Consider a queueing system in which customers arrive in accordance with a renewal process having an interarrival distribution F, and in which the service times of customers are independent and have distribution G. Moreover, suppose that there are k (k ≦ ∞) servers and that an arriving customer is immediately served if he finds one of the servers free, and if not then he joins the queue.

All sets mentioned in this paper will be assumed to be appropriately measurable without further comment, and events of probability zero will be neglected. We start with a fixed (large, convex) subset R of the plane, termed the “retina”.

The aim of the present note is to give an alternative simpler proof to a result of Belyaev [1], namely that in a loss system of n servers with recurrent input and negative exponential service times the intervals between losses, suitably scaled to have constant mean, tend to a negative exponential distribution as n tends to infinity.

In this note we adopt the notation and terminology of Kingman (1966) without further comment. For the general single server queue one has For the queueing systems GI/D/1 and D/G/1 we shall show that it is possible to make use of the special form of the service time and inter-arrival time distributions, respectively, to evaluate the right hand side of (1). A similar evaluation applies to the limiting distribution when it exists. The results obtained could also be obtained from those of Finch (1969) and Henderson and Finch (1970) by using suitable limiting arguments.

It is often convenient to construct a branching diagram to represent the relationships between languages belonging to the same language family (for example, between Romance languages such as Rumanian, Italian, French and Spanish). It is desirable that the diagram should illustrate both the closeness of the relationships (that is, which pairs of languages have an immediate common ancestor and which are more remotely related) and the relative chronology of the languages. In this paper, the problem of estimating the time since two languages had a common ancestor and were the same is considered. Interval estimates are sought since, in general, consistent point estimates may be impossible; a numerical example is given to illustrate this.

Both paper and author (D. R. McNeil, (1968)) will be referred to below as DRM. The said paper deals with the following situation: an intersection is controlled by a traffic light with a fixed cycle time, T; the possibility of other delays, e.g., due to turning vehicles, is ignored; arrivals at the light form a compound Poisson process; if vehicles arrive to find the light green and the queue empty they are not delayed, while in the contrary case they depart when they reach the head of the queue, providing the light is green, each vehicle taking a constant time to move off. The length of the effective red period is R. For further details and discussion, DRM may be consulted.

The problem of random n-fold coincidences for Geiger counters was considered by Janossy [1] and Schrödinger [2]. In this note we consider a more general problem related to the following model.

For given functions, g(x), h(x), F(x) and a constant, a, f(x) is said to be a solution of the optimal inventory equation if where f(y - z) = f(0) when z > y. (The same convention holds in similar equations below.) A method of deriving a solution to (1) is to define inductively a sequence of approximating functions, fn(x), by:

In this note we present results for the queueing system Ek/G/1 which are analogous to those obtained in Finch (1969) for the queueing system GI/Ek/1. As in that paper we adopt the notation and terminology of Kingman (1966) without further comment.

In an earlier contribution to this Journal, Ten Hoopen and Reuver [5] have studied selective interaction of two independent recurrent processes in connection with the unitary discharges of neuronal spikes. They have assumed that the primary process called excitatory is a stationary renewal point process characterised by the interval distribution ϕ(t). The secondary process called the inhibitory process also consists of a series of events governed by a stationary renewal point process characterised by the interval distribution Ψ(t). Each secondary event annihilates the next primary event. If there are two or more secondary events without a primary event, only one subsequent primary event is deleted. Every undeleted event gives rise to a response. For this reason, undeleted events may be called registered events. Ten Hoopen and Reuver have studied the interval distribution between two successive registered events. As is well-known, the interval distribution does not fully characterise a point process in general and in this case it would be interesting to obtain other statistical features like the moments of the number of undeleted events in a given interval as well as correlations of these events. The object of this short note is to point out that the point process consisting of the undeleted events can be studied directly by the recent techniques of renewal point processes ([1], [3]).

The selective interaction of a stationary point process and a renewal process is studied. Equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.

Two theorems on Markov chains, both of which already appear in the literature: the classification of the states into the set of all non-recurrent (transient) states and recurrent classes, and the corresponding classification for idempotent Markov chains due to Doob, are proved from the viewpoint of ergodic theory.

Recently Rao (1963) has considered discrete models where an original observation produced by nature is subjected to a destructive process and we observe the undestroyed part of the original observation. Suppose the original observation produced by nature is distributed according to a Poisson distribution with parameter λ and the probability that the original observation n is reduced to r due to the destructive process is Now if Y denotes the resulting random variable (r.v.), then it is easily seen that Let us call this condition the *-condition. Later Rao and Rubin ({1964),Theorem 1) proved that the *-condition is a characterizing property of the Poisson distribution.

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.