This note is concerned with the continuing misconception that a discrete-time network of queues, with independent customer routing and the number of arrivals and services in a time interval following geometric and truncated geometric distributions respectively, has a product-form equilibrium distribution.

An urn contains balls of different colours, which are randomly drawn one at a time. After each draw the number of balls in the urn with the same colour as the ball last drawn is changed. Special cases are sampling with and without replacement, and Pólya sampling. The drawing stops when a given number of colours has been drawn a given number of times. The number of times the different colours have been drawn is studied in this paper by imbedding the urn scheme in birth processes. Both exact and asymptotic results are obtained. In particular, waiting times for sampling with and without replacement and for Pólya sampling are considered.

A single machine is available to process a collection of stochastic jobs preemptively. Rewards are received at job completions. We seek policies for machine allocation which maximize the total reward. Application areas point to the need to study such models for resource allocation when job processing requirements are dependent. To this end, models are developed in which the nature of such dependence is derived from various notions of positive and negative dependence in common usage in reliability. Optimal policies for resource allocation of simple structure are obtained for a variety of such models.

The number of returns to the origin for a simple random walk starting and ending at the origin is reconsidered; closed-form expressions in μ are given for μ3 and μ4.

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

We consider a controlled i.i.d. process, where several i.i.d. sources are sampled sequentially. At each time instant, a controller determines from which source to obtain the next sample. Any causal sampling policy, possibly history-dependent, may be employed. The purpose is to characterize the extremal large deviations of the sample mean, namely to obtain asymptotic rate bounds (similar to and extending Cramér's theorem) which hold uniformly over all sampling policies. Lower and upper bounds are obtained, and it is shown that in many (but not all) cases stationary sampling policies are sufficient to obtain the extremal large deviations rates. These results are applied to a hypothesis testing problem, where data samples may be sequentially chosen from several i.i.d. sources (representing different types of experiments). The analysis provides asymptotic estimates for the error probabilities, corresponding both to optimal and to worst-case sampling policies.

The basis of this paper is the elementary observation that the n-step descendant distribution of any Galton–Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton–Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain continuous-state branching processes (CSBPs) satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall (2006).) Our analysis shows that the nonlinear scaling dynamics of CSBPs become linear and purely dilatational when expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove the existence of universal critical Galton–Watson processes and CSBPs analogous to Doeblin's `universal laws'. Namely, these universal processes generate all possible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix.

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

Suppose segments are distributed at random along a line, their locations being determined by a Poisson process. In the case where segment length is fixed, we compare efficiencies of several different estimates of Poisson intensity. The case of random segment length is also considered, and there we study estimation procedures based on empiric properties. The one-dimensional mosaic may be viewed as an M/G/∞ queue.

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision-maker is risk-aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

Under a fourth-order moment condition on the branching and a second-order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous-state and continuous-time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In the case of a non-trivial process, under a first-order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

Perturbation analysis estimators for expectations of possibly discontinuous functions of the time-stationary workload were derived in [2]. The expressions obtained may, however, not be valid if the customer-stationary distribution of the workload has atoms (at points other than zero). This was pointed out by Brémaud and Lasgouttes in [1]. In this note we clearly state the additional condition required for the validity of the expressions in [2]. We furthermore show how our approximation scheme can also be used to obtain the correct expressions for the right and left derivatives given in [1].

Sufficient conditions are given ensuring tightness of a suitably normalized stationary waiting time in the FIFO G/G/1 queue under heavy traffic.

This paper examines a problem of importance to the telecommunications industry. In the design of modern ATM switches, it is necessary to use simulation to estimate the probability that a queue within the switch exceeds a given large value. Since these are extremely small probabilities, importance sampling methods must be used. Here we obtain a change of measure for a broad class of models with direct applicability to ATM switches.

We consider a model with A independent sources of cells where each source is modeled by a Markov renewal point process with batch arrivals. We do not assume the sources are necessarily identically distributed, nor that batch sizes are independent of the state of the Markov process. These arrivals join a queue served by multiple independent servers, each with service times also modeled as a Markov renewal process. We only discuss a time-slotted system. The queue is viewed as the additive component of a Markov additive chain subject to the constraint that the additive component remains non-negative. We apply the theory in McDonald (1999) to obtain the asymptotics of the tail of the distribution of the queue size in steady state plus the asymptotics of the mean time between large deviations of the queue size.

The problem of finding bounds for P(A1 ∪ ⋯ ∪ An) based on P(Ak1 ∩ ⋯ ∩ Aki) (1 ≤ k1 < ⋯ < ki ≤ n, i = 1,…,d) goes back to Boole (1854), (1868) and Bonferroni (1937). In this paper upper bounds are presented using methods in graph theory. The main theorem is a common generalization of the earlier results of Hunter, Worsley and recent results of Prékopa and the author. Algorithms are given to compute bounds. Examples for bounding values of multivariate normal distribution functions are presented.

Let (φ(Xn))n be a function of a finite-state Markov chain (Xn)n. In this article, we investigate the conditions under which the random variables φ(n) have the same distribution as Yn (for every n), where (Yn)n is a Markov chain with fixed transition probability matrix. In other words, for a deterministic function φ, we investigate the conditions under which (Xn)n is weakly lumpable for the state vector. We show that the set of all probability distributions of X0, such that (Xn)n is weakly lumpable for the state vector, can be finitely generated. The connections between our definition of lumpability and the usual one (i.e. as the proportional dynamics property) are discussed.

We study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.