This paper is an extension of Davis (1965) by allowing immigration. Mean square convergence is proved for a random variable in a branching diffusion process allowing immigration.

A spatial epidemic process where the individuals are located at positions in the Euclidean space R2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R2 according to a Brownian motion with a diffusion coefficient σ2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

A stationary doubly stochastic Poisson sequence is considered. Asymptotic properties of the best linear estimates of the intensities are investigated.

We establish the moderate deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, we derive the moderate deviation principle for two important population models: super-Brownian motion and the Fleming–Viot process.

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. We use comparison relationships between networks of this class with independent and identically distributed driving sequences and the GI/GI/1/1 queue to obtain the tail asymptotics of the stationary maximal dater under light-tailed assumptions for service times. The exponential rate of decay is given as a function of a logarithmic moment generating function. We exemplify an explicit computation of this rate for the case of queues in tandem under various stochastic assumptions.

The lifelengths of components of a system are usually dependent due to the common random production and operating environments. In this paper, we introduce a multi-variate pure jump Markov process to describe a large class of damage processes on various system components driven by common environmental shocks, and establish some dependence properties (association) for such a process and its multivariate increment process. These strong association properties describe both spatial dependence and temporal dependence of a multivariate pure jump process, and also provide a vehicle to derive some structural properties of component lifelengths of the systems operating in such an environment. Some bounds for the joint survival functions of component lifelengths are also obtained.

Given a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

Examples of non-stationary random set processes are presented. The reconstruction of a pattern, generated by such a process, is studied; a particular arrangement of the sample-points is considered.

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

A large number n of sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional to n. In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections, and weak convergence, we investigate asymptotic behaviour of optimal costs as n increases to ∞. The introduced methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.

We consider a stochastic graph generated by a continuous-time birth-and-death process with exponentially distributed waiting times. The vertices are the living particles, directed edges go from mothers to daughters. The size and the structure of the connected components are investigated. Furthermore, the number of connected components is determined.

The equilibrium probability distribution over the set of absorbing states of a reducible Markov chain is specified a priori and it is required to obtain the constrained sub-space or feasible region for all possible initial probability distributions over the set of transient states. This is called the inverse problem. It is shown that a feasible region exists for the choice of equilibrium distribution. Two different cases are studied: Case I, where the number of transient states exceeds that of the absorbing states and Case II, the converse. The approach is via the use of generalised inverses and numerical examples are given.

In this paper, we present a perturbation method of solving a second-order difference equation with variable coefficients with some additional assumptions. This method can be used to devise an algorithmic solution for the moments of sojourn times in some processor sharing queues. In particular, we examine three queues. The first has exponential service and a fairly general interrupted arrival mechanism. The second is a cyclic queue. The third is a model for a computer system in which finite and infinite sources interact.

Cottrell et al. (1983) have indicated how ideas from the large deviations theory lead to fast simulation schemes that estimate the mean time taken by the slotted ALOHA protocol to saturate starting empty. Such fast simulation schemes are particularly useful when the attempt probability is small. The remaining time to saturation when the protocol has been operating for a time is more accurately described by the quasi-stationary exit time from the stable regime. The purpose of this article is to prove that the ratio of the quasi-stationary exit time to the exit time starting empty approaches 1 as the attempt probability becomes small.

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) < α for all y < x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.

It is shown that the two descriptions of the ages of alleles corresponding to the two formulations of the stationary infinitely-many-neutral-alleles diffusion model discussed by Ethier (1990a) are equivalent.

Expressions for the mean and variance of the recurrence time of non-overlapping draft-patterns of draft from a Moran Reservoir Model (discrete-state and discrete-time Markov chain) are derived using Feller's Renewal argument. In addition an expression for the mean recurrence time for self-overlapping patterns of draft is derived using run-theory.

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

In [1], [4] and [6] simple and intuitively obvious bounds were suggested for various overflow systems and claimed to be insensitive, that is to be valid also in the non-exponential case. This insensitivity was formalized in [4] and [6] when the service distributions have a monotone failure rate. This technical note provides another proof which for the case of a pure overflow system with one customer class extends the insensitivity to the general case.