Constrained semi-Markov Decision Process Research Articles

Constrained semi-Markov decision processes with ratio and time expected average criteria in Polish spaces

This paper deals with the ratio and time expected average criteria for constrained semi-Markov decision processes (SMDPs). The state and action spaces are Polish spaces, the rewards and costs are unbounded from above and from below, and the mean holding times are allowed to be unbounded from above. First, under general conditions we prove the existence of constrained-optimal policies for the ratio expected average criterion by developing a technique of occupation measures including the mean holding times for SMDPs, which are the generalizations of those for the standard discrete-time and continuous-time MDPs. Then, we give suitable conditions under which we establish the equivalence of the two average criteria by the optional sampling theorem, and thus we show the existence of constrained-optimal policies for the time expected average criterion. Finally, we illustrate the application of our main results with a controlled linear system, for which an exact optimal policy is obtained.

Reinforcement Learning for Call Admission Control and Routing under Quality of Service Constraints in Multimedia Networks

In this paper, we solve the call admission control and routing problem in multimedia networks via reinforcement learning (RL). The problem requires that network revenue be maximized while simultaneously meeting quality of service constraints that forbid entry into certain states and use of certain actions. The problem can be formulated as a constrained semi-Markov decision process. We show that RL provides a solution to this problem and is able to earn significantly higher revenues than alternative heuristics.

Constrained Semi-Markov decision processes with average rewards

This paper deals with constrained average reward Semi-Markov Decision Processes (SMDPs) with finite state and action sets. We consider two average reward criteria. The first criterion is time-average rewards, which equal the lower limits of the expected average rewards per unit time, as the horizon tends to infinity. The second criterion is ratio-average rewards, which equal the lower limits of the ratios of the expected total rewards during the firstn steps to the expected total duration of thesen steps asn → ∞. For both criteria, we prove the existence of optimal mixed stationary policies for constrained problems when the constraints are of the same nature as the objective functions. For unichain problems, we show the existence of randomized stationary policies which are optimal for both criteria. However, optimal mixed stationary policies may be different for each of these critria even for unichain problems. We provide linear programming algorithms for the computation of optimal policies.

Time-average optimal constrained semi-Markov decision processes

Optimal causal policies maximizing the time-average reward over a semi-Markov decision process (SMDP), subject to a hard constraint on a time-average cost, are considered. Rewards and costs depend on the state and action, and contain running as well as switching components. It is supposed that the state space of the SMDP is finite, and the action space compact metric. The policy determines an action at each transition point of the SMDP.Under an accessibility hypothesis, several notions of time average are equivalent. A Lagrange multiplier formulation involving a dynamic programming equation is utilized to relate the constrained optimization to an unconstrained optimization parametrized by the multiplier. This approach leads to a proof for the existence of a semi-simple optimal constrained policy. That is, there is at most one state for which the action is randomized between two possibilities; at all other states, an action is uniquely chosen for each state. Affine forms for the rewards, costs and transition probabilities further reduce the optimal constrained policy to ‘almost bang-bang’ form, in which the optimal policy is not randomized, and is bang-bang except perhaps at one state. Under the same assumptions, one can alternatively find an optimal constrained policy that is strictly bang-bang, but may be randomized at one state. Application is made to flow control of a birth-and-death process (e.g., an M/M/s queue); under certain monotonicity restrictions on the reward and cost structure the preceding results apply, and in addition there is a simple acceptance region.

Time-average optimal constrained semi-Markov decision processes

Optimal causal policies maximizing the time-average reward over a semi-Markov decision process (SMDP), subject to a hard constraint on a time-average cost, are considered. Rewards and costs depend on the state and action, and contain running as well as switching components. It is supposed that the state space of the SMDP is finite, and the action space compact metric. The policy determines an action at each transition point of the SMDP. Under an accessibility hypothesis, several notions of time average are equivalent. A Lagrange multiplier formulation involving a dynamic programming equation is utilized to relate the constrained optimization to an unconstrained optimization parametrized by the multiplier. This approach leads to a proof for the existence of a semi-simple optimal constrained policy. That is, there is at most one state for which the action is randomized between two possibilities; at all other states, an action is uniquely chosen for each state. Affine forms for the rewards, costs and transition probabilities further reduce the optimal constrained policy to ‘almost bang-bang’ form, in which the optimal policy is not randomized, and is bang-bang except perhaps at one state. Under the same assumptions, one can alternatively find an optimal constrained policy that is strictly bang-bang, but may be randomized at one state. Application is made to flow control of a birth-and-death process (e.g., an M/M/s queue); under certain monotonicity restrictions on the reward and cost structure the preceding results apply, and in addition there is a simple acceptance region.