Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.

Suppose that a certain population consists of N individuals. One member initially learns a rumour from an outside source, and starts telling it to other members, who continue spreading the information. A knower becomes inactive once he encounters somebody already informed. Daley and Kendall, who initiated the study of this model, conjectured that the number of eventual knowers is asymptotically normal with mean and variance linear in N. Our purpose is to confirm this conjecture.

We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index α and that E{Yα+ε} < ∞ for some ε > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.

An elegant time-dependent solution for the number in the M/M/c queueing system is derived in a direct way.

We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes 𝒱 for the items, and the allowed matchings depend on the classes, according to a matching graph on 𝒱. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is independent and identically distributed, the sequence of buffer-content is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is nonbipartite.

Consider first a Markov chain with two ergodic states E1 and E2, and discrete time parameter set {0, 1, 2, ···, n}. Define the random variables Z0, Z1, Z2, ···, Znby then the conditional probabilities for k = 1,2,···, n, are independent of k. Thus the matrix of transition probabilities is

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuously depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s, S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique that can be used to derive various further identities.

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

This paper deals with the recurrence of branching random walks on polynomially growing graphs. Amongst other things, we demonstrate the strong recurrence of tree indexed random walks determined by the resistance properties of spherically symmetric graphs. Several branching walk models are considered to show how the branching mechanism influences the recurrence behaviour.

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.

Let {X(t), V(t), t ≥ 0} be a telegraph process, with V(0+) = 1. The distribution of X(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levels a and −a are studied for M/G and for G/M processes.

Let X(t), – ∞ < t < ∞, be a stationary time series with mean cx. Let 0 < τ1 < τ2 < … < τN ≦ T denote A given sampling times in the interval (0, T]. We determine the asymptotic distribution of the estimate [X(τ1) + … + X(τN)]/N of cx when the sampling times are fixed, satisfying a form of generalised harmonic analysis requirement, and when the sampling times are the times of events of a stationary point process independent of the series X(t). The results obtained may be viewed as non-standard central limit theorems.

A family of random variables {X(θ)} parameterized by the parameter θ satisfies stochastic convexity (SCX) if and only if for any increasing and convex function f(x), Ef[X(θ)] is convex in θ. This definition, however, has a major drawback for the lack of certain important closure properties. In this paper we establish the notion of strong stochastic convexity (SSCX), which implies SCX. We demonstrate that SSCX is a property enjoyed by a wide range of random variables. We also show that SSCX is preserved under random mixture, random summation, and any increasing and convex operations that are applied to a set of independent random variables. These closure properties greatly facilitate the study of parametric convexity of many stochastic systems. Applications to GI/G/1 queues, tandem and cyclic queues, and tree-like networks are discussed. We also demonstrate the application of SSCX in bounding the performance of certain systems.

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

We consider an N-server queue with arbitrary arrivals and service times which are random but with differing rates for different servers. Customers arriving when all servers are occupied do not enter the system. We show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system. We then show that in a particular component-repair context with exponential repair times the policy of repairing failed components with the smallest failure rate stochastically maximises the number of working components.

The asymptotic behaviour of a superadditive bisexual Galton–Watson branching process is studied. Sufficient conditions for the almost sure and L1 convergence of the suitably normed process are given. Finally, a first approach to the study of the L1 convergence for a superadditive bisexual Galton–Watson branching process under the Z log+Z condition is considered.

The paper characterizes point processes with the order statistic property without the unnecessary condition of finiteness of the first moment of the process, a condition imposed by previous researchers. It shows that the class of these processes is composed only of mixed Poisson processes up to a time-scale transformation and of the mixed sample processes. It also introduces a multivariate analog of the order statistic property and characterizes completely the class of multivariate point processes with this property.

The purpose of this paper is to study two notions of stochastic comparisons of non-negative random variables via ratios that are determined by their Laplace transforms. Some interpretations of the new orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory are described.