A new condition for the existence of optimum stationary policies in average cost Markov decision processes - Unbounded cost case

1986 ◽  
Author(s):  
Linn Sennott
Keyword(s):  
Markov Decision Processes ◽  
Average Cost ◽  
Decision Processes ◽  
Markov Decision ◽  
Unbounded Cost

scholarly journals Denumerable state continuous time Markov decision processes with unbounded cost and transition rates under average criterion

2002 ◽  
Vol 43 (4) ◽  
pp. 541-557 ◽  
Author(s):  
Xianping Guo ◽  
Weiping Zhu
Keyword(s):  
Markov Decision Processes ◽  
Continuous Time ◽  
Decision Processes ◽  
Transition Rates ◽  
Birth And Death Processes ◽  
Optimality Equation ◽  
Average Criterion ◽  
Markov Decision ◽  
Unbounded Cost ◽  
Queue Model

AbstractIn this paper, we consider denumerable state continuous time Markov decision processes with (possibly unbounded) transition and cost rates under average criterion. We present a set of conditions and prove the existence of both average cost optimal stationary policies and a solution of the average optimality equation under the conditions. The results in this paper are applied to an admission control queue model and controlled birth and death processes.

scholarly journals New discount and average optimality conditions for continuous-time Markov decision processes

2010 ◽  
Vol 42 (04) ◽  
pp. 953-985 ◽  
Author(s):  
Xianping Guo ◽  
Liuer Ye
Keyword(s):  
Markov Decision Processes ◽  
Continuous Time ◽  
Average Cost ◽  
Nonnegative Solution ◽  
Decision Processes ◽  
Stationary Policy ◽  
Discounted Cost ◽  
Markov Decision ◽  
Optimal Stationary Policy ◽  
Bounded Below

This paper deals with continuous-time Markov decision processes in Polish spaces, under the discounted and average cost criteria. All underlying Markov processes are determined by given transition rates which are allowed to be unbounded, and the costs are assumed to be bounded below. By introducing an occupation measure of a randomized Markov policy and analyzing properties of occupation measures, we first show that the family of all randomized stationary policies is ‘sufficient’ within the class of all randomized Markov policies. Then, under the semicontinuity and compactness conditions, we prove the existence of a discounted cost optimal stationary policy by providing a value iteration technique. Moreover, by developing a new average cost, minimum nonnegative solution method, we prove the existence of an average cost optimal stationary policy under some reasonably mild conditions. Finally, we use some examples to illustrate applications of our results. Except that the costs are assumed to be bounded below, the conditions for the existence of discounted cost (or average cost) optimal policies are much weaker than those in the previous literature, and the minimum nonnegative solution approach is new.

scholarly journals New discount and average optimality conditions for continuous-time Markov decision processes

2010 ◽  
Vol 42 (4) ◽  
pp. 953-985 ◽  
Author(s):  
Xianping Guo ◽  
Liuer Ye
Keyword(s):  
Markov Decision Processes ◽  
Continuous Time ◽  
Average Cost ◽  
Nonnegative Solution ◽  
Decision Processes ◽  
Stationary Policy ◽  
Discounted Cost ◽  
Markov Decision ◽  
Optimal Stationary Policy ◽  
Bounded Below

This paper deals with continuous-time Markov decision processes in Polish spaces, under the discounted and average cost criteria. All underlying Markov processes are determined by given transition rates which are allowed to be unbounded, and the costs are assumed to be bounded below. By introducing an occupation measure of a randomized Markov policy and analyzing properties of occupation measures, we first show that the family of all randomized stationary policies is ‘sufficient’ within the class of all randomized Markov policies. Then, under the semicontinuity and compactness conditions, we prove the existence of a discounted cost optimal stationary policy by providing a value iteration technique. Moreover, by developing a new average cost, minimum nonnegative solution method, we prove the existence of an average cost optimal stationary policy under some reasonably mild conditions. Finally, we use some examples to illustrate applications of our results. Except that the costs are assumed to be bounded below, the conditions for the existence of discounted cost (or average cost) optimal policies are much weaker than those in the previous literature, and the minimum nonnegative solution approach is new.

Average Cost Semi-Markov Decision Processes and the Control of Queueing Systems

1989 ◽  
Vol 3 (2) ◽  
pp. 247-272 ◽  
Author(s):  
Linn I. Sennott
Keyword(s):  
Markov Decision Processes ◽  
Average Cost ◽  
Queueing Systems ◽  
Decision Processes ◽  
Single Server ◽  
Stationary Policy ◽  
Markov Decision ◽  
Optimal Stationary Policy ◽  
Poisson Arrivals ◽  
Action Spaces

Semi-Markov decision processes underlie the control of many queueing systems. In this paper, we deal with infinite state semi-Markov decision processes with nonnegative, unbounded costs and finite action sets. Axioms for the existence of an expected average cost optimal stationary policy are presented. These conditions generalize the work in Sennott [22] for Markov decision processes. Verifiable conditions for the axioms to hold are obtained. The theory is applied to control of the M/G/l queue with variable service parameter, with on-off server, and with batch processing, and to control of the G/M/m queue with variable arrival parameter and customer rejection. It is applied to a timesharing network of queues with a single server and finally to optimal routing of Poisson arrivals to parallel exponential servers. The final section extends the existence result to compact action spaces.

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