The least squares estimators βi(N), j = 1, …, p, from N data points, of the autoregressive constants for a stationary autoregressive model are considered when the disturbances have a distribution attracted to a stable law of index α < 2. It is shown that N1/δ(βi(N) – β) converges almost surely to zero for any δ > α. Some comments are made on alternative definitions of the βi(N).

An expression for the first-passage density of Brownian motion to a curved boundary is expanded as a series of multiple integrals. Bounds are given for the error due to truncation of the series when the boundary is wholly concave or wholly convex. Extensions to the Brownian bridge and to continuous Gauss–Markov processes are given. The series provides a practical method for calculating the probability that a sample path crosses the boundary in a specified time-interval to a high degree of accuracy. A numerical example is given.

Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.

A rigorous treatment is given for a construction via Markov chains of a binary (0–1) stationary homogeneous Markov random field on Z × Z. The resulting process possesses rather interesting properties. For example, its correlation structure is geometric and it may be easily simulated. Some of the results are rather unintuitive — indeed counter-intuitive — but their demonstration is straightforward involving only the most elementary properties of Markov chains.

In this paper we consider the initial and asymptotic behaviour of the failure rate function resulting from mixtures of subpopulations and formation of coherent systems. In particular, it is shown that the failure rate of a mixture has the same limiting behaviour as the failure rate of the strongest subpopulation. A similar result holds for systems except the role of strongest subpopulation is replaced by strongest min path set.

A non-homogeneous Poisson shock model has a continuous mean function Λ(t). The kth shock Sk causes simultaneous failure of the components j ∊ J ∊ {1, ···, n} with probability pJ(Sk). If Tj is the lifetime of component j, it is shown that (T1, · ··, Tn) belongs to various multivariate non-parametric life classes depending on the life class of .

We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T).

The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.

By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

We show, in the context of a neurobiological problem, that ζ a(α)−ζ a(β) increases with a, while ζ a (α)/ζ a (β) decreases with a, where 0 < β < α < 1 and ζ a (α) is the α th quantile of either the gamma distribution with shape parameter a, or a scaled F distribution with parameters 2a and 2m for any m.

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.

Using Bernstein's theorem concerning absolutely monotonic functions, Rao and Rubin (1964) have established a characterization of the Poisson distribution. The present note arrives at an extended version of this result using a technique existing in the renewal theory. A variant of a related result due to Srivastava and Srivastava (1970) and an extension of the result of Talwalker (1970) are also presented.

In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind.

Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

The Weibull distribution for the strength of a long fiber is reviewed, with emphasis on its derivation via extreme-value theory. It is shown that a corresponding Weibull distribution, but with different scale and shape parameters, is appropriate for the lower tail distribution of a bundle of parallel fibers under general load-sharing assumptions. This result is applied to series-parallel systems. The paper contains discussion of the application of these results to the strength of fibrous materials. It is concluded that the Weibull approximation, developed for series-parallel systems, is appropriate for long thin materials consisting of a small number of parallel fibers.

Expressions are obtained for the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process. Explicit formulae for the inverse of this matrix are given for the general autoregressive process of order p (n ≧ p), and for the first order mixed autoregressive moving average process.

Let X = (X1, X2, …, Xn) be an exchangeable random vector, and write X(1:i) = min{X1, X2, …, Xi}, 1 ≤ i ≤ n. In this paper we obtain conditions under which X(1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

The paper investigates some properties of the first-order bilinear time series model: stationarity and invertibility. Estimates of the parameters are obtained by a modified least squares method and shown to be strongly consistent.

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say is Laplace-smaller than for all s > 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.