The Percus–Yevick approximation of classical liquid theory is employed to obtain variances associated with random configurations of equal spheres in ℝ3. The idea is illustrated by considering the number of spheres, and also the amount of hard material, contained within a fixed cylinder — an application of some practical value.

Sufficient conditions are derived for ergodicity of the two-dimensional process of binary variables discussed by Galbraith and Walley (1976).

A model for predicting expected-value population distributions is developed, assuming that all movements are Markovian and time-homogeneous. Each individual is classified by the amount of time he has spent in the population and by which of a number of classes, of an unspecified nature, he inhabits. The limiting properties of the population distribution are derived, and, in particular, conditions for convergence to a stable distribution are given.

Some discussion of the relevance of the theory to practical applications is given, primarily to manpower planning when recruitment occurs purely to maintain a specified overall population size.

A security of supply model for energy is described. The energy demand is a non-random periodic function of time and the supply model is a continuous-time Markov chain. Asymptotic normality is proved for the two important reliability quantities: the amount of energy not supplied and the time the demand is not met within a time interval.

We derive the Laplace transforms of sums and weighted sums of random variables forming a Markov chain whose stationary distribution is gamma. Both seasonal and non-seasonal cases are considered. The results are applied to two problems in stochastic reservoir theory.

This work considers items (e.g. books, files) arranged in an array (e.g. shelf, tape) with N positions and assumes that items are requested according to a Markov chain (possibly, of higher order). After use, the requested item is returned to the leftmost position of the array. Successive applications of the procedure above give rise to a Markov chain on permutations. For equally likely items, the number of requests that makes this Markov chain close to its stationary state is estimated. To achieve that, a coupling argument and the total variation distance are used. Finally, for non-equally likely items and so-called p-correlated requests, the coupling time is presented as a function of the coupling time when requests are independent.

This paper compares estimators of u = |P2 — P1| based on partial knowledge about points P1, P2 which are chosen independently from the same probability distribution in the plane. Typical estimators are the conditional expectations E(u | r1, r2), E(u | s), E(u | rM), E(u | θ) where ri is the distance from Pi to the origin 0, s = r1 + r2, rM = Max{r1, r2}, θ = ∠ P10P2. These expectations are the least squares estimators, given the conditioned variables. Simpler linear estimators are also considered.

In the epidemic with removal with range r, each site z, once infected, remains so for a period of time Tz, the variables Tz being i.i.d. with mean μ. While infected, a site infects its healthy r-neighbours independently at total rate α. After infection, sites become immune. We show that the critical rate of infection αc (r), above which an epidemic starting from a single site may continue forever, converges to μ–1 as r →∞.

We give a new method for simulating the time average steady-state distribution of a continuous-time queueing system, by extending a ‘read-once’ or ‘forward’ version of the coupling from the past (CFTP) algorithm developed for discrete-time Markov chains. We then use this to give a new proof of the ‘Poisson arrivals see time averages’ (PASTA) property, and a new proof for why renewal arrivals see either stochastically smaller or larger congestion than the time average if interarrival times are respectively new better than used in expectation (NBUE) or new worse than used in expectation (NWUE).

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

In this note a characterization for a U-shaped probability density function is given using a well-known result for unimodal distributions due to Khinchine. As a corollary of this result, a characteristic property based on moments is found.

Let (X, Y) = (Xn, Yn)n≥1 be the output process generated by a hidden chain Z = (Zn)n≥1, where Z is a finite-state, aperiodic, time homogeneous, and irreducible Markov chain. Let LCn be the length of the longest common subsequences of X1,..., Xn and Y1,..., Yn. Under a mixing hypothesis, a rate of convergence result is obtained for E[LCn]/n.

This paper considers a fixed-cycle and a semi vehicular-actuated traffic light queue with strictly stationary arrivals and independent and identically distributed departure headways and lost times. These queues are reduced to a generalized model of Loynes and sufficient conditions are derived under which these queues have stationary distributions. Two typical examples of semi vehicular-actuated traffic light queues are discussed.

We obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g. Lp-distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.

The propagation rate of the one-dimensional stochastic simple epidemic converges almost surely to a front velocity for the epidemic. Percolation methods are used in the proof.

This paper studies a discrete-time single-server queue with two independent inputs and service interruptions. One of the inputs to the queue is an independent and identically distributed process. The other is a much more general process and it is not required to be Markov nor is it required to be stationary. The service interruption process is also general and it is not required to be Markov or to be stationary. This paper shows that a stochastic decomposition property for the virtual waiting-time process holds in the discrete-time single-server queue with service interruptions. To the best of the author's knowledge, no stochastic decomposition results for virtual waiting-time processes in non-work-conserving queues, such as queues with service interruptions, have been obtained before and only work-conserving queues have been studied in the literature.

In this paper, we show that the mean comparison theorem, which is valid for Brownian motion, cannot be extended to Poisson processes. A counterexample in the Poisson case for which the mean comparison theorem does not hold is provided.

We investigate default probabilities and default correlations of Merton-type credit portfolio models in stress scenarios where a common risk factor is truncated. For elliptically distributed asset variables, the asymptotic limits of default probabilities and default correlations depend on the max-domain of attraction of the asset variables. In the regularly varying case, we derive an integral representation for multivariate default probabilities, which turn out to be strictly smaller than 1. Default correlations are in (0, 1). In the rapidly varying case, asymptotic multivariate default probabilities are 1 and asymptotic default correlations are 0.