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 examine (for example, stock, production, and queueing systems), laptop technological know-how, communications engineering, keep watch over of populations (such as fisheries and epidemics), and administration technological know-how, between many different fields. This quantity offers a unified, systematic, self-contained presentation of contemporary advancements at the concept and purposes 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 premiums. a lot of the fabric appears to be like for the 1st time in publication form.

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6(b) we have ∞ g0 (f ) − g0 f ∗ = P (t, f )v dt + P ∗ (f )u ≥ 0. 43) give (b). 23) (with n = 0) is 0-bias optimal. We now wish to present a policy iteration algorithm for computing one such a policy. To state this algorithm we will use the following notation. 44) q(j |i, a)g0 (f )(j ). j ∈S Let f B1 (i) := ⎧ ⎪ ⎨ wf (i, a) > g−1 (f )(i); f a ∈ A0 (i) : ⎪ ⎩ or ⎫ ⎪ ⎬ q(j |i, a)g1 (f )(j ) > g0 (f )(i) . 45) We now define an improvement policy h ∈ F (depending on f ) as follows: f h(i) ∈ B1 (i) f if B1 (i) = ∅ and h(i) := f (i) f if B1 (i) = ∅.

Suppose that the algorithm stops at a policy denoted f ∗ . 60) j ∈S ∀i ∈ S. 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.

Therefore, the vectors g−1 (f ) and Hfn r(f ) (for 2 ≤ n ≤ |S| + 1) belong to the null space of Q(h) − Q(f ) (that is, the space {u : (Q(h) − Q(f ))u = 0}). Since Q(h) − Q(f ) = 0, we see that the rank of Q(h) − Q(f ) is at least 1, and so the dimension of the null space of Q(h) − Q(f ) is at most |S| − 1. Hence, g−1 (f ) and Hfn r(f ) (for 2 ≤ n ≤ |S| + 1) are linearly dependent. 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.