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.

We consider a stochastic fluid network with independent subordinator inputs to the various stations and deterministic internal flow which is of feed-forward type. We show that under suitable conditions the process of fluid contents in the station has a limiting distribution, where the limit holds in total variation and is independent of the initial condition. We also show that this limiting distribution is of product form only for trivial setups.

This paper considers a class of epidemic models in which susceptibles may enter or leave the population according to a general continuous time density dependent Markov chain. A sequence of such epidemics indexed by N, the initial number of susceptibles, is constructed on the same probability space as a time-inhomogeneous birth-and-death process. A coupling argument is then used to demonstrate the strong convergence of the sequence of infectives to the birth-and-death process. This result is used to provide a threshold analysis of the epidemic model in question.

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 establish a necessary condition for any importance sampling scheme to give bounded relative error when estimating a performance measure of a highly reliable Markovian system. Also, a class of importance sampling methods is defined for which we prove a necessary and sufficient condition for bounded relative error for the performance measure estimator. This class of probability measures includes all of the currently existing failure biasing methods in the literature. Similar conditions for derivative estimators are established.

In this paper we study the problem of computing the downtime distribution of a parallel system comprising stochastically identical components. It is assumed that the components are independent, with an exponential life-time distribution and an arbitrary repair time distribution. An exact formula is established for the distribution of the system downtime given a specific type of system failure scenario. It is shown by performing a Monte Carlo simulation that the portion of the system failures that occur as described by this scenario is close to one when we consider a system with quite available components, the most common situation in practice. Thus we can use the established formula as an approximation of the downtime distribution given system failure. The formula is compared with standard Markov expressions. Some possible extensions of the formula are presented.

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.

Let , be a recurrent Markov renewal process and Mik (t) be the expected value of Nk (t) provided that at the initial moment the system is in state i. It is shown that when the mean recurrence times μ ii are finite, the differences μ ij Mki (t) – t behave asymptotically the same for all states i and k.

Vasicek (1977) proved that among all queueing disciplines that do not change the departure process of the queue, FIFO and LIFO yield, respectively, the smallest and the largest expectation of any given convex function of the service delay. In this note we further show that, if arriving customers join the queue stochastically ‘closer' to the server(s), then the expected value of any convex function of service delay is larger. As a more interesting result, we also show that if the function under consideration is concave, then the conclusion will be exactly the opposite. This result indicates that LIFO will be the best discipline if the delay cost is an increasing function but at a diminishing rate.

We consider percolation on the graph of the product of a regular tree T with degree d and the line ℤ, in which each tree edge is open with probability 1 – exp(–JTß) and each line edge is open with probability . Let C(o, 0) be the open cluster for . Denote by θ (β) the percolation probability. Here we show that θ (β) is continuous when ß > ßc , where ßc = sup{ß : θ(β) = 0}.

Recently new partial orderings s-SFR, s-FR, s-ST, s-CV and s-CX have been defined in the literature. These are more general in the sense that most of the earlier known partial orderings reduce to particular cases of these orderings. Moreover, these orderings have helped in defining a new and useful ageing criterion. In this paper, we discuss preservation properties of these orderings under mixtures. Results are expected to be useful in engineering and reliability, where mixtures of distributions arise naturally in numerous situations. Moreover, existing results, specifically for LR, FR and ST orderings under the operation of mixtures, reduce to special cases.

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[W t/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions limL→x L –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→x b –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.

In this paper, transient characteristics related to excursions of the occupation process of M/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

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.

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

For (marked) Poisson point processes we give, for increasing events, a new proof of the analog of the BK inequality. In contrast to other proofs, which use weak-convergence arguments, our proof is ‘direct' and requires no extra topological conditions on the events. Apart from some well-known properties of Poisson point processes, the proof is self-contained.

This paper considers a modified block replacement with two variables and general random minimal repair cost. Under such a policy, an operating system is preventively replaced by new ones at times kT (k= 1, 2, ···) independently of its failure history. If the system fails in [(k − 1)T, (k − 1)T+ T0 ) it is either replaced by a new one or minimally repaired, and if in [(k − 1) T + T 0 , kT) it is either minimally repaired or remains inactive until the next planned replacement. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the system at age y depends on the random part C(y) and the deterministic part ci (y). The expected cost rate is obtained, using the results of renewal reward theory. The model with two variables is transformed into a model with one variable and the optimum policy is discussed.

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.

In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.