The solution of a hierarchical n-compartmental homogeneous process with multiple-valued migration rates is obtained using Laplace transform techniques. Such models are suggested for modelling the infection process of endemic diseases because of their property of gamma-distributed compartmental residence times in each of n compartments. Its relevance to the onchocerciasis infection process (with and without the imposition of controls) is discussed and application to age-prevalence data (having only two infective states into which an individual can be classified) is undertaken.

A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered.

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.

Solutions of the Burgers equation with a stationary (spatially) stochastic initial condition are considered. A class of limit laws for the solution which correspond to a scale renormalization is considered.

We obtain a sufficient condition for the transience of a Markov chain, and a sufficient condition for its null recurrence. These are applied to characterize the stability of a multiple-access communication system. Performance bounds for the system are also obtained.

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density.

In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

As has been shown by de Maré, in a stationary Gaussian process the length of the successive zero-crossing intervals cannot be independent, except for the degenerate case of a pure cosine process. However, no closed-form expression of the distribution of these quantities is known at present. In this paper we present an accurate explicit approximative formula, derived by replacing the Slepian model process by its regression curve.

We consider the extension of optimal stopping problems to non-zero-sum strategic settings called stopping games. By imposing a monotone structure on the pay-offs of the game we establish the existence of a Nash equilibrium in non-randomized stopping times. As a corollary, we identify a class of games for which there are Nash equilibria in myopic stopping times. These games satisfy the strategic equivalent of the classical ‘monotone case' assumptions of the optimal stopping problem.

Non-randomness that arises in the shuffling of multiple numbers of decks of playing cards is modeled. An efficient way to exploit the non-randomness is derived, and its effect on the win rate for a simple game of chance is derived.

Many mean-value quantities of stationary random tessellations can be expressed in terms of three fundamental mean-value quantities. In this note we characterize the set of triples of mean values that can be realized, and show that every possible triple can arise from a suitable ergodic stationary isotropic tessellation.

Arcs of length lk, 0 < lk < 1, k = 1, 2, ···, n, are thrown independently and uniformly on a circumference having unit length. Let P(l1, l2, · ··, ln) be the probability that is completely covered by the n random arcs. We show that P(l1, l2,· ··, ln) is a Schur-convex function and that it is convex in each argument when the others are held fixed.

n jobs are to be preemptively scheduled for processing on n machines. The machines may have differing speeds and the jobs have processing requirements which are distributed as independent exponential random variables with different means. Holding cost g(U) is incurred per unit time that the set of uncompleted jobs is U and it is desired to minimize the total expected holding cost which is incurred until all jobs are complete. We show that if g satisfies certain simple conditions then the optimal policy is one which takes the jobs in the order 1, 2, ···, n and assigns each uncompleted job in turn to the fastest available machine. In the special case in which the objective is to minimize the expected weighted flowtime, where there is a holding cost of wi while job i is incomplete, the sufficient condition is simply w1 ≧ … ≧ wn and λ1 w1 ≧ … ≧ λn wn.

A general shock model in which the time intervals between shocks have infinite expectation is considered. Limit theorems for the first time the magnitude of a shock exceeds a and the historical maximum magnitude are given.

The long-run average cost per unit time of operating a finite dam controlled by a policy (Lam Yeh (1985)) is determined when the cumulative input process is the integral of a Markov chain. A penalty cost which accrues continuously at a rate g(X(t)), where g is a bounded measurable function of the content, is also introduced. An example where the input rate is a two-state Markov chain is considered in detail to illustrate the computations.

Multiple channel queues with Poisson arrivals, exponential service distributions, and finite capacity are studied. A customer who finds the system at capacity either leaves the system for ever or may return to try again after an exponentially distributed time. Steady state probabilities are approximated by assuming that the returning customers see time averages. The approximation is shown to result in an upper bound on system performance.

In this paper we investigate the server's busy period in a single-server queueing situation in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A closed-form expression is obtained for the Laplace transform bn(z) of the joint probability and probability density function of the busy period duration and the number of customers served in it. Some numerical values are given showing the effect of correlation between T and S.

In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi : i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti : i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.