We examine the existence of limiting behavior, or stability, for storage models with shot noise input and general release rules. The shot noise feature of the input process allows the individual inputs to gradually enter the store.

We first show that a store under the unit release rule is stable if and only if the traffic intensity is less than one; this extends the classic result of Prabhu (1980) to the case of shot noise input. The stability of the unit release rule store and various stochastic orderings are then used to derive a sufficient condition for a store with a general release rule to be stable. Finally, we show that when restricted to a compact state space, our storage model is always stable.

An important component of the paper is the methodology employed: coupling and stochastic monotonicity play a key role in analyzing the non-Markov processes encountered.

We analyse the queue QL at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[Wt/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions limL→xL–1 log P[QL > Lb] = – I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings v(t) = tv, a(t) = ta (such as occur in FBM) we analyse the asymptotics of the shape function limb→xb–u/a(I(b) – δbv/a) = vu for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[QL > Lb] ≈ exp[−δLbv/a] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone, and also perturbed by an Ornstein–Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.

Consider a queueing network with batch services at each node. The service time of a batch is exponential and the batch size at each node is arbitrarily distributed. At a service completion the entire batch coalesces into a single unit, and it either leaves the system or goes to another node according to given routing probabilities. When the batch sizes are identical to one, the network reduces to a classical Jackson network. Our main result is that this network possesses a product form solution with a special type of traffic equations which depend on the batch size distribution at each node. The product form solution satisfies a particular type of partial balance equation. The result is further generalized to the non-ergodic case. For this case the bottleneck nodes and the maximal subnetwork that achieves steady state are determined. The existence of a unique solution is shown and stability conditions are established. Our results can be used, for example, in the analysis of production systems with assembly and subassembly processes.

We study a model of a stochastic transportation system introduced by Crane. By adapting constructions of multidimensional reflected Brownian motion (RBM) that have since been developed for feedforward queueing networks, we generalize Crane's original functional central limit theorem results to a full vector setting, giving an explicit development for the case in which all terminals in the model experience heavy traffic conditions. We investigate product form conditions for the stationary distribution of our resulting RBM limit, and contrast our results for transportation networks with those for traditional queueing network models.

As a model for an ATM switch we consider the overflow frequency of a queue that is served at a constant rate and in which the arrival process is the superposition of N traffic streams. We consider an asymptotic as N → ∞ in which the service rate Nc and buffer size Nb also increase linearly in N. In this regime, the frequency of buffer overflow is approximately exp(–NI(c, b)), where I(c, b) is given by the solution to an optimization problem posed in terms of time-dependent logarithmic moment generating functions. Experimental results for Gaussian and Markov modulated fluid source models show that this asymptotic provides a better estimate of the frequency of buffer overflow than ones based on large buffer asymptotics.

As well as having complete knowledge of the future, a superprophet can also alter the order of observation as it is presented to a player without foresight, whose strategy is known to the prophet. It is shown that a superprophet can only do twice as well as his counterpart, if the underlying random sequence is independent.

The so-called ‘Swiss Army formula', derived by Brémaud, seems to be a general purpose relation which includes all known relations of Palm calculus for stationary stochastic systems driven by point processes. The purpose of this article is to present a short, and rather intuitive, proof of the formula. The proof is based on the Ryll–Nardzewski definition of the Palm probability as a Radon-Nikodym derivative, which, in a stationary context, is equivalent to the Mecke definition.