An extended version, called collective, of the randomized Reed-Frost processes is considered where each infective during his survival time fails to transmit the infection within any given set of susceptibles with a probability depending only on the size of that set. Our purpose is to provide a unified analysis of the distribution of the final size and severity, the two main components of the cost generated by the infection process. The method developed relies on the construction of a family of martingales and the use of a family of polynomials studied recently by the authors (Lefèvre and Picard (1990)). The results generalize a number of earlier ones and are derived in a more direct and systematic way than before.

This paper presents a simple model of the decision problems which face many animals in the wild. Animals have an upper limit to the amount of energy reserves which can be stored as body fat. They constantly use energy and find food as a stochastic process. Death occurs if energy reserves fall to zero or if the animal is taken by a predator. Typically animals have a range of behavioural options which differ in the distribution of food gained and in the associated predation risk. There is often a trade-off between starvation and predation in that animals have to expose themselves to higher predation risks in order to gain more food.

The above situation is modelled as a finite-state, finite-time-horizon Markov decision problem. The policy which maximises long-term survival probability is characterised. As special cases two trade-offs are analysed. It is shown that an animal should take fewer risks in terms of predation as its reserves increase, and that an animal should reduce the variability of its food supply at the expense of its mean gain as its reserves increase.

Let (Y(t), t > 0) be a d-dimensional non-homogeneous multivariate extremal process. We suppose the ith component of Y describes time-dependent behaviour of random utilities associated with the ith choice. At time t we choose the ith alternative if the ith component of Y(t) is the largest of all the components. Let J(t) be the index of the largest component at time t so J has range {1, …, d} and call {J(t)} the leader process. Let Z(t) be the value of the largest component at time t. Then the bivariate process (J(t), Z(t)} is Markov. We discuss when J(t) and Z(t) are independent, when {J(s), 0<s≦t} and Z(t) are independent and when J(t) and {Z(s), 0<s≦t} are independent. In usual circumstances, {J(t)} is Markov and particular properties are given when the underlying distribution is max-stable. In the max-stable time-homogeneous case, {J(et)} is a stationary Markov chain with stationary transition probabilities.

Most theoretical studies of image processing employ discrete image models. While that might be a good approximation to digital analysis, it severely restricts the class of tractable models for the blur component of image degradation, and concentrates excessive attention on specialized features of the pixel lattice. It is analogous to modelling all real statistical data using discrete distributions, which is clearly unnecessary. In this paper we study a continuous model for image analysis, in the presence of systematic degradation via a point spread function and stochastic degradation by a second-order stationary random field. Thus, we depart from the restrictive white-noise models which are commonly used in the theory of image analysis. We establish a general result which describes the performance of optimal image processing methods when the noise process has short-range dependence. Concise limits to resolution are derived, depending on image type, point spread function and noise correlation. These results are developed in important special cases, giving explicit formulae for optimal smoothing sets and convergence rates.

The approximation of sums of independent random variables by compound Poisson distributions with respect to stop-loss distances is investigated. These distances are motivated by risk-theoretic considerations. In contrast to the usual construction of approximating compound Poisson distributions, the method suggested in this paper is to fit several moments. For two moments, this can be achieved by scale transformations. It is shown that the new approximations are more stable and improve the usual approximations by accompanying laws in examples where the probability 1 – pi that the ith summand is zero is not too large.

This paper deals with the thrown string problem posed by Synge [12]. The models studied are those of Willenborg [14] and Kingman [7]. The convergence results obtained are complementary to those of Kingman, although they are derived by quite different methods involving the use of martingales.

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

The problem of optimally allocating partially effective, defensive weapons against randomly arriving enemy aircraft so that a bomber maximizes its probability of reaching its designated target is considered in the usual continuous-time context, and in a discrete-time context. The problem becomes that of determining the optimal number of missiles K(n, t) to use against an enemy aircraft encountered at time (distance) t away from the target when n is the number of remaining weapons (missiles) in the bomber's arsenal. Various questions associated with the properties of the function K are explored including the long-standing, unproven conjecture that it is a non-decreasing function of its first variable.

Reversible spatial birth-death processes are studied with simultaneous jumps of multi-components. A relationship is established between (i) a product-form solution, (ii) a partial symmetry condition on the jump rates and (iii) a solution of a deterministic concentration equation. Applications studied are reversible networks of queues with batch services and blocking and clustering processes such as those found in polymerization chemistry. As illustrated by examples, known results are hereby unified and extended. An expectation interpretation of the transition rates is included.

For the queue M/G/1 under the discipline SRPT (shortest remaining processing time) the system state is taken to be the counting measure N which assigns to each Borel set A of R+ the number N(A) of customers present with residual service times taking values in A. A steady-state analysis is given for the corresponding Laplace functional. As a corollary, the steady-state number in queue is obtained in terms of its generating function.

Birth processes with piecewise linear birth rates are analysed, and numerical results suggest that, relative to the linear case, convex birth rates increase variability and concave birth rates decrease variability.

A Fourier series is obtained for the variance of the number of points lying in a parallelogram thrown randomly on to a square lattice at a fixed angle.

A constant regression of the quotient on the sum of two i.i.d. non-degenerate positive random variables is a characteristic property of the gamma distribution.

A consecutive-k-out-of-n: F system consists of n components ordered on a line. Each component, and the system as a whole, has two states: it is either functional or failed. The system will fail if and only if at least k consecutive components fail. The components are not necessarily equal and we assume that components' failures are stochastically independent. Using a result of Barbour and Eagleson (1984) we find a bound for the distance of the distribution of system's lifetime from the Weibull distribution. Subsequently, using this bound limit theorems are derived under quite general conditions.

In this paper, we study a similar replacement model in which the successive survival times of the system form a process with non-increasing means, whereas the consecutive repair times after failure constitute a process with non-decreasing means. The system is replaced at the time of the Nth failure since the installation or last replacement. Based on the long-run average cost per unit time, we determine the optimal replacement policy N∗ and the maximum of the long-run average reward explicitly. Under additional conditions, the policy N∗ is even optimal among all replacement policies.

The flexibility of a notion considered by Lindley and Singpurwalla is pointed out. It is shown that their set-up can be generalized by looking at systems whose component life lengths are a priori dependent.

We study the relative errors in reliability measures for the series system, under the wrong assumption of independence incorporating environmental effects, when the Marshall and Olkin model with environmental effects should be used.

We show that a theory for stopped two-dimensional random walks is well suited to describe cumulative shock models. Limit theorems for the lifetime/failure time of a system are provided.

Singh and Jain (1989) have proved some preservation results for partial orderings of life distributions assuming that shocks occur according to a homogeneous Poisson process. It is shown that their results hold under less restrictive conditions.