We consider a stochastic input–output system with additional total clearings at certain random times determined by its own evolution (and specified by a controller). Between two clearings, the stock level process is a superposition of a Brownian motion with drift and a compound Poisson process with positive jumps, reflected at zero. We introduce meaningful cost functionals for this system and determine them explicitly under several (classical and new) clearing policies.

This article presents Monte Carlo techniques for estimating network reliability. For highly reliable networks, techniques based on graph evolution models provide very good performance. However, they are known to have significant simulation cost. An existing hybrid scheme (based on partitioning the time space) is available to speed up the simulations; however, there are difficulties with optimizing the important parameter associated with this scheme. To overcome these difficulties, a new hybrid scheme (based on partitioning the edge set) is proposed in this article. The proposed scheme shows orders of magnitude improvement of performance over the existing techniques in certain classes of network. It also provides reliability bounds with little overhead.

We relate two variants of the inspection paradox by showing that both represent the length of a renewal interval that contains a chosen point, the difference being in the way that point is chosen. We show, in both cases, that the length of the interval is likelihood ratio larger than that of an ordinary renewal interval.

The multiarmed-bandit problem is often taken as a basic model for the trade-off between the exploration and utilization required for efficient optimization under uncertainty. In this article, we study the situation in which the unknown performance of a new bandit is to be evaluated and compared with that of a known one over a finite horizon. We assume that the bandits represent random variables with distributions from the one-parameter exponential family. When the objective is to maximize the Bayes expected sum of outcomes over a finite horizon, it is shown that optimal policies tend to simple limits when the length of the horizon is large.

Very few stochastic systems are known to have closed-form transient solutions. In this article we consider an immigration birth and death population process with total catastrophes and study its transient as well as equilibrium behavior. We obtain closed-form solutions for the equilibrium distribution as well as the closed-form transient probability distribution at any time t ≥ 0. Our approach involves solving ordinary and partial differential equations, and the method of characteristics is used in solving partial differential equations.

In this article, we introduce a new dispersion order weaker than the classic dispersion order discussed by Lewis and Thompson (1981). We study the equivalence of this order with the majorization order under the assumption of unimodality. Finally, we use this equivalence to characterize the IFR aging notion for unimodal distributions by means of the notion of decreasing in randomness.

This article is concerned with the problem of controlling a simple immigration process, which represents a pest population, by the introduction of a predator. It is assumed that the cost rate caused by the pests is an increasing function of their population size and that the cost rate of the controlling action is constant. The existence of a control-limit policy that minimizes the expected long-run average cost per unit time is established. The proof is based on the variation of a fictitious parameter over the entire real line.

Let τk|n denote the lifetime of a k-out-of-n system, where the n components have independent lifetimes Ti with completely arbitrary distribution Fi, i = 1,..., n. It is shown that τk+1|n ≤hr τk|n, τk|n ≤hr τk−1|n−1, and τk|n−1 ≤hr τk|n if Ti ≤hrTn, i = 1,..., n − 1; τk+1|n ≤rh τk|n, τk−1|n ≤rh τk|n, and τk|n ≤rh τk−1|n−1 if Tn ≤rhTi, i = 1,..., n − 1. These results are available in the literature for the special case of Fi's being absolutely continuous. Also, even in this case, the proofs are often tedious and use the concept of “totally positive of order infinity in differences of k.” In contrast, the proofs given here are simple and elegant and do not use the above concept.

External and internal monotonicity properties for Jackson networks have been established in the literature with the use of coupling constructions. Recently, Lopez et al. derived necessary and sufficient conditions for the (strong) stochastic comparison of two-station Jackson networks with increasing service rates, by constructing a certain Markovian coupling. In this article, we state necessary and sufficient conditions for the stochastic comparison of L-station Jackson networks in the general case. The proof is based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.

The above-mentioned article by James Lynch was published in Probability in the Engineering and Informational Sciences (1999), 13: 33–36.

It has recently been brought to the author's attention that the results in that paper were preceded and superseded by the results in Thomas H. Savits' paper, “A multivariate IFR class,” which appeared in the Journal of Applied Probability (1985), 22: 197–204. This acknowledgment is to correct this contretemps.