A recurrence process is the process of forward recurrence times from the points of one renewal process to the next subsequent event in a second independent renewal process. Univariate and multivariate recurrence processes are examined. Exponential recurrence processes are investigated in greater detail.

Consider a network of nodes (switches) and connecting links. Each link consists of a group of channels (trunks). A call instantaneously seizes channels along a route between the originating and terminating node, holds them for a randomly distributed length of time and frees them instantaneously at the end of the call. If no channels are available, the call is blocked. For special networks with exponential call holding times, Erlang has shown that the steady-state probabilities are in product form. In this paper, we extend this work to general networks and show that if for each pair of nodes there is a unique route, then the blocking probabilities are in product form and are insensitive to the call holding-time distribution, which means that they depend on the call duration only through its mean.

Using a result of Bikjalis (1971) concerning the rate of convergence in the multidimensional central limit theorem we obtain informations about some limit distributions in multivariate renewal theory.

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.

We generalize Jackson's theorem to the non-ergodic case. Here, despite the fact that the entire Jackson network will not achieve steady state, it is still possible to determine the maximal subnetwork that does. We do so by formulating and algorithmically solving a new non-linear throughput equation. These results, together with the ergodic results and the ones for closed networks, completely characterize the large-time behavior of any Jackson network.

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

We consider degenerate limit laws for the sequence {Xn, n}n(N of successive maxima of identically distributed random variables. It turns out that the concentration of Xn, n for large n can be determined in terms of a tail ratio of the underlying distribution function F. Applications to the outlier-behaviour of probability distributions are given.

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

This note shows how the initial density of the volume of a random simplex in a body, K, depends upon the fourth moment of sectional area and so rejects the conjecture that this initial density is independent of K.

This paper introduces a property of entropies called polynomial additivity, which includes the additivity properties of the Shannon entropy and of entropies of all degrees. The forms of all such entropies with the sum property and range of infinite cardinality are found. Of these, the only ones with the measurable sum property are affine transformations of the Shannon entropy, the entropies of all degrees, and the length of the distribution.

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.

Upper and lower bounds are obtained for the integral giving the probability that n invariant chords to a circle all intersect each other.

A random mapping (T,Pj) of a finite set V into itself is studied. We give a new proof of the fundamental lemma of [6]. Our method leads to the derivation of several results which cannot be deduced from [6]. In particular we determine the distribution of the number of components, cyclical points and ancestors of a given point.

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S′−x)+ ≦ E(S″−x)+ (all x >0), are stochastically ordered as W′≦dW″. The weaker conclusion, that E(W′−x)+ ≦ E(W″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T′)+ ≦ E(x−T″)+ (all x). A sufficient condition for wk≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

The similarity between order statistics and record values motivated us, in this paper, to investigate some relationships between the order statistics and the record values. These relationships are employed to characterize a continuous distribution by some moment properties of the spacings of the record values, and hence obtain characterizations of the exponential distribution. Some of the well-known results follow trivially.

It is known that for a GI/G/1 queue, if the interarrival time has finite first moment, then the (k + 1)th moment of the service time is finite if the kth moment of the stationary workload is finite. This result is extended to certain tandem and multiple-server queues.

We extend recent results of Schuh on the convergence of , where α ≧ 0 and Sn is the sum of n i.i.d. positive random variables to sums of the form for a large class of functions g, give simpler proofs than those of Schuh, and derive reformulations of the explosion criteria for Markov branching processes with discrete and continuous state space.

In this paper we study a series of servers with exponentially distributed service times. We find that the sojourn time of a customer at any server depends on the customer's past history only through the customer's interarrival time to that server. A method of calculating the conditional probabilities of sojourn times is developed.

This paper considers a problem of determining the optimal number of units for a parallel redundant system. The optimal number N∗ to minimize the expected cost is given by a unique solution of equations. When the system is replaced before failure at time T, the optimal N∗ and T∗ are discussed, and a computing procedure for obtaining these values is specified.