By G. George Yin, Qing Zhang

ISBN-10: 1461443458

ISBN-13: 9781461443452

ISBN-10: 1461443466

ISBN-13: 9781461443469

Prologue and Preliminaries: advent and review- Mathematical preliminaries.- Markovian models.- Two-Time-Scale Markov Chains: Asymptotic Expansions of ideas for ahead Equations.- profession Measures: Asymptotic houses and Ramification.- Asymptotic Expansions of strategies for Backward Equations.- Applications:MDPs, Near-optimal Controls, Numerical equipment, and LQG with Switching: Markov selection Problems.- Stochastic regulate of Dynamical Systems.- Numerical tools for keep watch over and Optimization.- Hybrid LQG Problems.- References.- Index

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Prologue and Preliminaries: advent and evaluation- Mathematical preliminaries. - Markovian versions. - Two-Time-Scale Markov Chains: Asymptotic Expansions of strategies for ahead Equations. - profession Measures: Asymptotic houses and Ramification. - Asymptotic Expansions of options for Backward Equations.

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**Example text**

It begins with the birth and death processes. 3, we treat a number of Markov chains with ﬁnitestate spaces, including queueing systems, seasonal variation, and system reliability. 4 gives examples arising from the context of stochastic optimization involving Markovian structures. 5 deals with jump linear systems, in particular, the optimal control formulation of linear quadratic control problems with Markovian jumps and large-scale systems via aggregation and decomposition. One of the main ideas that underlies the basis of the asymptotic results throughout the book is the time-scale separation.

2) If pε (t) converges, how can one determine the limit? (3) What is the convergence rate? (4) Suppose pε (t) → ν(t) = (ν1 (t), . . , νm (t)), a probability distribution as ε → 0. Deﬁne χε (t) = (I{αε (t)=1} , . . , I{αε (t)=m} ). Consider the centered and scaled occupation measure 1 nε (t) = √ ε t 0 (χε (s) − ν(s))ds. As ε → 0, what is the limit distribution of the random process nε (·)? (5) Will the results carry over to singularly perturbed Markov chains with weak and strong interactions (when the states of the Markov chain belong to multiple irreducible classes)?

Suppose that a manufacturing system consists of two components, one on-line and one backup. As discussed in Taylor and Karlin [204], the assumption of exponentially distributed operating time does not reﬂect reality well due to the burn-in phenomenon. , the Markov chain is stationary. Introduce the hazard rate function r(t) = f (t)/(1 − F (t)), where f (t) is the probability density function of the failure time, and F (t) is the corresponding distribution function. In [204] the following mixed exponential distribution is introduced: f (t) = pαe−αt + qβe−βt , and p > 0, q > 0 with p + q = 1.

### Continuous-time Markov chains and applications : a two-time-scale approach by G. George Yin, Qing Zhang

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