chains, commonly used for the description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularized by adding a special damping matrix, multiplied by a small damping (perturbation) parameter ε. In this chapter, we present the results of detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.

2.1. Introduction

Perturbed Markov chains is a popular and important aspect in the study of the theory of Markov processes and their applications to stochastic networks, queuing and reliability models, bio-stochastic systems and many other stochastic models.

We refer here to some recent books and papers devoted to perturbation problems for Markov-type processes: Stewart (1994, 1998, 2001), Hartfiel and Meyer (1998), Korolyuk and Korolyuk (1999), Englund (2001); Konstantinov et al. (2003), Korolyuk and Limnios (2005), Mitrophanov (2005), Bini et al. (2005), Yin and Zhang (2005), Gambini et al. (2008), Gyllenberg and Silvestrov (2008), Ni et al. (2008), Ni (2011), Avrachenkov et al. (2013, 2018), Silvestrov and Petersson (2014), Petersson (2016), Silvestrov and Silvestrov (2016, 2017a, b, c, d), Silvestrov et al. (2018) and Yin and Zhang (2013). In particular, we would like to mention ...

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