In this paper we present a simple combinatorial approach for the derivation of zero-avoiding transition probabilities in a Markovian r-node series Jackson network. The method we propose offers two advantages: first, it is conceptually simple because it is based on transition counts between the nodes and does not require a tensor representation of the network. Second, the method provides us with a very efficient technique for numerical computation of zero-avoiding transition probabilities.

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

We study the buffer allocation problem in a two-stage cyclic queueing system. First, we show that transposing the number of buffers assigned to each queue does not affect the throughput. Second, we prove that the optimal buffer allocation scheme, in the sense of maximizing the system's throughput, is the one for which the absolute difference between the number of buffers, assigned to each queue, is minimized, i.e., it becomes either 0 or 1. This optimal allocation is insensitive to the general-type service-time distributions. These two distributions may be different and service times may even be correlated.

In this note, the strong limiting behaviour of busy periods in GI/G/1 queues is studied under the condition that the traffic intensity equals unity.

We show that for a large class of one-dimensional interacting particle systems, with a finite initial configuration, any limit measure , for a sequence of times tending to infinity, must be invariant. This result is used to show that the one-dimensional biased annihilating branching process with parameter > 1/3 converges in distribution to the upper invariant measure provided its initial configuration is almost surely finite and non-null.

The generalized queueing networks (G-networks) which we introduce in this paper contain customers and signals. Both customers and signals can be exogenous, or can be obtained by a Markovian movement of a customer from one queue to another after service transforming itself into a signal or remaining a customer. A signal entering a queue forces a customer to move instantaneously to another queue according to a Markovian routing rule, or to leave the network, while customers request service. This synchronised or triggered motion is useful in representing the effect of tokens in Petri nets, in modelling systems in which customers and work can be instantaneously moved from one queue to the other upon certain events, and also for certain behaviours encountered in parallel computer system modelling. We show that this new class of network has product-form stationary solution, and establish the non-linear customer flow equations which govern it. Network stability is discussed in this new context.

This paper addresses random packing of equal-sized disks in a manner such that no disk has a gap on its circumference large enough to accommodate an extra touching neighbour. This structure generalises the deterministic packing models discussed in classical geometry (Coxeter (1961), Hilbert and Cohn-Vossen (1952)). Relationships with the dual mosaic formed by joining the centres of touching disks are established. Constraints on the neighbourhood of disks and on the packing density are established.

For the original Moran dam with independent and identically distributed inputs a representation of the stationary distribution is given which readily provides a geometric rate of convergence to this distribution. For the integer-valued case the stationary distribution can be expressed in terms of simple boundary crossing probabilities for the underlying random walk.

A random smooth path of infinite length crossed a square lattice. Intersections with the lattice were censored if they lay within a threshold distance of a preceding uncensored intersection, defined by tracking along the path in one direction. The distribution of distances between consecutive uncensored intersections is derived.