We show that natural estimators occurring in certain simulation settings have convergence rates less than the canonical rate usually associated with simulation. These natural estimators use replication schemes that attenuate bias. For some important examples, we find alternative estimators that converge at the canonical rate. The implications of these asymptotic comparisons for choosing good strategies when the computer-time budget is modest are discussed.

An estimator, based on a simulation, is given for the expected number of events by time t of a renewal process. The estimator is obtained by combining the techniques of control variates and conditional expectation.

We wish to schedule heterogeneous jobs on a single machine to stochastically maximize the number of jobs that are correctly completed before a random due date. Jobs arrive according to an arbitrary arrival process. Job i has probability ci, that it will be correctly completed, and a processing time that has distribution Fi. Jobs that are incorrectly processed cannot be reprocessed, and jobs that are not completed by the due date are assumed to be incorrectly processed. The machine is subject to breakdowns and repairs. Both premptive and non-preemptive policies are considered. We give conditions under which simple index rules are optimal. This generalizes and extends the results of Derman et al. [3], Pinedo and Rammouz [9], and Buyukkoc et al. [2], who derive policies that maximize the expected number of jobs correctly completed by the due date. We also consider policies that stochastically minimize the number of correct job completions. For single class systems, we give conditions under which FCFS (first-come-first-served) or LAST (least-attained-service-time) service disciplines stochastically minimize or maximize the number in the system.

As the formal methods of proving correctness of a computer program are still very inadequate, in practice when a new piece of software is developed and all obvious errors are removed, it is tested with different (random) inputs in order to detect the remaining errors and assess its quality. We suppose that whenever the program fails the error causing the failure can be detected and removed correctly. Thus, the quality of the software increases as testing goes on. In this paper, we consider two different models and present the minimum variance unbiased estimators of the expected failure rate of the revised software at any time of testing t, based on the data generated up to that point.

A two-unit parallel system with cold standby, Markovian degradation of the working unit and one repair facility is considered. The calculation of the availability of this system under a control-limit repair policy is discussed. Iterative schemes for computing the availability in case of Erlangian repair-time distributions are presented along with numerical results.

Mixture distributions are a frequently used tool in modelling random phenomena. We consider mixtures of densities from a one-parameter exponenvial family of distributions. Using the tools of totally positive functions and the variation-diminishing property of such, we study the effect of sign-crossing properties of two mixing densities μ1 and μ2 on the resulting mixture distributions f1 and f2. The results enable us to make stochastic and variability cornparisons for binomial-beta, mixed Weibull, and mixed gamma distributions.

This paper motivates and introduces a number of new concepts of positive dependence among sets of random variables. In particular, setwise positive upper (and lower) orthant dependence, setwise association, and other related concepts are studied and their relationships are explored. These new concepts are applied to various multivariate normal distributions, some of which have both positive and negative covariances. A variety of new and interesting inequalities are also obtained.

Some properties of repair age (the elapsed time since the last repair) and the residual repair life (the time until the next repair) are studied for the minimal repair process under various assumptions on the aging characteristics of the equipment under maintenance. Bounds on expected repair age and expected residual repair life are obtained. Some characterizations of the Poisson process are also proved.

We consider a renewal reward process in continuous time. The supremum average reward, γ* for this process can be characterised as the unique root of a certain function. We show how one can apply the Newton–Raphson algorithm to obtain successive approximations to γ*, and show that the successive approximations so obtained are the same as those obtained by using the policy improvement technique.

A condition expressed in Eq. (7) is given which, with one simplifying regularity condition, ensures that the policy-improvement algorithm is equivalent to application of the Newton–Raphson algorithm to an optimality condition. It is shown that this condition covers the two known cases of such equivalence, and another example is noted. The condition is believed to be necessary to within transformations of the problem, but this has not been proved.

This paper is concerned with a certain property of the stationary distribution of a generalized semi-Markov process (GSMP) known as insensitivity. It is well-known that the so-called Matthes' conditions form a necessary and sufficient algebraic criterion for insensitivity. Most proofs of these conditions are basically algebraic. By interpreting a GSMP as a simple queueing network, we are able to show that Matthes' conditions are equivalent to the quasi-reversibility of the network, thus obtaining another simple proof of the sufficiency of these conditions. Furthermore, we apply our method to find a simple criterion for the insensitivity of GSMP's with generalized routing (in a sense that is introduced in the paper).

Our primary aim in this paper is to study a functional equation that arises in a problem of queueing. Consider a queue with compound Poisson arrivals and general service times with a gating mechanism. The gating mechanism takes in at most m(≤∞) customers at a time for service and serves these customers according to the processor-sharing discipline. In this paper, we examine various performance characteristics for this queue. The characteristics include waiting-time distribution, queue-length distribution, time spent with the server, and batch-size distribution for service.

Given a stochastic activity network in which the length of some or all of the arcs are random variables with known probability distributions. This paper concentrates on identifying the shortest path and the M shortest paths in the network and on using the M paths to identify surrogate stochastic networks which are amenable for deriving analytical solutions. First, it identifies the M shortest paths using a certain form of stochastic dominance. Second, it identifies the M shortest paths by applying the deterministic methods to the network resulting from replacing the random length of every arc by its mean value. The two sets of the M paths are compared with those obtained by Monte Carlo sampling. Finally, the paper investigates how the distributional properties of the shortest path in the surrogate network compare with those of the shortest path in the original stochastic network.