A generalization of Downton's (1968) carrier-borne epidemic process is considered in which the assumption of homogeneous mixing is replaced by a general infection mechanism. Using a method and notation similar to that of Gani and Purdue (1984), a recursive algorithm for the joint distribution of the total size and the total observed size of the epidemic is obtained.

Let {X n , n = 0, 1,…} be the sequence of the lower records for an arbitrary underlying distribution μ on [0, ∞). We show that is equal in distribution to where {τ i , i = 1, 2,…} is a Poisson flow of unit intensity and g is a right-continuous and nonincreasing function defined by μ. This observation allows us to extend results of Bose et al. and simplify their proofs.

We consider an infinite server queueing system. An examination of sample path dynamics allows a straightforward development of integral equations having solutions that give time-dependent occupancy (number of customers) and backlog (unfinished work) distributions (conditioned on the time of the first arrival) for the GI/G/∞ queue. These integral equations are amenable to numerical evaluation and can be generalized to characterize GIX/G/∞ queue. Two examples are given to illustrate the results.

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

We derive an integral expression for the expected coverage of an n-dimensional target when the value density function of the target, the damage function, and the distribution of the damage function center around the target center are spherically symmetric. This integral expression is transformed by means of Parseval's theorem for Hankel transforms to an integral of the product of the Hankel transforms of order ½n − 1 of all underlying functions.

We analyse the expected performance of various group testing, or pooling, designs. The context is that of identifying characterized clones in a large collection of clones. Here we choose as performance criterion the expected number of unresolved ‘negative’ clones, and we aim to minimize this quantity. Technically, long inclusion–exclusion summations are encountered which, aside from being computationally demanding, give little inkling of the qualitative effect of parametric control on the pooling strategy. We show that readily-interpreted re-summation can be performed, leading to asymptotic forms and systematic corrections. We apply our results to randomized designs, illustrating how they might be implemented for approximating combinatorial formulae.

A method is provided for numerical evaluation, with any given accuracy, of the probability that at least p% of the genetic material from an individual's chromosomal segment survives to the next generation. Relevant MAPLE® V codes, for automated implementation of such evaluation, are also provided. The genomic continuum model, with Haldane's model for the crossover process, is assumed.

Several stochastic models have been proposed to describe the kinetic theory of reversible chemical reactions. However, in macroscopic systems the effects of stochastic variability are often outweighed by mean effects. In the present paper we show that some observed phenomena can be explained quite adequately by a stochastic model in which the stochastic variability is not negligible in comparison with mean effects. Our argument involves approximations to a stochastic model for competing chemical reactions.

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

Cars entering and leaving a parking place in a forest are registered automatically in connection with an investigation of forest recreational use. By subtraction the number of cars parked is computed. Due to erroneous and missing counts, errors occur and may for instance indicate a negative number. The theory of incompletely observed Markov chains is used to obtain an estimate of the number of cars parked.

It is proved that, in a non-homogeneous Poisson cluster process whose cluster centre rate is λ(t), the long-term behaviour of several associated quantities is equivalent to their behaviour in a homogeneous process with centre rate Λ and the same cluster structure if λ(t)→ λ as t→∞.

This is a discussion of the place of empirical methods in the study of probability models. It is argued that, for models of great complexity, careful experimentation may be a legitimate source of factual information on the behavior of such models. The need for rigorous standards of empirical methodology and of appropriate reporting is stressed. A simple, but interesting question on the distribution of the maximal eigenvalue of positive matrices with random elements is used as an example of an experimental study. In the course of the discussion of that example, several operational rules of sound computer experimentation are stated.

The object of this paper is to present a probabilistic model for analyzing changes through time in a binary dyadic relation on a finite set of points. The total relation on the set takes the form of an aggregate of directed binary dyadic relations between ordered pairs of points belonging to the set; equivalently, the total relation on the set can be represented by means of a digraph or an incidence matrix isomorphic with the total relation. Such a relation, changing in its structure as time proceeds, is a reasonable mathematical model, for example, for the evolution of inter-relationships of the members of a social or any other group. A group of this kind is organized for a specific activity involving some sort of “communication” from one member to the other, and may be observed at successive discrete points in time generating statistics on the evolutionary process. (For a detailed treatment, see [3].) As a matter of fact, under suitable assumptions, the model presented here has potentialities for application in those situations which can be represented mathematically in terms of a finite set of points and an all-or-none relationship between ordered pairs of these points. Some of the other examples are communication networks, ecology, animal sociology, and management sciences (see [5]).

The reversed strict ascending ladder epoch v1 + ··· +vm of the random walk S(n) with drift to ∞ is the m th time n at which the event S(n) < S(k), k > n, occurs and the corresponding ladder height is H1 + ··· + Hm = S(v1 + ··· + vm). It is shown that the random vectors (vi, Hi) are independent and for i ≧ 2 have the same distribution as the first strict ascending ladder time and height in the usual sense. This leads to an equality between the distribution of the first strict ladder height and P{min S(n) ≧ x} for x > 0. Reversed weak ladder points are defined analogously.

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.

For some three-dimensional random walks on the cubic lattice, the probability of the walk returning to its starting point is given numerically.

This paper presents new methods for selecting the best population value under the classical secretary problem sequential selection rules. Unlike the classical case, the possible population arrival sequences considered here are not necessarily equally likely. When arrival orders in the sequence are rank dependent, the performance of the optimal solution to the standard problem (totally random arrival order assumed) is very poor, compared with procedures designed to guard against non-uniform arrival orders.

The two-dimensional periodogram has been proposed as an estimator of the spectral density of a real, homogeneous, random field defined over a regular lattice on the plane. In the present paper, results pertaining to the asymptotic distributional properties of such a periodogram are obtained. These results generalize some of the work of Hannan (1960), Walker (1965) and more directly Olshen (1967a,b) concerning asymptotic theory for the periodogram of a stationary time series. Although extension of asymptotic theory for one-dimensional periodograms to a parallel theory for two-dimensional periodograms is not completely straightforward (one runs into problems akin to the problems encountered in extending the theory of one-dimensional trigonometric series to two dimensions), further extensions to asymptotic theory for p-dimensional periodograms (p > 2) is easily accomplished by an obvious mimicry of the definitions, theorems and proofs for the two-dimensional case. Since the notation required for the p-dimensional case is rather cumbersome, we have chosen to give results only for two-dimensional periodograms.