It is shown that the gamma distribution is the unique mixing distribution of exponentials that leads to a linearly increasing mean residual lifetime function. A corollary of this result is that regardless of the mixing process the Pareto distribution is the unique aggregate distribution mixture with this property.

In this paper we consider a set of denumerable stochastic matrices where the parameter set is a compact metric space. We give a number of simultaneous recurrence conditions on the stochastic matrices and establish equivalences between these conditions. The results obtained generalize corresponding results in Markov chain theory to a considerable extent and have applications in stochastic control problems.

For a single-server system having several Poisson streams of customers with exponentially distributed service times, busy period densities, waiting time densities, and idle state probabilities are completely monotone. The exponential spectra for such densities are of importance for understanding the transient behavior of such systems. Algorithms are given for the computation of such spectra. Applications to heavy traffic situations and priority systems are also discussed.

We prove and extend Burke's output theorem for queues in equilibrium with the help of a filtering formula giving the time evolution of the least square estimate of the state with respect to the output.

Conditions are given on a process so that a shock model governed by the process will have nbue survival time. A related condition is given so that survival will be NBU and it is shown that a pure birth process satisfies this condition. A result of A-Hameed and Proschan is obtained as a corollary.

A finite dam is studied in which the net input in each period, i.e. the excess of input over demand, is a function of a Markov chain. By using a martingale, the joint distribution of the time to first overflow and the cumulative unsatisfied demand until overflow is investigated. The probability of overflow before the unsatisfied demand exceeds a fixed level is also considered.

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which yi → ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk < yo, · · ·, yN–1 for some k ≧ N, is necessary and sufficient for transience.

We consider a queuing system with several identical servers, each with its own queue. Identical customers arrive according to some stochastic process and as each customer arrives it must be assigned to some server's queue. No jockeying amongst the queues is allowed. We are interested in assigning the arriving customers so as to maximize the number of customers which complete their service by a certain time. If each customer's service time is a random variable with a non-decreasing hazard rate then the strategy which does this is one which assigns each arrival to the shortest queue.

It is demonstrated that standard Brownian motion in the tangent plane at the centroid of frequency space does not well approximate the discrete Wright—Fisher process for more than 2N generations where N is the effective population size. This result is obtained using Wright's concept of negligible mutation rate for the study of systematic evolutionary effects together with Ludwig's notion of the persistence of a dynamical system. This work may be viewed as a mathematical elaboration of a portion of Wright's shifting balance theory of evolution.

This paper is concerned with the expected ultimate size of the Downton carrier-borne epidemic (1968). The solution obtained is slightly recursive and it may be expressed as a power series in π, the proportion of the infected susceptibles becoming carriers. It generalizes the results of Dietz (1966) and Abakuks (1973) for some special cases of this model.

We examine a failure model for a system existing in a random environment. The system accumulates damage through a shock process and the failure time depends on the accumulated damage in the system. The cumulative damage process is assumed to be a semi-Markov process. Upon failure the system must be replaced by a new identical one and a failure cost is incurred. If the system is replaced before failure, a smaller cost is incurred. We allow a controller to replace the system at any stopping time before failure time. We consider the problem of specifying a replacement rule which minimizes the total long-run average cost per unit time.

It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.

The utilisation factor for a discrete dam, defined as the stationary probability of non-emptiness of the dam just before release, is obtained for a class of models that includes the Odoom–Lloyd model, the Anis–Lloyd model, the model of Herbert, etc. The method used points to a different approach to measuring dam utilisation via the actual utilisation, which appears to be more fruitful, and this is discussed in the last section.

Stochastic models in population genetics which lead to diffusion equations are considered. A geometric formula for the asymptotic expansions of the fundamental solutions of these equations is presented. Specifically, the random genetic drift process of one-locus theory and the Ohta–Kimura model of two-locus di-allelic systems with linkage are studied. Agreement with the work of Keller and Voronka for the two-allele one-locus case is obtained. For the general n-allele problem, the formulas obtained here are apparently new.

Given a stationary Markov process with s transient states and r absorbing states, a matrix infinite series solution is presented for the variance of duration of stay in state j within the interval [0, t), given initial state i. Closed forms are derived for absorbing states, and for transient states if eigenvalues are real and distinct. Several relationships among Markov matrices are presented.

The value of a process on C[0,∞) when it starts its first one-sided excursion of width ≧ α is related to first hitting one of two boundaries. Explicit results for brownian motion are applied to a finite dam model.

Let Xn1 ≦ Xn2 ≦ ··· ≦ Xnn denote the order statistics from a sample of n independent, identically distributed random variables, and suppose that the variables Xnn, Xn, n–1, ···, when suitably normalized, have a non-trivial limiting joint distribution ξ1, ξ2, ···, as n → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn, n ≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems for ξ n as n →∞.

We show, in the context of a neurobiological problem, that ζ a(α)−ζ a(β) increases with a, while ζ a (α)/ζ a (β) decreases with a, where 0 < β < α < 1 and ζ a (α) is the α th quantile of either the gamma distribution with shape parameter a, or a scaled F distribution with parameters 2a and 2m for any m.

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.