Birth processes with piecewise linear birth rates are analysed, and numerical results suggest that, relative to the linear case, convex birth rates increase variability and concave birth rates decrease variability.

This paper studies the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, modified by having the server take random vacations. In the first model, there is a vacation each time the queue becomes empty, as occurs for high-priority customers with a non-preemptive priority service discipline. Approximations for both the transient and steady-state behavior are developed for the case of relatively long vacations by proving a heavy-traffic limit theorem. If the vacation times increase appropriately as the traffic intensity increases, the workload and queue-length processes converge in distribution to Brownian motion with a negative drift, modified to have a random jump up whenever it hits the origin. In the second model, vacations are generated exogenously. In this case, if both the vacation times and the times between vacations increase appropriately as the traffic intensity increases, then the limit process is reflecting Brownian motion, modified by the addition of an exogenous jump process. The steady-state distributions of these two limiting jump-diffusion processes have decomposition properties previously established for vacation queueing models, i.e., in each case the steady-state distribution is the convolution of two distributions, one of which is the exponential steady-state distribution of the reflecting Brownian motion obtained as the heavy-traffic limit without vacations.

This note is concerned with a system of non-renewable components in parallel subjected to a common environment which is described by a real-valued external stochastic process. Given the environment process, the components are supposed to work independently, and the corresponding failure rates are all increasing (or decreasing) functions of the observed current state. It is then proved that, under these assumptions, the association of the external process implies the association of the component lifelengths. Connection with existing results is underlined.

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

This paper identifies a probability structure for queues that belong to the class of vacation systems operating according to Markov schedules, admitting a wide variety of server scheduling disciplines including most disciplines associated with those M/GI/1/L vacation systems reported in the literature. The conditions that define these schedules are identified, and it is shown that when these conditions are satisfied, queueing behavior is governed by an underlying Markov renewal/semi-regenerative structure. A simple example is examined (the M/GI/1 vacation system with limited batch service) to demonstrate the usefulness and generality of the underlying probability structure.

A star-like circuit-switching network with multistage service is considered. Likelihood ratio ordering is used to show that there exists an optimal threshold for the mean number of messages processed through the network as a function of the input traffic intensity.

A method originating from statistical mechanics (low-density and high-temperature expansions) is used to prove the existence and uniqueness of a stationary regime for switching networks on finite or infinite graphs. The main assumption is that the message (customer) flows circulating through the network are ‘localized' in the sense that, for any message, the probability of having a long path is rapidly decreasing (and, moreover, a path of a ‘typical' message consists of one line). The switching rule combines message-switching and circuit-switching principles. The stationary regime for the network under consideration may be treated as a ‘small perturbation' of the ‘idealized' regime in the totally decoupled network where all the messages have single line paths.

A Fourier series is obtained for the variance of the number of points lying in a parallelogram thrown randomly on to a square lattice at a fixed angle.

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

It is shown that a distribution with complete monotone derivative is geometrically infinitely divisible and that the class of distributions with complete monotone derivative is a proper subclass of the class of geometrically infinitely divisible distributions.

A difficulty in the application [5] of the method of polynomials as exposited by Galambos is investigated. The method, recast as the method of indicators, in a form due originally to Rényi [6], is applied to the situation of non-constant coefficients.

A constant regression of the quotient on the sum of two i.i.d. non-degenerate positive random variables is a characteristic property of the gamma distribution.

Limiting distribution results are obtained for the sufficient statistics of an asymmetric Ising model on a torus. Applications are discussed.

A consecutive-k-out-of-n: F system consists of n components ordered on a line. Each component, and the system as a whole, has two states: it is either functional or failed. The system will fail if and only if at least k consecutive components fail. The components are not necessarily equal and we assume that components' failures are stochastically independent. Using a result of Barbour and Eagleson (1984) we find a bound for the distance of the distribution of system's lifetime from the Weibull distribution. Subsequently, using this bound limit theorems are derived under quite general conditions.