The use of alternating renewal processes to model the behaviour of a repairable system is discussed, and properties of two measures of availability are reviewed. It is shown how the introduction of forward recurrence times to failure enables these measures to be generalised.

We consider an infinite-server queueing system in an extraneous environment. Initially it is shown that the systems of interest can be decomposed into a two-stage system. The primary system is an infinite-server queue with many customer types subject to a clearing mechanism. The secondary system is a special type of bulk-arrival, infinite-server queue. We derive results for the primary and secondary systems separately and combine the results to find the mean steady-state behavior of the original system.

The existence of a weak limit birth-and-death process on the natural integers for the simple birth-and-death process conditional on non-extinction up to time t as t→∞ is proved. Starting from the latter a new weak limiting procedure yields a diffusion Markov process on the positive infinite semi-axis.

A branching process is described which contains some better-known simple branching processes as special cases. Variables examined are the rate of development, the number of individuals currently alive, and the duration of the process. The apparent dynamics of collapsing supercritical realisations are isolated.

An upper bound for the asymptotic velocity in various directions of first-passage percolation on the square lattice is derived. In particular this gives a lower bound for the so-called time constant. The result is generalized to other lattices. Numerical examples are included.

Earlier stochastic analyses of chemical reactions have provided formal solutions which are unsuitable for most purposes in that they are expressed in terms of complex algebraic functions. Normal approximations are derived here for solutions to a variety of reactions. Using these, it is possible to investigate the level at which the classical deterministic solutions become inadequate. This is important in fields such as radioimmunoassay.

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.

The distribution and its moments are given of the length of the intercept made by a random probe through a box having perpendicular edge lengths a, b and c. Three types of random probe are considered.

If a rumour is spread in a population through random contacts between its members, to how many people, on average, will the rumour be told? What can be said about the variations in the number of people who have heard the rumour? Does it matter very much whether a single person or a group of persons starts spreading the rumour? Questions such as these are discussed below, using the framework of a simple stochastic model for snowball sampling.

Asymptotic normality and the weak invariance principle are studied for an iterated renewal process of order A martingale decomposition is used in this context to provide a simple proof.

In this note the joint distribution for the times in an interval [0, t] spent in the states 1, 2, ···, N in a standard quasi-Markov process of order N is considered. An expression for the Laplace transform with respect to t of the Laplace–Stieltjes transform of this joint distribution is derived.

We consider the estimation of the parameter of the best interpolator from a restricted class of linear, time-invariant interpolators for the weakly stationary process {xt}. The estimation is supposed to be based on a part-realisation x1, · ··, xn and on certain prior information regarding the spectral density of the process. Two methods of estimation are proposed — one which uses x1, · ··, xn and the prior information and another which uses only x1, · ··, xn. For the methods proposed it is reasonable to conjecture that the method which utilises the prior information as well as x1, · ··, xn, will be more efficient. Reasonable as this conjecture may seem, its proof is by no means obvious and is the main subject of the paper.

We consider an unordered graph where there is one arc emanating from each node. We suppose that the arc that emanates from i will go to j with probability Pj. The probability that the resultant graph is connected and a recursive formula for the distribution of the number of components it possesses are derived.

Let N, X1, X2, · ·· be non-constant independent random variables with X1, X2, · ·· being identically distributed and N being non-negative and integer-valued. It is shown that the independence of and implies that the Xi's have a Bernoulli distribution and N has a Poisson distribution. Other related characterization results are considered.

Under certain conditions it is shown that for jobs with increasing completion rates the expected flow time is minimised by a non-pre-emptive service discipline.