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In this chapter, we apply the RG-factorizations to provide a unified algorithmic framework for dealing with transient solution in stochastic models. The transient solution includes the transient probability, the first passage time, the sojourn time and time-inhomogeneous Markov chains. Based on the first passage time, we extends the PH distribution and the MAP to the GPH distribution and the GMAP...
In this chapter, we study quasi-stationary distributions for blockstructured Markov chains with either finitely-many levels or infinitely-many levels. We derive the RG-factorizations for the β -discounted block-structured transition matrix. We provide conditions for the state α -classification, and derive two sets of expressions for the quasi-stationary distribution. As important examples, we analyze...
In this chapter, we consider reward processes of an irreducible continuous-time block-structured Markov chain. By using the RG-factorizations, we provide a unified algorithmic framework to derive expressions for conditional distributions and conditional moments of the reward processes. As an important example, we study the reward processes for an irreducible continuous-time level-dependent QBD process...
In this chapter, we present sensitivity analysis for performance measures of an irreducible perturbed Markov chain which is either discretetime or continuous-time. By using the UL- and LU-types of RG-factorizations, we can express the nth derivatives of the performance measures, including the stationary, transient and discounted cases. Furthermore, we apply the sensitivity analysis to study symmetric...
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