By Xianping Guo

ISBN-10: 3642025463

ISBN-13: 9783642025464

Continuous-time Markov determination techniques (MDPs), often referred to as managed Markov chains, are used for modeling decision-making difficulties that come up in operations examine (for example, stock, production, and queueing systems), computing device technology, communications engineering, keep watch over of populations (such as fisheries and epidemics), and administration technological know-how, between many different fields. This quantity presents a unified, systematic, self-contained presentation of contemporary advancements at the thought and functions of continuous-time MDPs. The MDPs during this quantity comprise many of the instances that come up in functions, simply because they enable unbounded transition and reward/cost premiums. a lot of the cloth looks for the 1st time in booklet form.

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9(b) we have g−1 (h) − g−1 (f ) = P ∗ (h)vfh + P ∗ (h) − I g−1 (f ) ≥ 0, and so the first part of (a) follows. (b) From the proof of (a) we have (P ∗ (h) − I )g−1 (f ) ≥ 0 and vfh (i) ≥ 0 for all the recurrent states i under Q(h). On the other hand, by (a) and the condition in (b) we have P ∗ (h)vfh 0, and so g−1 (h) − g−1 (f ) = P ∗ (h)vfh + P ∗ (h) − I g−1 (f ) This inequality gives (b). 0. 38 3 Average Optimality for Finite Models (c) By (a), it suffices to prove that g−1 (h) = g−1 (f ). Suppose that g−1 (h) = g−1 (f ).

2 n-bias Optimality Criteria In this section we introduce the n-bias optimality criteria. In what follows, we suppose that all operations on matrices or vectors, such as limits of sequences, are component-wise. Without risk of confusion, we denote by “0” both the matrix and the vector with all zero components, by e the vector with all components 1, and by I the identity matrix. Also, for each vector u, we denote by u(i) the ith component of u. 5 implies that the corresponding transition function pf (s, i, t, j ) is stationary, that is, pf (s, i, t, j ) = pf (0, i, t − s, j ) for all i, j ∈ S and t ≥ s ≥ 0.

53). 21 below). Otherwise, increment k by 1 and return to Step 2. 19, we now obtain the following. 21 Fix n ≥ 1. 23). 23). 20 we see that gn (fk ) either increases or remains the same. 20(b), when gn (fk ) remains the same, gn+1 (fk ) increases in k. Thus, any two policies in the sequence {fk } either have different n-biases or have different (n + 1)-biases. Thus, every policy in the iteration sequence is dif∗ ferent. Since the number of policies in Fn−1 is finite, the iteration must stop after a finite number of iterations; otherwise, we can find the next improved policy in the policy iteration.

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