This paper derives an expression for the correlation function of a continuous parameter branching process, and obtains conditions for the convergence of this function to a zero or positive value.

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

In his paper ‘Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test’ (J. Appl. Prob.8, 431–453), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.

An infinite capacity dam subject to an additive input process and a general release rule is considered. The input process can have infinitely many jumps in any finite interval, and the rate of release is r(x) when the dam content is x. The content is constructed as the increasing limit of a sequence of Markov processes and the convergence is shown to be almost surely uniform over finite intervals. The limit process is shown to be a standard Markov process and its characteristic equation is computed.

This paper aims at showing that for the discrete time analogue of the M/G/l queueing model with service in random order and with a traffic intensity ρ > 0, the condition ρ < ∞ is sufficient in order that every customer joining the queue be served eventually, with probability one (Theorem 2).

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits: where Xj(t) is the amount of time prior to time t spent in state j.

The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.

It has been demonstrated by Takács in a series of papers and in his book that combinatorial methods can be successfully applied to derive certain probability distributions in queueing processes. In this paper, we further illustrate the usefulness of combinatorial techniques and determine the stochastic law of the busy period in two queueing systems particularly involving batches. It may be of interest to note that queues involving batches have been dealt with in [3] and [7].

It is proved that in a GI/G/s queue with the traffic intensity ρ < (S – m + l)/S, m ≧ 1, at least m servers are simultaneously idle infinitely often with probability one.

For a particular kind of finite-server loss system in which the number and identity of servers depends on the type of the arriving call and on the state of the system, the limits of the state probabilities (as t → ∞) are found for an arbitrary service-time distribution.

The higher-order transition probability generating functions for a random-walk with correlation between steps is calculated as a discrete-domain Green's function.

A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.

The paper develops the steady-state solution of a finite space queueing system wherein each of the two non-serial servers is separately in series with two non-serial servers. It is assumed that the arriving units of the same type may demand a different number of service phases. Poisson arrivals and exponential service times are assumed at all the four channels of the system. Service of units is completed on a first-come, first-served basis at each channel. The steady-state solution for infinite queueing space is obtained as a special case of finite queueing space.

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.

Determination of the limiting distributions for a class of mixed-type stochastic processes with state-dependent rates of decline is reduced to the solution of a class of integral equations. For the case where the rate of decline is proportional to the state, some results are obtained by solving the integral equation of the process through Fuchs' method.

The purpose of this note is to extend results on critical Galton-Watson branching processes, due to Feller and Lamperti: if the number of individuals at time zero tends to infinity, the process converges, with two different normalizations, to certain diffusion processes in the functional form, discussed in, e.g., Billingsley.

In this note a characterization for a U-shaped probability density function is given using a well-known result for unimodal distributions due to Khinchine. As a corollary of this result, a characteristic property based on moments is found.

In a recent paper (Downton (1972)) an attempt was made to use the combinatorial approach of Downton (1967) to obtain an expression for the Laplace transform (and hence the moments) of the area under the infectives trajectory of the general stochastic epidemic. The basic property of that area, which makes this possible, is that it is composed of a random number of rectangles, each of whose area has a distribution depending on the infection rate and the number of susceptibles present at the appropriate stage in the development of the epidemic, but which is independent of the number of infectives. However, Mr A. Abakuks has pointed out that in one respect the argument used was faulty. It was argued that the contribution to the area under the infectives curve of an interval ending in a reduction in the number of infectives from i to i – 1 had an exponential distribution with parameter p (the infection rate); and if the interval ended in a reduction in the number of susceptibles from s to s – 1 the distribution was exponential with parameter s. While this is true unconditionally, the combinatorial situation described was essentially a conditional one, in which each of these contributions to the total area had an exponential distribution, independent of i, but with parameter (ρ + s) regardless of whether the interval ended in a reduction in the number of infectives or of susceptibles.