By Xianping Guo

Continuous-time Markov determination tactics (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), desktop technological know-how, communications engineering, keep watch over of populations (such as fisheries and epidemics), and administration technology, between many different fields. This quantity presents a unified, systematic, self-contained presentation of modern advancements at the thought and functions of continuous-time MDPs. The MDPs during this quantity comprise lots of the situations that come up in purposes, simply because they permit unbounded transition and reward/cost charges. a lot of the cloth seems for the 1st time in ebook form.

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4 Characterization of n-bias Policies 29 Hfn r(f ) for all 2 ≤ n ≤ k, and so the desired conclusion is proved. 6) we have Q(h) − Q(f ) g−1 (f ) = 0 and Q(h) − Q(f ) gn (f ) = 0 ∀n ≥ |S|. 6(c) give that, for each n ≥ |S|, gn (h) − gn (f ) ∞ = P (t, h) Q(h) − Q(f ) gn (f ) dt − P ∗ (h) Q(h) − Q(f ) gn+1 (f ) 0 = 0. 7. 7 shows that to obtain a policy that is n-bias optimal for all n ≥ −1, it suffices to find an |S|-bias optimal policy. 7 we only need to focus on the existence and calculation of an |S|-bias optimal policy.

Now we give some results on the difference of the n-biases of two policies. 6 Suppose that f and h are both in F . Then (a) g−1 (h) − g−1 (f ) = P ∗ (h)[r(h) + Q(h)g0 (f ) − g−1 (f )] + [P ∗ (h) − I ]g−1 (f ). (b) If g−1 (h) = g−1 (f ), then ∞ P (t, h) r(h) + Q(h)g0 (f ) − g−1 (f ) dt g0 (h) − g0 (f ) = 0 + P ∗ (h) Q(h) − Q(f ) g1 (f ) ∞ = P (t, h) r(h) + Q(h)g0 (f ) − g−1 (f ) dt 0 + P ∗ (h) g0 (h) − g0 (f ) . 26 3 Average Optimality for Finite Models (c) For some n ≥ 0, if gn (h) = gn (f ), then ∞ gn+1 (h) − gn+1 (f ) = P (t, h) Q(h) − Q(f ) gn+1 (f ) dt 0 + P ∗ (h) Q(h) − Q(f ) gn+2 (f ).

4. 17 Let F−1 ∗ and f ∈ F , if the following two conditions hold (a) For all f ∗ ∈ F−1 (i) Q(f )g−1 (f ∗ ) = 0 (ii) r(f ) + Q(f )g0 (f ∗ ) ≥ g−1 (f ∗ ) then g−1 (f ) = g−1 (f ∗ ). (b) Under the two conditions in (a), if, in addition, Q(f )g1 (f ∗ )(i) ≥ g0 (f ∗ )(i) for all states i such that [r(f ) + Q(f )g0 (f ∗ )](i) = g−1 (f ∗ )(i), then g−1 (f ) = g−1 f ∗ and g0 (f ) ≥ g0 f ∗ . 10)), and v := r(f ) + Q(f )g0 (f ∗ ) − g−1 (f ∗ ) ≥ 0 (by condition (ii)). 1) and a straightforward calculation that [P ∗ (f ) − I ]g−1 (f ∗ ) = 0.