By Xianping Guo

ISBN-10: 3642025463

ISBN-13: 9783642025464

Continuous-time Markov selection approaches (MDPs), often referred to as managed Markov chains, are used for modeling decision-making difficulties that come up in operations study (for example, stock, production, and queueing systems), machine technology, communications engineering, keep watch over of populations (such as fisheries and epidemics), and administration technology, between many different fields. This quantity offers a unified, systematic, self-contained presentation of modern advancements at the conception and functions of continuous-time MDPs. The MDPs during this quantity comprise many of the instances that come up in purposes, simply because they permit unbounded transition and reward/cost charges. a lot of the fabric appears to be like for the 1st time in publication form.

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**Extra resources for Continuous-Time Markov Decision Processes: Theory and Applications**

**Example text**

3) and r(f ) ≥ 1), there exists an integer 2 ≤ k ≤ |S| such that Hfk+1 r(f ) is a linear combination of g−1 (f ) and Hfn r(f ) for all 2 ≤ n ≤ k. We now show by induction that, for each m ≥ 2, Hfm r(f ) is a linear combination of g−1 (f ) and Hfn r(f ) for all 2 ≤ n ≤ k. To see this, suppose that this conclusion holds for some m (≥ k + 1). 4) and Hf g−1 (f ) = 0, gives k−1 Hfm+1 r(f ) = λl Hfl+1 r(f ) + λk Hfk+1 r(f ) . 4 Characterization of n-bias Policies 29 Hfn r(f ) for all 2 ≤ n ≤ k, and so the desired conclusion is proved.

2. 3. 4. ∗ . 5) gl (fk ) for all 0 ≤ l ≤ n + 1. 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.

23) hold for all 0 ≤ k ≤ n when f ∗ there is replaced with f above. 11(a), f is n-bias optimal. 45) respectively. Furthermore, for all i ∈ S and n ≥ 0, let ⎧ ⎫ ⎪ q(j |i, a)gn+1 (f )(j ) > gn (f )(i), or ⎪ j ∈S ⎪ ⎪ ⎨ ⎬ f f a ∈ A q(j |i, a)g (f )(j ) > g (f )(i) (i) : . 52) We then define an improvement policy h ∈ F (depending on f ) as follows: f h(i) ∈ Bn+2 (i) f if Bn+2 (i) = ∅, and h(i) := f (i) f if Bn+2 (i) = ∅. 53) that f (i) is not in Bn+2 (i) for any i. 23) for some n ≥ 0. 53). Then, (a) gk (h) = gk (f ) for all −1 ≤ k ≤ n, and gn+1 (h) ≥ gn+1 (f ).

### Continuous-Time Markov Decision Processes: Theory and Applications by Xianping Guo

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