We consider a fork-join network with arrival and service times forming a stationary and ergodic process. The usual stability condition, namely that the input rate is strictly less than all the service rates, is proved to be valid in this general case. Finally we extend the result to the case where there is random routing.

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk}and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

Let {Zφ (t)} be a multitype finite-mean supercritical (CMJ) process with general characteristic φ and malthusian parameter α. It is shown that there exist some constants {c(t} such that {c–1(t)Zφ (t)} converges almost surely to a non-degenerate limit Wφ as t →∞. Characterizations of {c(t)} and Wφ are also given.

Suppose the payoffs (aij) in the n × n matrix A are drawn independently from some continuous probability distribution. The number of tactics used in an ESS is investigated. Asymptotic results on the size of the ESS with largest support are given, using the work of Karlin and Kingman on the size of polymorphisms in one-locus multi-allele diploid selection models.

Equations are derived for the distribution of the busy period of the GI/G/2 queue. The equations are analyzed for the M/G/2 queue, assuming that the service times have a density which is an arbitrary linear combination, with respect to both the number of stages and the rate parameter, of Erlang densities. The coefficients may be negative. Special cases and examples are studied.

It is shown that, in a renewal process with inter-arrival distribution F, an observation from the asymptotic (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and W are independent random variables with U ~ uniform(0, 1) and W distributed according to the length-biased version of F.

Realization probability is a new concept pertaining to perturbation analysis of closed queuing networks. The sensitivities of throughputs in a closed single-class Jackson network can be expressed in terms of realization probabilities. In this paper, based on a discussion of perturbation analysis for networks with state-dependent service rates, we derive some new formulas for sensitivities of throughputs using realization probability.

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.

This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

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.

A population process {Zn} is defined, with offspring distribution depending on both population size and generation number. As is well known, in models where the offspring distribution depends only on the size, under most conditions extinction surely results. The model we now introduce would mean that if history has an influence on evolution of the population, it is possible to obtain models for which a probability as high as desired exists that the population size balances between two given values.

This paper is concerned with the problem of controlling a simple immigration–birth process, which represents a pest population, by the introduction of catastrophes which, when they occur, reduce the population size to zero. The optimality criterion is that of minimising the long-term average cost per unit time of the process. Firstly, an optimal policy is found within a restricted class of stationary policies, which introduce catastrophes if and only if the population size is greater than or equal to some critical value x. The optimality of this policy within the wider class of all stationary policies is then verified by applying the general results of Bather (1976).

We consider a supercritical, p-dimensional Markov branching process (MBP). Based on the finite and the infinite lines of descent of particles of this p-dimensional MBP, we construct an associated 2p-dimensional process. We show that such a process is a 2p-dimensional, supercritical MBP. This 2p-dimensional process retains the branching property when conditioned on the sets of extinction and non-extinction. Asymptotic results and central limit theorems for the associated process and the original process are established by using the results of Gadag and Rajarshi (1987).

We develop a representation for general queueing networks, and study some stochastic monotonicity properties that have been previously established in Jackson networks (e.g. Shanthikumar and Yao (1986), (1987)).

In this paper we study the maximum of Gaussian Fourier series emerging in the analysis of random vibrations of a finite string. Evaluating the distribution of the maximal displacement corresponds to the analysis of the maximum for the related Gaussian Fourier series with independent coefficients.

When vibrations are triggered by an initial white noise disturbance (where the instantaneous form of the vibrating string is composed of three processes pieced together) we give upper bounds for the maximal displacement.

In the last section we consider forced vibrations at special instants where the instantaneous form of the vibrating string has the structure of a Brownian bridge. This enables us to give the exact distribution of the maximum for the related sine series.

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

In this paper a bisexual Moran model is introduced. The population consists of N pairs of individuals. At times t = 1, 2, ·· ·two individuals are born, who ‘choose their parents randomly' and independently of each other. Then one of the pairs is removed and replaced by the two individuals born at that instant.

The extinction probability of the descendants of a single pair and the number of ancestors of a whole generation are studied. A limit result for large population sizes has been derived by diffusion approximation methods.

The superposition of independent, discrete, renewal processes produces a counting process which is also a discrete time series. The conditional distribution and correlation structure of this kind of time series may be obtained. In suitable conditions the conditional distribution has a spectrum which is exactly or approximately rational. When this is so, an ARMA can be found which matches the spectrum of the superposition.