We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).

The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

This paper discusses the joint distribution of the maximum and the time at which it is attained, of a random walk whose mean path is a curvilinear trend which itself has a maximum. A typical example of such a problem is the distribution of the maximum number of infectives present during the course of an epidemic. Another example where the random walk is constrained to terminate at 0 after a given time is provided by the distribution of the strength and breaking extension of a bundle of fibres.

A diffusion approximation to the joint distribution is obtained for the general case of a Brownian bridge. In the commonest class of cases which includes the two examples mentioned, a certain integral equation has to be solved. Its solution enables the marginal distribution of the time to reach the maximum to be tabulated, and the marginal distribution of the maximum confirms the results previously obtained by Daniels (1974) and Barbour (1975). Of particular interest is the conditional expectation of the maximum for a given time of attainment which behaves asymmetrically.

Consider a Poisson point process of density 1 in Rd, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e0, · ··, en, in Rd and a variety of distances, we obtain the asymptotic behavior of Ndn, the number of points which have e0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-lp distances related to correlation coefficients.

Let (I(t))∞t = () be a semi-Markov process with state space II and recurrent probability transition kernel P. Subject to certain mixing conditions, where Δis an invariant probability measure for P and μb is the expected sojourn time in state b ϵΠ. We show that this limit is robust; that is, for each state b ϵ Πthe sojourn-time distribution may change for each transition, but, as long as the expected sojourn time in b is µb on the average, the above limit still holds. The kernel P may also vary for each transition as long as Δis invariant.

The maximum-entropy approach to the estimation of the spectral density of a time series has become quite popular during the last decade. It is closely related to the fact that an autoregressive process of order p has maximal entropy among all time series sharing the same autocovariances up to lag p. We give a natural generalization of this result by proving that a mixed autoregressive-moving-average process (ARMA process) of order (p, q) has maximal entropy among all time series sharing the same autocovariances up to lag p and the same impulse response coefficients up to lag q. The latter may be estimated from a finite record of the time series, for example by using a method proposed by Bhansali (1976). By the way, we give a result on the existence of ARMA processes with prescribed autocovariances up to lag p and impulse response coefficients up to lag q.

A static model is proposed for a string thrown at random on a horizontal plane. The curvature of such a string is described by a Markov field that is derived from the Brownian bridge.

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

Suppose Xi≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t)k=1Xk> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.

The aim of this paper is to present a few techniques which may be useful in the analysis of time series when a failure is suspected. We present two categories of tests and investigate their asymptotic properties: one, of nonparametric type, is intended to detect a general failure in spectrum; the other investigates the properties of likelihood ratio tests in parametric models which have a non-standard behaviour in this situation. Finally, we obtain the asymptotic distribution of the likelihood estimators of the change parameters.

Sufficient conditions for a sum of dissociated random variables to be approximately normally distributed are derived. These results generalize the central limit theorem for U-statistics and provide conditions which can be verified in a number of applications. The method of proof is that due to Stein (1970).

In this paper upper and lower bounds for the availability and unavailability, to any level, in a fixed time interval are arrived at for multistate monotone systems based on corresponding information on the multistate components. These are assumed to be maintained and interdependent. Such bounds are of great interest when trying to predict the performance process of the system, noting that exact expressions are obtainable just for trivial systems. The bounds given generalize the existing bounds known in traditional binary theory, and represent improvements of the ones now being developed in multistate theory.

The concept of a characterization and its stability for queueing models is introduced. The principle of two stages in the study of the stability property is formulated. A series of results concerning the G/G/1 model is obtained.

We have shown that within the setting of a difference equation it is possible to link ergodicity with stability via the physical notion of energy in the form of a Lyapunov function.

A relation between the distributions of the workload of an arbitrary server and the waiting time is obtained for a GI/G/s queue in steady state. An expression for the total workload of the system (total unprocessed work) is also found. Several well-known results emerge as special cases. Various useful relations, for instance bounds and approximations for the mean waiting time, are also given.

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of closed queueing networks in heavy usage, which in practical terms means that the processor is utilized more than about 80 per cent. This paper extends recent work by Mitra and Morrison [10] on the same system in normal usage. The closed system has a CPU operating under the processor-sharing (‘time-slicing’) discipline and a bank of terminals. The presence of multiple job-classes allows distinctions in the user’s behavior in the terminal and in the service requirements. This work is primarily applicable to the case of large numbers of terminals. We give an effective method for calculating, for the equilibrium waiting time, the first and second moments and the leading term in the asymptotic approximation to the distribution. Our results are in the form of asymptotic expansions in inverse powers of , where N is a large parameter. The expansion coefficients depend on the classical parabolic cylinder functions.