By Xianping Guo

Continuous-time Markov determination strategies (MDPs), often referred to as managed Markov chains, are used for modeling decision-making difficulties that come up in operations learn (for example, stock, production, and queueing systems), laptop technology, communications engineering, regulate 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 situations that come up in purposes, simply because they enable unbounded transition and reward/cost charges. a lot of the cloth appears to be like for the 1st time in publication form.

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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 ). 39) which contradicts the condition in (c). (d) We first prove that g0 (h) ≥ g0 (f ). To do so, let g := g0 (h) − g0 (f ). 39), we can see that Q(h)g−1 (f ) = 0. 38) we have vfh ≥ 0, and, moreover, vfh (i) = f 0 for all recurrent states i under Q(h), and so h(i) is not in B0 (i) for any recurrent i under Q(h). 36) we have that I − P ∗ (h) ∞ g= 0 P (t, h)vfh dt ≥ 0.

This means that (a) holds for n = 0. We now consider the case n ≥ 1. By induction, let us suppose that f ∗ is in Fm∗ for ∗ , we need some 0 ≤ m ≤ n − 1. 4, to show that f ∗ is in Fm+1 ∗ ∗ to prove that gm+1 (f ) ≥ gm+1 (f ) for all f ∈ Fm . To this end, first note that, for each f ∈ Fm∗ , the definition of Fm∗ and f ∗ ∈ Fm∗ (the induction hypothesis) give that gl (f ∗ ) = gl (f ) for all −1 ≤ l ≤ m. 5(b), P ∗ (f ) Q(f )gm+1 f ∗ − gm f ∗ = 0. 10) it follows that Q(f )gm+1 f ∗ ≤ gm f ∗ = Q f ∗ gm+1 f ∗ .

20), gives Al (i) = Al 0 (i) for all 0 ≤ l ≤ n + 1 and i ∈ S. 23) because f0 does. 23). 11, f ∗ is n-bias optimal. 21 we conclude that we can use policy iteration algorithms to obtain a policy that is n-bias optimal. In particular, in a finite number of iterations, we can obtain a policy that is n-bias optimal for all n ≥ −1 by using the |S|-bias policy iteration algorithm. 6 The Linear Programming Approach We cannot close this chapter on finite MDPs without mentioning the linear programming formulation, which was one of the earliest solution approaches.