Conditions are given on a process so that a shock model governed by the process will have nbue survival time. A related condition is given so that survival will be NBU and it is shown that a pure birth process satisfies this condition. A result of A-Hameed and Proschan is obtained as a corollary.

We consider the limiting behaviour of a k-type (k < ∞) Galton-Watson process which is augmented at each generation by a stochastic immigration component. In Section 2, conditions for ergodicity are found for a subclass of such processes. In Section 3, expressions are derived for the first two moments of the nth generation (by way of a recurrence relation) and for the first two asymptotic moments, in a manner which to some extent generalises previous results.

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.

We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.

A generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function. The notions of duality, critical elements and strong coherency for such a structure are discussed and the functional form of the structure function is analysed. An expression is derived for the distribution function of the state of the system, given the distributions of the states of the components, and generalisations of the Moore–Shannon and IFRA and NBU closure theorems are proved. The states of the components are then permitted to vary with time and a first-passage-time distribution is discussed. A simple model for such a process, based on the concept of partial availability, is then proposed. Lastly, an alternative continuum structure function is introduced and discussed.

The problem discussed is that of determining optimal release rates for finite dams when the release rate may take one of two values and specific costs are involved in increasing and decreasing this release rate. From another point of view, this is a problem of controlling the output simply by switching on and off. A diffusion model is suggested and by considering a family of plausible output policies an optima procedure, which minimises the long-term average cost of operating the dam, is established. Various properties of this output policy are given.

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

In the .author and van Eeden considered, as prior distributions for the cumulative, F, of the bio-assay problem, processes whose sample functions are, with probability one, distribution functions. The example we considered there had the undesirable property that its mean, E(F), was singular with respect to Lebesgue measure. In fact, Dubins and Freedman have shown that a class of such processes, which includes the example we considered, has sample functions F which are, with probability one, singular.

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

For a distribution function F on (0,∞), regular variation of its tail is known to imply that F is subexponential. Let be merely of dominated variation. This note shows that F need not be subexponential, and investigates which of the known necessary conditions for subexponentiality become sufficient when insisted upon for such an F.

This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.

This note deals with extinction criteria for bisexual Galton–Watson processes with arbitrary mating functions in terms of the averaged reproduction mean per mating unit. It gives a satisfactory answer to a question put forward by Hull (1982).

It is common to represent taxonomic hierarchies of related objects (such as similar plant or animal species or languages of the same family) by rooted trees with labelled terminal vertices which represent the objects. The multivariate data comparing numerous characteristics of the objects is first reduced to indices of similarity (or more often of dissimilarity) between each pair of objects. These are used to classify the objects into groups which are then depicted on a tree.

This paper shows that an unrooted tree with labelled terminal vertices may provide a better representation of the relationships between the objects because the similarity indices are required to conform to fewer restrictions. Also for a given number of terminal vertices, there are fewer unrooted than rooted trees so that studies using probability distributions of trees or seeking the most suitable tree to represent the data are more practicable.

We consider the problem of how to schedule n tasks on to several identical processors to meet the objective of minimising the expected flow-time. The strategy which always serves those tasks whose processing-time distributions have the highest hazard rates is shown to be optimal when these distributions are all exponential.

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

If Δr denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δr satisfies the weak law of large numbers and a central limit theorem.

This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record.

Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

In the present paper we demonstrate that the concept of a list code is from a mathematical point of view a more canonical notion than the classical code concept (list size one) in that it allows a unified treatment of various coding problems. In particular we determine for small list sizes the capacities of arbitrarily varying channels.

We establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with non-zero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin.

In this paper we study the equilibrium behavior of the queue lengths in a product-form closed queueing network when the service rates at a subset of stations (nodes) are increased. Univariate and multivariate likelihood ratio orderings as well as multivariate stochastic ordering of the queue lengths are established to indicate the effect of increasing the service rates. Relations among these orderings are also developed.

A new burn-in procedure for a repairable component is proposed. During a burn-in period, the failed component is only minimally repaired rather than being completely repaired. This procedure is shown to be economical and efficient when the minimal repair method is applicable during a burn-in process. The properties of the optimal burn-in time b∗ and block replacement policy T∗ are also given.