When starting from 0, a standard M/M/k queueing process has a second-order stochastic monotonicity property of a strong kind: its increments are stochastically decreasing (the SDI property). A first attempt to generalize this to the Jackson queueing network fails. This gives us reason to reexamine the underlying theory for stochastic monotonicity of Markov processes starting from a zero-point, in order to find a condition on a function of a Jackson network process to have the SDI property. It turns out that the total number of customers at time t has the desired property, if the network is idle at time O. We use couplings in our analysis; they are also of value in the comparison of two networks with different parameters.

Recent developments in stochastic modeling show that enormous analytical advantages can be gained if a general cumulative distribution function (c.d.f.) can be approximated by generalized hyperexponential distributions. In this paper, we introduce a procedure to explicitly construct such approximations of an arbitrary c.d.f. Although our approach can be used in different types of stochastic models, the main motivation comes from queueing theory in obtaining approximations of the idle-period distribution and other performance measures in GI/G/1 queues.

Recently, some authors have suggested usage models of Markov type as a technique of specifying the estimated operational use distribution of a given program. A main purpose of such models is the derivation of random test cases allowing unbiased estimates on the (un)reliability of the program in its intended environment. In this article, we show that by a shift of the transition probabilities of the Markov chain corresponding to such a model, prior information on the errorjproneness of single-program operations can be taken into account. An unbiased unreliability estimator with reduced variance is obtained. Furthermore, it is shown that minimization of the variance leads to a special stochastic optimization problem that can be demonstrated to be convex, such that efficient solution techniques apply. Some related questions are also treated in a more general, non-Markovian framework.

Recently, Derman and Ross (1995, An improved estimator of a in quality control, Probability in the Engineering and Informational Sciences 9: 411–415) derived an estimator of the standard deviation in the standard quality control model and showed that it had smaller mean squared error than the usual estimator. The new estimator was also shown to be consistent even when the underlying distribution deviates from normality, unlike the usual estimator. In this note, the mean squared error is further improved via shrinkage of the Derman-Ross estimator, and a consistent minimum variance unbiased estimator is presented. Finally, by making use of additional subgroup statistics, a minimum variance unbiased estimator is derived and further improved via shrinkage.

The convex (concave) parameterization of a generalized renewal process is considered in this paper. It is shown that if the interrenewal times have log concave distributions or have log concave survival functions (i.e., an increasing failure rate distribution), then the renewal process is convexly (concavely) parameterized in its mean parameterization.

A hunter hunts over a planning horizon t with i bullets. The distribution of the value of targets appearing and the probability of a bullet hitting are known. Encountering a target of value w, he has a choice of retiring from hunting completely, shooting a single bullet, or moving on to the next period to find another target. If he retires, he immediately obtains a terminal reward dependent on the remaining periods and bullets. If he fires and succeeds, he obtains the reward w and decides whether to retire or move on to the next epoch. If he misses and the target remains stationary, he must immediately decide whether to fire an additional bullet, retire, or move on to the next period. The objective in this paper is to examine the optimal decision rules that would maximize the total expected reward. We reveal that the optimal decision rule for firing is not always monotone in the number of remaining bullets and the optimal decision rule for retirement may become possibly the following form. It is better to continue hunting than retire, only if the hunter has neither too few bullets nor too many.

A single server processes jobs that can yield rewards but expire on predetermined dates. Expected immediate rewards from each job are deteriorating. The instance is formulated as a multiarmed bandit problem, and an index-based scheduling policy is shown to maximize the expected total reward.

We consider preemptive and nonpreemptive scheduling of partially ordered tasks on parallel processors, where the precedence relations have an intervalorder, an in-forest, or a uniform out-forest structure. Processing times of tasks are random variables with an increasing in likelihood ratio distribution in the nonpreemptive case and an exponential distribution in the preemptive case. We consider a general cost that is a function of time and of the uncompleted tasks and show that the most successors (MS) policy stochastically minimizes the cost function when it satisfies certain agreeability conditions. A consequence is that the MS policy stochastically minimizes makespan, weighted flowtime, and the weighted number of late jobs.

Simple optimal policies are known for the problem of scheduling jobs to minimize expected makespan on two parallel machines when the job running-time distribution has a monotone hazard rate. But no such policy appears to be known in general. We investigate the general problem by adopting two-point running-time distributions, the simplest discrete distributions not having monotone hazard rates. We derive a policy that gives an explicit, compact solution to this problem and prove its optimality. We also comment briefly on first-order extensions of the model, but each of these seems to be markedly more difficult to analyze.

We consider a single-item, periodic-review inventory model with uncertain demands in which each period's production volume is limited by a capacity level. The demand distributions, capacity levels, and cost parameters vary according to a periodic pattern. We prove that modified base-stock policies are optimal for the finite-horizon planning model and for both the infinite-horizon discounted and undiscounted cost criterion. We further show that the optimal base-stock levels can be calculated via a simple but efficient value-iteration method. Finally, we have conducted a comprehensive numerical study to ascertain the efficiency of this solution method as well as various qualitative properties of the performance of capacitated production/inventory systems under periodically varying demand and cost patterns.