134 research outputs found
Discounted continuous-time constrained Markov decision processes in Polish spaces
This paper is devoted to studying constrained continuous-time Markov decision
processes (MDPs) in the class of randomized policies depending on state
histories. The transition rates may be unbounded, the reward and costs are
admitted to be unbounded from above and from below, and the state and action
spaces are Polish spaces. The optimality criterion to be maximized is the
expected discounted rewards, and the constraints can be imposed on the expected
discounted costs. First, we give conditions for the nonexplosion of underlying
processes and the finiteness of the expected discounted rewards/costs. Second,
using a technique of occupation measures, we prove that the constrained
optimality of continuous-time MDPs can be transformed to an equivalent
(optimality) problem over a class of probability measures. Based on the
equivalent problem and a so-called -weak convergence of probability
measures developed in this paper, we show the existence of a constrained
optimal policy. Third, by providing a linear programming formulation of the
equivalent problem, we show the solvability of constrained optimal policies.
Finally, we use two computable examples to illustrate our main results.Comment: Published in at http://dx.doi.org/10.1214/10-AAP749 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Maximal reliability of controlled Markov systems
This paper concentrates on the reliability of a discrete-time controlled
Markov system with finite states and actions, and aims to give an efficient
algorithm for obtaining an optimal (control) policy that makes the system have
the maximal reliability for every initial state. After establishing the
existence of an optimal policy, for the computation of optimal policies, we
introduce the concept of an absorbing set of a stationary policy, and find some
characterization and a computational method of the absorbing sets. Using the
largest absorbing set, we build a novel optimality equation (OE), and prove the
uniqueness of a solution of the OE. Furthermore, we provide a policy iteration
algorithm of optimal policies, and prove that an optimal policy and the maximal
reliability can be obtained in a finite number of iterations. Finally, an
example in reliability and maintenance problems is given to illustrate our
results
ON THE FIRST PASSAGE g-MEAN-VARIANCE OPTIMALITY FOR DISCOUNTED CONTINUOUS-TIME MARKOV DECISION PROCESSES
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