We consider a queuing system with a workload-dependent service rate. We specifically assume that the service rate is first increasing and then decreasing as a function of the amount of work. The latter qualitative behavior is quite common in practical situations, such as production systems. The admission of work into the system is controlled by a policy for accepting or rejecting jobs, depending on the state of the system. We seek an admission control policy that maximizes the long-run throughput. Under certain conditions, we show that a threshold policy is optimal, and we derive a criterion for determining the optimal threshold value.

We start with a simple derivation of an identity connecting the conditional expected residual service time as seen by an arrival and the steady-state tail distribution function of the number of customers in the system, which was previously proven by Mandelbaum and Yechiali. We then show how to use it to obtain bounds on the the stationary distribution of the number of customers in the M/G/1 queue.

In this article we introduce isotone differences stochastic ordering of Markov processes on lattice ordered state spaces as a device to compare the internal dependencies of two such processes. We derive a characterization in terms of intensity matrices. This enables us to compare the internal dependency structure of different degradable Jackson networks in which the nodes are subject to random breakdowns and repairs. We show that the performance behavior and the availability of such networks can be compared.

For the GI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

This article discusses a real-world application of a terminating two-person stochastic game. The problem comes from a Dutch television game show in which two finalists play a dice game. Each player chooses a number of dice to be rolled. The score of the roll is added to the player's total provided that none of the dice showed the outcome one. The first player reaching a prespecified number of points is the winner. This article discusses the computation and the structure of an optimal strategy.

We consider a multiclass make-to-stock system served by a single server with adjustable capacity (service rate). At any point in time, the decision-maker must determine the capacity level, make a production decision (i.e., whether to produce an item to stock or to satisfy a backorder), and make a rationing decision (i.e., whether to satisfy a new order from stock or place it on backorder). In this article we characterize the structure of optimal capacity adjustment, production, and stock rationing policy for both finite- and infinite-horizon problems. We show that an optimal policy is monotone in current inventory and backorder levels, and we characterize its properties. In a numerical study we compare the optimal policy with heuristic policies and show that the savings from using an optimal policy can be significant.

We study a stochastic scheduling problem of processing a set of jobs on a single machine. Each job has a random processing time Pi and a random due date Di, which are independently and exponentially distributed. The machine is subject to stochastic breakdowns in either preempt-resume or preempt-repeat patterns, with the uptimes following an exponential distribution and the downtimes (repair times) following a general distribution. The problem is to determine an optimal sequence for the machine to process all jobs so as to minimize the expected total cost comprising asymmetric earliness and tardiness penalties, in the form of E[[sum ]αi max{0,Di − Ci} + βi max{0,Ci − Di}]. We find sufficient conditions for the optimal sequences to be V-shaped with respect to {E(Pi)/αi} and {E(Pi)/βi}, respectively, which cover previous results in the literature as special cases. We also find conditions under which optimal sequences can be derived analytically. An algorithm is provided that can compute the best V-shaped sequence.

Recently, Bartoszewicz [5,6] considered mixtures of exponential distributions treated as the Laplace transforms of mixing distributions and established some stochastic order relations between them: star order, dispersive order, dilation. In this article the preservation of the likelihood ratio, hazard rate, reversed hazard rate, mean residual life, and excess wealth orders under exponential mixtures is studied. Some new preservation results for the dispersive order are given, as well as the preservation of the convex transform order, and the star one is discussed.

In this article, we study the problem of finding tight bounds on the expected value of the kth-order statistic E [Xk:n] under first and second moment information on n real-valued random variables. Given means E [Xi] = μi and variances Var[Xi] = σi2, we show that the tight upper bound on the expected value of the highest-order statistic E [Xn:n] can be computed with a bisection search algorithm. An extremal discrete distribution is identified that attains the bound, and two closed-form bounds are proposed. Under additional covariance information Cov[Xi,Xj] = Qij, we show that the tight upper bound on the expected value of the highest-order statistic can be computed with semidefinite optimization. We generalize these results to find bounds on the expected value of the kth-order statistic under mean and variance information. For k < n, this bound is shown to be tight under identical means and variances. All of our results are distribution-free with no explicit assumption of independence made. Particularly, using optimization methods, we develop tractable approaches to compute bounds on the expected value of order statistics.