In this paper two burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure) which can be removed by a minimal repair or a complete repair and the other is Type II failure (catastrophic failure) which can be removed only by a complete repair. During a burn-in process, with burn-in Procedure I, the failed component is repaired completely regardless of the type of failure, whereas, with burn-in Procedure II, only minimal repair is done for the Type I failure and a complete repair is performed for the Type II failure. In field use, the component is replaced by a new burned-in component at the ‘field use age’ T or at the time of the first Type II failure, whichever occurs first. Under the model, the problems of determining optimal burn-in time and optimal replacement policy are considered. The two burn-in procedures are compared in cases when both the procedures are applicable.

The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharya et al. (1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modified R/S test. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool the R/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger than N-½, where N is the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.

For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. It is shown that with initial configuration ℤ the distribution of particles at all times is a renewal process and that the probability that a site remains occupied for all time tends to 1/e. Time-dependent behaviour is also calculated for the tree 𝕋r.

The paper proposes a hidden semi-Markov model for breakpoint rainfall data that consist of both the times at which rain-rate changes and the steady rates between such changes. The model builds on and extends the seminal work of Ferguson (1980) on variable duration models for speech. For the rainfall data the observations are modelled as mixtures of log-normal distributions within unobserved states where the states evolve in time according to a semi-Markov process. For the latter, parametric forms need to be specified for the state transition probabilities and dwell-time distributions.

Recursions for constructing the likelihood are developed and the EM algorithm used to fit the parameters of the model. The choice of dwell-time distribution is discussed with a mixture of distributions over disjoint domains providing a flexible alternative. The methods are also extended to deal with censored data. An application of the model to a large-scale bivariate dataset of breakpoint rainfall measurements at Wellington, New Zealand, is discussed.

In this paper we utilize a particular transformation of i.i.d. exponential random variables to derive two distributional identities. Throughout the analysis we discover some peculiar properties of exponentials. We also discuss possible generalizations and applications of the results.

The tapered (or generalized) Pareto distribution, also called the modified Gutenberg-Richter law, has been used to model the sizes of earthquakes. Unfortunately, maximum likelihood estimates of the cutoff parameter are substantially biased. Alternative estimates for the cutoff parameter are presented, and their properties discussed.

We present, discuss and prove an apparently unfamiliar result in the convergence of probability measures.

The present paper provides simple inequalities for the number of lost customers during a busy period of a GI/M/1/n queueing system.

Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

In this note, we give some preservation results for the classes IFR(2), NBU(2) and their dual classes under the formation of special coherent systems. Further, we show with examples that the relationships among these aging classes and others are strictly one-way implications.

The paper reviews the formulation of the linked stress release model for large scale seismicity together with aspects of its application. Using data from Taiwan for illustrative purposes, models can be selected and verified using tools that include Akaike's information criterion (AIC), numerical analysis, residual point processes and Monte Carlo simulation.

It is well known that a simple closed-form formula exists for the stationary distribution of the workload in M/GI/1 queues. In this paper, we extend this to the general stationary framework. Namely, we consider a work-conserving single-server queueing system, where the sequence of customers’ arrival epochs and their service times is described as a general stationary marked point process, and we derive a closed-form formula for the stationary workload distribution. The key to our proof is two-fold: one is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The other is the preemptive-resume last-come, first-served discipline.

Consider a Navier-Stokes incompressible turbulent fluid in R2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

For many years the modified Mercalli (MM) scale has been used to describe earthquake damage and effects observed at scattered locations. In the next stage of an analysis involving MM data, isoseismal lines based on the observations have been added to maps by hand, i.e. subjectively. However a few objective methods have been proposed (by e.g. De Rubeis et al., Brillinger, Wald et al. and Pettenati et al.). The work presented here develops objective methods further. In particular the ordinal character of the MM scale is specifically taken into account. Numerical smoothing is basic to the approach and methods involving splines, local polynomial regression and wavelets are illustrated. The approach also allows the inclusion of explanatory variables, for example site effects. The procedure is implemented for data from the 17 October 1989 Loma Prieta earthquake.

Products of independent identically distributed random stochastic 2 × 2 matrices are known to converge in distribution under a trivial condition. Rates for this convergence are estimated in terms of the minimal Lp-metrics and the Kolmogoroff metric and applications to convergence rates of related interval splitting procedures are discussed.

Let Xt be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose Xt is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X1) | Xs1Xs∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.

As models for molecular evolution, immune response, and local search algorithms, various authors have used a stochastic process called the evolutionary walk, which goes as follows. Assign a random number to each vertex of the infinite N-ary tree, and start with a particle on the root. A step of the process consists of searching for a child with a higher number and moving the particle there; if no such child exists, the process stops. The average number of steps in this process is asymptotic, as N → ∞, to log N, a result first proved by Macken and Perelson. This paper relates the evolutionary walk to a process called random bisection, familiar from combinatorics and number theory, which can be thought of as a transformed Poisson process. We first give a thorough treatment of the exact walk length, computing its distribution, moments and moment generating function. Next we show that the walk length is asymptotically normally distributed. We also treat it as a mixture of Poisson random variables and compute the asymptotic distribution of the Poisson parameter. Finally, we show that in an evolutionary walk with uniform vertex numbers, the ‘jumps’, ordered by size, have the same asymptotic distribution as the normalized cycle lengths in a random permutation.

Some new results about the NBU(2) class of life distributions are obtained. Firstly, it is proved that the decreasing with time of the increasing concave ordering of the excess lifetime in a renewal process leads to the NBU(2) property of the interarrival times. Secondly, the NBU(2) class of life distributions is proved to be closed under the formation of series systems. Finally, it is also shown that the NBU(2) class is closed under convolution operation.