Average optimality for continuous-time Markov decision processes in polish spaces

This paper is devoted to studying the average optimality in continuous-time Markov decision processes with fairly general state and action spaces. The criterion to be maximized is expected average rewards. The transition rates of underlying continuous-time jump Markov processes are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. We first provide two optimality inequalities with opposed directions, and also give suitable conditions under which the existence of solutions to the two optimality inequalities is ensured. Then, from the two optimality inequalities we prove the existence of optimal (deterministic) stationary policies by using the Dynkin formula. Moreover, we present a ``semimartingale characterization'' of an optimal stationary policy. Finally, we use a generalized Potlach process with control to illustrate the difference between our conditions and those in the previous literature, and then further apply our results to average optimal control problems of generalized birth--death systems, upwardly skip-free processes and two queueing systems. The approach developed in this paper is slightly different from the ``optimality inequality approach'' widely used in the previous literature.Comment: Published at http://dx.doi.org/10.1214/105051606000000105 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Discrete-time controlled markov processes with average cost criterion: a survey

This work is a survey of the average cost control problem for discrete-time Markov processes. The authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades. The exposition ranges from finite to Borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies. The authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography. The authors have also identified several important questions that are still open to investigation

A discounted model for a repairable system with continuous state space

We examine repairable systems with a continous state space and partial repair options, carried out at fixed times $n=1,2,...$. Every time interval $[n,n+1)$ there is a manufacturing cost and a repair cost. These cost functions are not restricted to the class of bounded functions in this study. Conditions are found under which a control-limit replacement policy minimizes the discounted cost. Hence these conditions guarantee that there is an optimal policy under the discounted cost criterion which does not use partial repairs. We explicitly explain how to derive this optimal policy

Optimality of mixed policies for average continuous-time Markov decision processes with constraints

This article concerns the average criteria for continuous-time Markov decision processes with N constraints. We show the following; (a) every extreme point of the space of performance vectors corresponding to the set of stable measures is generated by a deterministic stationary policy; and (b) there exists a mixed optimal policy, where the mixture is over no more than N + 1 deterministic stationary policies

Discrete-Time Control with Non-Constant Discount Factor

This paper deals with discrete-time Markov decision processes (MDPs) with Borel state and action spaces, and total expected discounted cost optimality criterion. We assume that the discount factor is not constant: it may depend on the state and action; moreover, it can even take the extreme values zero or one. We propose sufficient conditions on the data of the model ensuring the existence of optimal control policies and allowing the characterization of the optimal value function as a solution to the dynamic programming equation. As a particular case of these MDPs with varying discount factor, we study MDPs with stopping, as well as the corresponding optimal stopping times and contact set. We show applications to switching MDPs models and, in particular, we study a pollution accumulation problem