A consecutive−k−out−of−n:F system is an n−vertex graph where the system fails if and only if some k consecutive vertices all fail. Assuming that the n vertices have, independently, the respective failure probabilities q1,…, qn, the problem is to find, subject to πqi = Q (a constant), a set of q1,…, qn so as to maximize the system reliability. In the case that the graph is a circle (or cycle), Chang and Hwang [1] conjectured that q1 =… = qn, = Q1/n is optimal if n and k are relatively prime. In this paper, it is shown that the conjecture is true for sufficiently small Q. It is also shown by counterexample that the conjecture is not true in general.

Discretization is a simple, yet powerful tool in obtaining time-dependent probability distribution of continuous-time Markov chains. One of the most commonly used approaches is uniformization. A recent addition to such approaches is an external uniformization technique. In this paper, we briefly review these different approaches, propose some new approaches, and discuss their performances based on theoretical bounds and empirical computational results. A simple method to get lower and upper bounds for first passage time distribution is also proposed.

The standard single-machine scheduling problem is modified to take into account unsuccessful job completions. We use a result due to Nash to analyze problems in which some jobs are alternative to one another in the sense that only one of a set of alternative jobs need be completed successfully. Conditions are proposed under which a nonpreemptive strategy is optimal for processing such a system.

For a two-station tandem system with a general arrival process, exponential ervice times, and blocking, we show that the distribution of the departure process does not change when the two stations are interchanged. Blocking here means that for some fixed b≥1, any customer completing service at the first station when b customers are at the second station cannot enter the second queue, and the first station cannot start serving a new customer until a service completion occurs at the second station. This result remains true if arrivals are lost when there are a(a≥0) customers in the system. We also show that when the sum of the service rates is held constant, each departure epoch is stochastically minimized if the two rates are equal. For a and b infinite, our results reduce to those given by Weber [14] and Lehtonen [8]. Our proof is based on a coupling method first used by Lehtonen. The same results hold for a blocking mechanism in which a customer completing its service at the first station must restart service when b customers are present at the second.

A continuum structure function (CSF) is a nondecreasing mapping from the unit hypercube to the unit interval. The Kim-Baxter definition of the reliability importance of component i in a CSF at system level α, Ri(α), say, is reviewed. Conditions under which Ri(α) is positive, under which Ri(α) is a continuous function of α, and under which Ri(α) ≥ Rj(α) uniformly in α are presented. A simple algorithm for evaluating Ri(α) is described.

Semi-Markov decision processes underlie the control of many queueing systems. In this paper, we deal with infinite state semi-Markov decision processes with nonnegative, unbounded costs and finite action sets. Axioms for the existence of an expected average cost optimal stationary policy are presented. These conditions generalize the work in Sennott [22] for Markov decision processes. Verifiable conditions for the axioms to hold are obtained. The theory is applied to control of the M/G/l queue with variable service parameter, with on-off server, and with batch processing, and to control of the G/M/m queue with variable arrival parameter and customer rejection. It is applied to a timesharing network of queues with a single server and finally to optimal routing of Poisson arrivals to parallel exponential servers. The final section extends the existence result to compact action spaces.

In this paper, a moment-iteration method is introduced. The method is used to solve Lindley's integral equation for the GI/G/l queue. From several forms of this integral equation, we derive the first two moments of the waiting-time distribution, the waiting probability, and the percentiles of the conditional waiting time. Numerical evidence is given that the method yields excellent results. The flexibility of the method provides the opportunity to solve the GI/G/l queue for all interarrival time distributions of practical interest. To show that the moment-iteration method is generally applicable, we give some results for an (s, S)-model with order-size-dependent lead times and finite production capacity of the supplier.

Models of life time observations that have conditionally independent generalized gamma distributions are investigated. Positive dependency among the unconditional joint distribution is discussed. General formula for moment inequalities are given. Conditional hazard rate, mean residual life functions, and bounds for system reliabilities are also discussed. An application was given.