A sufficiently large finite second-order stationary time series process on a line has approximately the same eigenvalues and eigenvectors of its dispersion matrix as its counterpart on a circle. It is shown here that this result can be extended to second-order stationary processes on a d-dimensional lattice.

A doubly stochastic switching model previously used by the author to check against doubly stochastic ‘white-noise' models is generalized in the case of Poisson processes to any distribution of fluctuating intensity rates. The resulting Cox process is used in place of a simple Poisson process as the basis of a mixed model, which is then fitted to some data on automobile accidents previously analysed by Seal (1980) in terms of a mixed Poisson process. It is shown that there is just significant evidence in favour of some additional heterogeneity ‘within' each year for individual drivers; the nature of the data analysed (yearly time intervals) does not, however, allow discrimination between heterogeneity which is definitely attributable to a common fluctuating environment, and that which is in effect independent for each driver.

An almost sure convergence result for the normed population size of a supercritical controlled branching process is proved.

The paper considers the GI/G/1 queueing system under the assumption of a last-come–first-served queue discipline, where each customer begins service immediately upon his arrival. At the next arrival, the previous service is interrupted but no loss of service is involved. It has been shown that when the system is considered exclusively at arrival epochs or exclusively at departure epochs, then the equilibrium distribution of the queue-size is geometric, while the remaining durations of the corresponding services are independent random variables each one distributed as the idle period in the dual (inverse) queue. In this paper alternative simpler proofs of the above results are given.

This paper considers the two-sex birth-death model {X(t), Y(t); t ≧ 0}; an explicit solution is obtained for its probability generating function. It is shown that moments of the process can be found directly from the Kolmogorov forward equations for the probabilities. An integral equation approach is also used to throw light on the structure of the process.

A simple coupling argument is used to obtain a new proof of a result of Daniels (1967) concerning the total size distribution of the general stochastic epidemic. The proof admits a straightforward generalisation to multipopulation epidemics and indicates that similar results are unlikely to be available for epidemics with non-exponential infectious periods.

This paper considers periodic and sequential preventive maintenance (PM) policies for the system with minimal repair at failure: the PM is done (i) at periodic times kx and (ii) at constant intervals xk (k = 1, 2, ···, N). The system has a different failure distribution between PM'S and is replaced at the Nth PM. The optimal policies which minimize the expected cost rates are discussed. The optimal x and N of periodic PM and {xk} of sequential PM are easily computed in a Weibull distribution case.

The problem of determining the probability of j departures during the time interval (0, t) from an M/M/l queue empty at t = 0 is considered. A closed-form solution is obtained. It is shown that this solution is unique and invariant under interchanging the arrival rate and service rate. Finally, sample computational representations of the solution are developed and results of a simple computation are provided.

Many population models which are far from stationarity can nevertheless be written in autoregressive format, perhaps with random coefficient. It is the thesis of this paper that procedures developed for stationary time series models are a useful guide to inferential results for population processes and may indeed be directly applicable. The illustrations concentrate on estimation of the matrix of mean vital rates in an age-structured population.

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.

Place n arcs of equal length uniformly at random on the circumference of a circle. We give a simple probabilistic derivation of the moments of the uncovered portion of the circle.

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.

Conditions for finiteness of moments of the following quantities have been found: the duration of a busy period of an Μ /G/∞ system; the duration of a partial busy period of an M/G/C loss system, and the duration of a partial busy period of an M/G/C queue.

We provide a term-by-term interpretation of Beneš's well-known but mysterious inversion of the Pollaczek–Khintchine formula. The strategy is to recognize the equality of waiting time in M/G/1–FIFO with remaining work in M/G/1–LIFO–preemptive resume. In the process, we give a new and simple derivation of some known results for M/G/1–LIFO–preemptive resume.

The standard ANOVA models with random effects for multi-indexed arrays of random variables with an arbitrary nesting structure on the indices are considered from the viewpoint of symmetry. It is found that the covariance matrix of such an array has sufficient symmetry to permit viewing the usual components of variance as a generalised spectrum and the linear models of random effects as a generalised spectral decomposition.