By G. George Yin, Qing Zhang

This e-book specializes in two-time-scale Markov chains in discrete time. Our motivation stems from current and rising functions in optimization and keep watch over of complicated platforms in production, instant verbal exchange, and ?nancial engineering. a lot of our e?ort during this publication is dedicated to designing approach types coming up from numerous functions, studying them through analytic and probabilistic thoughts, and constructing possible compu- tionalschemes. Ourmainconcernistoreducetheinherentsystemcompl- ity. even though all the functions has its personal certain features, them all are heavily comparable in the course of the modeling of uncertainty as a result of leap or switching random procedures. Oneofthesalientfeaturesofthisbookistheuseofmulti-timescalesin Markovprocessesandtheirapplications. Intuitively,notallpartsorcom- nents of a large-scale procedure evolve on the similar cost. a few of them swap quickly and others fluctuate slowly. The di?erent charges of adaptations let us lessen complexity through decomposition and aggregation. it'd be perfect if shall we divide a wide process into its smallest irreducible subsystems thoroughly separable from each other and deal with each one subsystem indep- dently. despite the fact that, this is infeasible actually because of a number of actual constraints and different concerns. hence, we need to take care of occasions within which the platforms are just approximately decomposable within the feel that there are susceptible hyperlinks one of the irreducible subsystems, which dictate the oc- sional regime adjustments of the method. An e?ective method to deal with such close to decomposability is time-scale separation. that's, we manage the structures as though there have been time scales, quickly vs. gradual. xii Preface Followingthetime-scaleseparation,weusesingularperturbationmeth- ology to regard the underlying structures.

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5 Gaussian, Diﬀusion, and Switching Diﬀusion Processes A Gaussian random vector x = (x1 , x2 , . . , xr ) is one whose characteristic function has the form φ(y) = exp i y, µ − 1 Σy, y 2 , where µ ∈ Rr is a constant vector, y, µ is the usual inner product, i denotes the pure imaginary number satisfying i2 = −1, and Σ is a symmetric nonnegative deﬁnite r × r matrix. In the above, µ and Σ are the mean vector and covariance matrix of x, respectively. Let x(t), t ≥ 0, be a stochastic process. It is a Gaussian process if for any 0 ≤ t1 < t2 < · · · < tk and k = 1, 2, .

4 Discrete-Time vs. Continuous-Time Models 17 important classes of ﬁnite-capacity queues; see for example Sharma [137] and the references therein. The two-time-scale formulation can be naturally imbedded in these queueing applications. In addition, for computational purpose, one often has to use ﬁnite-capacity queues to approximate queues with inﬁnitely many waiting rooms. Thus, the problem reduces to that of a ﬁnite-state Markov chain with a large state space. 4 Discrete-Time vs. Continuous-Time Models There is a close connection between continuous-time, singularly perturbed Markov chains and their discrete-time counterparts.

Criteria on recurrence can be found in most standard textbooks of stochastic processes or Markov chains. Note that (see Karlin and Taylor [79, p. 4]) if P is a transition matrix for a ﬁnite-state Markov chain, the multiplicity of the eigenvalue 1 is equal to the number of recurrent classes associated with P . A row vector π = (π 1 , . . , π m0 ) with each π i ≥ 0 is called a stationary distribution of αk if it is the unique solution to the system of equations πP = π, π i = 1. i As demonstrated in [79, p.