We consider the optimal scheduling problem for a single-server queue without arrivals. We allow preemptions, and our purpose is to minimize the expected flow time. The optimal nonanticipating discipline is known to be the Gittins index policy, which, however, is defined in an implicit way. Until now, its general behavior in this specific problem has been characterized only in a few special cases. In this article, we give as complete a characterization as possible. It turns out that the optimal policy always belongs to the family of multilevel processor sharing disciplines.

We consider the dynamic pricing problem of perishable products in a system with a constant production rate. Potential demands arrive according to a compound Poisson process, and are price-sensitive. We carry out the sample path analysis of the inventory process and by using level-crossing method, we derive its stationary distribution given a pricing function. Based on the distribution, we express the average profit function. By a stochastic comparison approach, we characterize the pricing strategy given different customers willingness-to-pay functions. Finally, we provide an approximation algorithm to calculate the optimal pricing function.

In this article we study Markov decision process (MDP) problems with the restriction that at decision epochs, only a finite number of given Markov decision rules are admissible. For example, the set of admissible Markov decision rules could consist of some easy-implementable decision rules. Additionally, many open-loop control problems can be modeled as an MDP with such a restriction on the admissible decision rules. Within the class of available policies, optimal policies are generally nonstationary and it is difficult to prove that some policy is optimal. We give an example with two admissible decision rules—={d1, d2} —for which we conjecture that the nonstationary periodic Markov policy determined by its period cycle (d1, d1, d2, d1, d2, d1, d2, d1, d2) is optimal. This conjecture is supported by results that we obtain on the structure of optimal Markov policies in general. We also present some numerical results that give additional confirmation for the conjecture for the particular example we consider.

Consider a group of players playing a sequence of games. There are k players, having arbitrary initial fortunes. Each game consists of each remaining player putting 1 in a pot, which is then won (with equal probability) by one of them. Players whose fortunes drop to 0 are eliminated. Let T(i) be the number of games played by i, and let T=max iT(i). For the case k=3, martingale stopping theory can be used to derive E[T] and E[T(i)]. When k>3, we obtain upper bounds on E[T] and, in the case in which all players have the same initial fortune, on E[T(i)]. Efficient simulation methods for estimating E[T] and E[T(i)] are discussed.

In this article, we establish some results concerning the likelihood ratio order of random vectors of order statistics in the case of independent but not necessarily identically distributed observations and for the case of possible dependent observations. Applications of these results to provide comparisons of conditional order statistics are also given.

In this article, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of the likelihood ratio order and the hazard rate order. Let X1 and X2 be two independent gamma random variables with Xi having shape parameter r>0 and scale parameter λi, i=1, 2, and let X*1 and X*2 be another set of independent gamma random variables with X*i having shape parameter r and scale parameter λ*i, i=1, 2. Denote by X2:2 and X*2:2 the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ1, λ2) weakly majorize (λ*1, λ*2), then X2:2 is stochastically greater than X*2:2 in the sense of likelihood ratio order. We also establish, among others, that if 0<r≤1 and (λ1, λ2) is p-larger than (λ*1, λ*2), then X2:2 is stochastically greater than X*2:2 in the sense of hazard rate order. The results derived here strengthen and generalize some of the results known in the literature.

We consider the filtering problem for a doubly stochastic Poisson or Cox process, where the intensity follows the Cox–Ingersoll–Ross model. In this article we assume that the Brownian motion, which drives the intensity, is not observed. Using filtering theory for point process observations, we first derive the dynamics for the intensity and its moment-generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment-generating function allows us to solve in closed form the filtering problem, between the jumps of the Cox process as well as at the jumps, which constitutes the main contribution of the article. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the article with the observation that the resulting conditional moment-generating function at time t, after Nt jumps, corresponds to a mixture of Nt+1 Gamma distributions. Currently, the model that we analyze has become popular in credit risk modeling, where one uses the intensity-based approach for the modeling of default times of one or more companies. In this approach, the default times are defined as the jump times of a Cox process. In such a model, one only has access to observations of the Cox process, and thus filtering comes in as a natural technique in credit risk modeling.

We propose a generalized extreme shock model with a possibly increasing failure threshold. Although standard models assume that the crucial threshold for the system might only decrease over time, because of weakening shocks and obsolescence, we assume that, especially at the beginning of the system's life, some strengthening shocks might increase the system tolerance to large shock. This is, for example, the case of turbines’ running-in in the field of engineering. On the basis of parametric assumptions, we provide theoretical results and derive some exact and asymptotic univariate and multivariate distributions for the model. In the last part of the article we show how to link this new model to some nonparametric approaches proposed in the literature.