The classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary (s, S) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an (s, S) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an (s, S) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.

A system is subject to a sequence of shocks occurring randomly at times n = 1, 2, ···; each shock causes a random amount of damage. The system might fail at any point in time n, and the probability of a failure depends on the history of the system. Upon failure the system is replaced by a new and identical system and a cost is incurred. If the system is replaced before failure a smaller cost is incurred. We study the problem of specifying a replacement rule which minimizes the long-run (expected) average cost per unit time. A special case, in which the system fails when the total damage first exceeds a fixed threshold, is analysed in detail.

For a large class of replacement models for stochastically deteriorating systems the optimality criteria of total expected discounted cost and long-run (expected) average cost per unit time have a common structure. In the present paper a formal description of this structure is given and the optimal rule is determined. A so-called ‘λ -minimization technique' is applied. This method is discussed in general terms.

A semi-Markov decision process, with a denumerable multidimensional state space, is considered. At any given state only a finite number of actions can be taken to control the process. The immediate reward earned in one transition period is merely assumed to be bounded by a polynomial and a bound is imposed on a weighted moment of the next state reached in one transition. It is shown that under an ergodicity assumption there is a stationary optimal policy for the long-run average reward criterion. A queueing network scheduling problem, for which previous criteria are inapplicable, is given as an application.

We study a production-inventory system in which input is deterministic, and its rate is the controlled parameter. The output is a compound Poisson process with exponential jump-size distribution. The parameters of the output process are unknown. The following adaptive policy is used: At each time t the unknown parameters are estimated by maximum likelihood method and the input rate which would be optimal if these were the true values of the parameter is used. The issues investigated are (1) the convergence of maximum likelihood estimates to the true value, and (2) the asymptotic properties of the cost under the adaptive policy.

In a preceding paper [2] we have introduced a new approach for solving a wide class of Markov decision problems in which the state-space may be general and the system may be continuously controlled. The criterion is the average cost. This paper discusses two applications of this approach. The first application concerns a house-selling problem in which a constructor builds houses which may be sold at any stage of the construction and potential customers make offers depending on the stage of the construction. The second application considers an M/M/c queueing problem in which the number of operating servers can be controlled by turning servers on or off.