Random arcs having random sizes are placed on a circle. Let V be the length of the uncovered portion of the circle and G be the number of uncovered gaps on the circle. Results are presented concerning the joint moments of V and G and the conditional distribution of V given G.

We consider a population of n projects which in general continue to evolve whether in operation or not (although by different rules). It is desired to choose the projects in operation at each instant of time so as to maximise the expected rate of reward, under a constraint upon the expected number of projects in operation. The Lagrange multiplier associated with this constraint defines an index which reduces to the Gittins index when projects not being operated are static. If one is constrained to operate m projects exactly then arguments are advanced to support the conjecture that, for m and n large in constant ratio, the policy of operating the m projects of largest current index is nearly optimal. The index is evaluated for some particular projects.

A sequence of independent experiments is performed, each producing a letter from a given alphabet. Using a result by Barbour and Eagleson (1984) we prove that under general conditions the number of non-overlapping occurrences of long recurrent patterns has approximately a Poisson distribution.

A circle is covered by random arcs with a given length a and endpoints chosen (independently and uniformly) among m equispaced points on the circle. The asymptotic distribution as a → 0 and m → ∞of the number of arcs required for complete coverage is given. The result connects earlier results for the cases ma = 1 (a discrete problem) and m = ∞ (the continuous limiting case).

A method is derived for choosing the width of a spectral window in spectrum estimation. The method is based upon the idea of minimising complexity, i.e. the length, in bits, of a code for the data.

A shock model in which the time intervals between shocks are in the domain of attraction of a stable law of order less than 1 or relatively stable is considered. Weak limit theorems are established for the cumulative magnitude of the shocks and the first time the cumulative magnitude exceeds z without any assumption on the dependence between the intershock interval and shock magnitude.

The familiar limiting distributions of excess, age, and spread are derived on sample paths, where these distributions are fractions of time. Similar results are obtained for the distribution of remaining service at a queue. For stochastic processes, where specified limits exist as finite constants with probability 1, the derived sample-path results are shown to imply that the same limiting distributions hold, where in a stochastic setting, these distributions may be defined in a more conventional way.

In this paper we trace the development of the asymptotic analysis of autocorrelations for stationary purely non-deterministic time series. We emphasize the interplay between mathematical requirements and modelling philosophy. We then proceed to extend the theory to the case where only a certain weak form of asymptotic independence of the linear prediction errors is needed rather than the earlier martingale difference or independence requirements.

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

We consider single-server loss systems with exponential service times and non-stationary Poisson input. We prove that if the arrival rate is given by a periodic function, the proportion of lost customers is convex increasing in the amplitude.

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.