Let ℰ be an event governed by a homogeneous second-order two-state Markov chain. Assume the steady state has been achieved. Consider Nm the number of times that ℰ occurs in m successive trials. The limiting distribution of P(Nm = k) is obtained under the condition that mP(ℰ)=λ.

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx. In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

In evolutionarily stable strategy models, transmission of one's strategy to one's offspring has routinely been assumed to be exact, whether this transmission is genetic or by instruction. If inexact transmission of strategy is possible, however, the nature of the mode of transmission becomes important. This paper demonstrates analytically that the possibility of learning one's strategy can increase rather than decrease eventual strategy diversity and that the resulting population mean strategy can differ appreciably from an evolutionarily stable strategy. These results are found to be consistent with simulation results reported in the literature.

The expected response time of a job that requires processing time t and meets n jobs on arrival in the M/G/1 processor-sharing system is derived.

In this paper, we show that the expected number in an M/M/c queue is convex with respect to the traffic intensity. The proof is conducted by expressing the second derivative of the expected queue size as the sum of non-negative terms.

A non-analytical proof of recurrence is obtained via an embedding procedure for a two-dimensional correlated lattice random walk.

The probability of attaining a goal structure in a partially stochastic manpower model is discussed in relation to the position of the structure described in Euclidean space. It is found that, within the regions of structures attainable with non-zero probability, the probabilities decrease on lines emanating from the structure attainable with probability 1.

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

This paper establishes the existence of the limit as t → ∞ of the distribution of the residual lifetime at time t of a two-unit warm-standby reliability system supported by a single repair facility, conditioned on the event that the system has not been down at any time in (0, t). The conditioned limit distribution is proved to be exponential irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.

The question as to what grade structures are maintainable in Markov manpower models, with or without the constraint that the total size be kept constant, is examined. Known results are collected together, examples and further results are given, and an old conjecture is generalised.

It has been recently proved that if N, X1, X2, … are non-constant mutually independent random variables with X1,X2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.

Coupling methods are applied to the study of the storage equation with stationary input. Results on the existence of statistical equilibria are obtained. Means of obtaining rate-of-convergence estimates are indicated.

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).

This paper studies the equilibrium behaviour of the M/G/k loss system with servers subject to breakdown. Such a system has k servers, whose customers arrive in a Poisson process. They are served if there is an idle server, otherwise they leave and do not return. Each server is subject to breakdown with probability of occurrence depending on the length of the time the server has been busy since his last repair. On a breakdown the customer waits and his service is continued just after the repair of the server. Among other things, a generalization of the Erlang B-formula is given and it is shown that the equilibrium departure process is Poisson. In fact these results are obtained for the more general case where customers may balk and service and repair rates are state dependent.

We propose a simple proof of Norton's theorem for multiclass quasireversible networks.

Binomial moments of the time-dependent and limiting distributions of inventory deficit are derived for a continuous-review (s, S) inventory system under the assumption that interarrival times, demand sizes, and lead times are independent sequences of independent, identically distributed random variables. Explicit expressions for the limiting distribution are also given in some special cases of practical interest. The approach used follows that of Finch [2] who investigated the system under the additional assumption of unit demands.