Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

In a cellular radio system a limited number of channels are available and neighbouring cells may not be allocated the same channel simultaneously because of the possibility of interference. Under heavy traffic, therefore, the average rate of losing calls may be reduced if certain calls are rejected as a matter of policy because of the potential inference they may cause. A fixed channel assignment policy is one in which each cell is allocated a fixed set of channels, this set typically being made smaller for cells likely to cause most interference. Sufficient conditions are found for the optimality of fixed channel assignment policies for a variety of layouts, and optimal and ‘good' policies are found in a number of other cases.

We have furnished further examples on the connection between some standard one-dimensional chaotic deterministic models and stochastic time series models via time reversal.

The purpose of this note is to point out the connection between the invariance property of M/M/1 and GI/G/1 queues (which has been reported in several papers) and the interchangeability and reversibility properties of tandem queues. This enables us to gain new insights for both problems and obtain stronger invariance results for M/M/1, GI/G/1, as well as loss systems M/M/1/N, GI/G/1/N and tandem systems.

The existence of a class of multitype measure branching processes is deduced from a single-type model introduced by Li [8], which extends the work of Gorostiza and Lopez-Mimbela [5] and shows that the study of a multitype process can sometimes be reduced to that of a single-type one.

We give a finite form for the probability mass function of the wrapped Poisson distribution, together with a probabilistic proof. We also describe briefly its connection with existing results.

Pitman has shown that if X is Brownian motion with maximum process M, then 2M – X is a BES0(3) process. We show that this can be seen by looking at finite-dimensional densities.

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk) and H(t) is established.

We show that the stationary version of the queueing relation H = λG is equivalent to the basic Palm transformation for stationary marked point processes.

Two types of minimal repair are discussed. After statistical minimal repair, the state of a system is statistically identical to what it was just before the failure. Physical minimal repair restores the failed unit to its exact physical condition before the failure. Several examples are given.

One-dimensional random packing, known as the car-parking problem, was first analyzed by Rényi (1958). A stochastic version of Kakutani's (1975) interval splitting is another typical model on a one-dimensional interval. We consider a generalized car-parking problem which contains the above two models as special cases. In the generalized model, one can park a car of length l, if there is a space not less than 1. We give the limiting packing density and the limiting distribution of the length of randomly selected gaps between cars. Our results bridge the two models of Rényi and Kakutani.

The bivariate characterization of stochastic ordering relations given by Shanthikumar and Yao (1991) is based on collections of bivariate functions g(x, y), where g(x, y) and g(y, x) satisfy certain properties. We give an alternate characterization based on collections of pairs of bivariate functions, g1(x, y) and g2(x, y), satisfying certain properties. This characterization allows us to extend results for single machine scheduling of jobs that are identical except for their processing times, to jobs that may have different costs associated with them.

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.