This paper uses martingale calculus in order to study the Markov properties of diffusion local time first discovered by Ray and Knight. The approach enables us to calculate the laws of the processes involved and is easily modified to deal with conditioning with respect to the excursion σ-fields.

A fiber bundle is a parallel arrangement of fibers. Under a steady tensile load, fibers fail randomly in time in a manner that depends on how they share the applied load. The bundle fails when all its fibers have failed in a specified region.

In this paper we consider the fatigue failure of such a bundle in a fiber load-sharing setting appropriate for composite materials, that is, to bundles impregnated with a flexible matrix. The bundle is actually modelled as a chain of short bundles, and local load sharing is assumed for the fibers within each short bundle. The chain of bundles fails once all the fibers in one of the short bundles have failed.

Reasonable assumptions are made on the stochastic failure of individual fibers. A general framework for describing fiber bundles is developed and is used to derive the limiting distribution of the time to the first appearance of a set of k or more adjacent failed fibers as the number of fibers in the bundle grows large. These results provide useful bounds on the distribution of the time to total bundle failure. Some implications and extensions of these results are discussed.

Correction terms are obtained for the Brownian motion approximation to one- and two-barrier first-passage probabilities. These approximations are given in terms of their Laplace transforms, which are formally (and non-rigorously) inverted. Applications to the one-sample Kolmogorov-Smirnov statistic are discussed.

Let K and L be convex bodies in where L can roll freely in K. Suppose that Borel sets are painted in the boundaries of K and L. The probability that after a random rolling of L in K the contact is paint-to-paint is determined and expressed by curvature measures of K and L.

A new device in the optimization of queuing systems is introduced by using the method of phases. Non-exponential queues under control are considered with respect to the expected discounted reward criterion. For models with hyper-Erlang distributions equivalent phase-type systems are established. Approximation results for Markov decision models allow the extension to the case of general distribution functions. The approach is demonstrated by finding the form of an optimal policy for the GI/M/c queue with customer admission and batch arrival as well as for the GI/M/1 queue with interarrival time control.

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Let Sn = X1 + · · · + Xn be a random walk with negative drift μ < 0, let F(x) = P(Xk ≦ x), v(u) =inf{n : Sn > u} and assume that for some γ > 0 is a proper distribution with finite mean Various limit theorems for functionals of X1,· · ·, Xv(u) are derived subject to conditioning upon {v(u)< ∞} with u large, showing similar behaviour as if the Xi were i.i.d. with distribution For example, the deviation of the empirical distribution function from properly normalised, is shown to have a limit in D, and an approximation for by means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and the GI/G/1 queue considered either within a busy cycle or in the steady state. The methods produce an alternate approach to known asymptotic formulae for ruin probabilities as well as related waiting-time approximations for the GI/G/1 queue. For example uniformly in N, with WN the waiting time of the Nth customer.

A general method based on a limit theorem for generation of random numbers from infinitely divisible distributions with essentially given Lévy measure is studied. Some classes of infinitely divisible distributions that appear in a natural way in this context are paid particular attention.

Motivated by a problem in PERT networks we consider the question of construction of random variables with maximum sums when the marginals are fixed.

A revealing alternate proof is provided for the Iglehart (1965), (1973)–Borovkov (1967) heavy-traffic limit theorem for GI/G/s queues. This kind of heavy traffic is obtained by considering a sequence of GI/G/s systems with the numbers of servers and the arrival rates going to ∞ while the service-time distributions are held fixed. The theorem establishes convergence to a Gaussian process, which in general is not Markov, for an appropriate normalization of the sequence of stochastic processes representing the number of customers in service at arbitrary times. The key idea in the new proof is to consider service-time distributions that are randomly stopped sums of exponential phases, and then work with the discrete-time vector-valued Markov chain representing the number of customers in each phase of service at arrival epochs. It is then easy to show that this sequence of Markov chains converges to a multivariate O–U (Ornstein–Uhlenbeck) diffusion process by applying simple criteria in Stroock and Varadhan (1979). The Iglehart–Borovkov limit for these special service-time distributions is the sum of the components of this multivariate O–U process. Heavy-traffic convergence is also established for the steady-state distributions of GI/M/s queues under the same conditions by exploiting stochastic-order properties.

It is shown that the heavy-traffic Gaussian limit for GI/G/∞ queues is Markovian if and only if the service-time distribution H(t) is of the form 1-H(t) = pe–αt for α > 0 and 0 < p ≦ 1.

A second-order asymptotic result for the probability of occurrence of a persistent and aperiodic recurrent event is given if the tail of the distribution of the waiting time for this event is regularly varying with index −1.

Messages are to be transmitted through a series of n nodes linked by communication channels. The lengths of successive messages are independent identically distributed random variables, and the time taken to transmit a message through a channel is determined by its length. Each node has a finite buffer, and when the number of messages at a node reaches the buffer size transmission from the preceding node is interrupted. This paper is concerned with the maximum rate at which messages can pass through the system, called the throughput. We investigate the asymptotic behaviour of throughput as the series length increases, and determine the rate at which buffer sizes should grow to ensure that throughput does not decline to 0.