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.

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.

Several models of maintenance of repairable multicomponent systems, subjected to a random environment, are introduced in this paper. For each model, it is shown how various probabilistic quantities of interest are “monotone” as a function of the probability law of the environment or as a function of other parameters of the model. Also, conditions are given which imply the positive dependence of the failure times, or of the counters of the number of failures, in these models.

Two queues forming two independent Poisson processes are served by one server with exponential service time. The server always works on the longer queue and, in case that they are of equal length, chooses either one with probability ½. Let πij be the probability that the two queue lengths equal i andj at equilibrium and π(z, w) = ∑πi j Ziwj. We determine π(z, w) and derive from this asymptotic formulas forπij as i, j → ∞. These asymptotic formulas are used to study the interdependence of the queue lengths. In particular, we obtain limit laws for the queue lengths conditioned on each other.

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.

Recently, the conjecture that the expected number of tests is nondecreasing in the failure probability for binomial group testing has been proved. The proof has also been extended to three models of multiaccess systems. However, probabilistic algorithms are used as a crucial part of these proofs. In this paper, we give conceptually simpler new proofs without using probabilistic algorithms. We also extend the result to a more general model where the number of tests is replaced by a cost function.

Two classes of inventory systems are considered, and modeled under a control policy of the bang-bang type. For the first, items and demands for those items arrive independently and singly at a system. Unsatisfied demands are lost, and items staying on the shelf too long are considered unsuitable for use and are rejected. For the second, items are produced continuously over time with fixed rate in a storage/production system with capacity k, in which demands of exponential size arrive according to a Poisson process, and no backorders are allowed. The problem of controlling the demand rate in the first model is shown to be equivalent to the problem of controlling the production rate in the second model. Optimization methods are then considered for both models employing discounted cost and averaged cost rate criteria for the optimal selection of switchover levels for arrival and demand rates.

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.

We define two new classes of multistate coherent systems by requiring, among other conditions, that their structure functions be Schur-concave (Schurconvex). The M + 1 performance levels of both the systems and their components are represented by the set [0, 1,…, M]. We present basic structural properties of the new classes. In particular, we study in some detail the number and form of the critical upper (lower) vectors to the various performance levels. We also present some probabilistic aspects of the new classes.

Consider the problem of simulating w independent Bernoulli random variables X1 X2, …, Xw, so that {Xk = 1 = zk}, using a computer with a w–bit word length. A conventional method would be to generate w pseudorandom numbers uniform in [0,1], then “threshold” them with the values zk. This paper proposes an alternative method that only requires c pseudorandom numbers, where c is large enough so that the numbers zk are approximated satisfactorily with c bits. In many cases, c ≪ w and so fewer pseudorandom numbers are required. The new method is tested against the conventional method by employing both in simulations to estimate the reliabilities of several coherent systems. The results show that the new method is much faster.

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.

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.

Compensator or dual predictable projection is one of the main concepts of the general theory of stochastic processes, established by a French probabilistic school. In general, it is a considerably abstract concept and difficult to grasp. But in the Poisson case, it affords very powerful tools to calculate the expectation of certain stochastic integrals with respect to Poisson processes and random measures. Naturally, compensators can be used to solve certain problems of applied probability, involved in Poisson processes and random measures. In this direction, Brémaud and Jacod [1] had used the compensators of Poisson processes to solve the optimal dispatching problem, investigated by Ross [4] originally. In this note, we use the compensators of Poisson random measures to solve two optimal stopping problems: house selling and the burglary problem. The solutions are simple and completely rigorous as well. Our objective is to attract more attention to this method of using compensators. To facilitate reading, we introduce some fundamental results on marked point processes and Poisson random measures and their compensators.

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.

Previous studies have examined the behavior of outlier detection rules for symmetric distributions that label as “outside” any observations that fall outside the interval [FL – k(Fu – FL), Fu + k(Fu – FL)], where FL and FU are functions of the order statistics estimating the 0.25 and 0.75 quantiles of the distribution underlying the i.i.d. sample. A measure of the performance of this type of rule is the “some-outside rate” per sample computed with respect to a given (usually Gaussian) null distribution. The “some-outside rate” (SOR) per sample is the probability that the sample will contain one or more observations labeled as “outside,” given that the null distribution is the true distribution. In this paper, asymptotic expansions of k = kn as a function of n that guarantee an asymptotically constant, prespecified SOR are given for a variety of symmetric null distributions including the Gaussian, double exponential, logistic, and Cauchy distributions. The main theorem also applies to the case of a nonsymmetric null distribution by slightly modifying the labeling rule.