A k-out-of-n system consisting of n components is one that works if and only if at least k of the n components work. Suppose there are n+r (1 ≤ r ≤ n) components available of which r will be used for active redundancy. From the given n+r components, r components are chosen to be used as active redundancies, and another r components receive active redundancies (i.e. these r components are bolstered). Consider a k-out-of-n system consisting of the r bolstered and the other n – r original components. The problem of which r components should be used for active redundancy, and where to allocate them in order to maximize the lifetime of the resulting k-out-of-n system is studied. It is shown that under the usual stochastic ordering ‘≤st’ the first r weakest components should be used for active redundancy and allocated in reverse order to the next r weakest components.

In statistical diagnostics and sensitivity analysis, the local influence method plays an important rôle. In the present paper, we use this method to study financial time series data and conditionally heteroskedastic models under elliptical distributions. We start with a likelihood displacement, and consider data- and model-perturbation schemes. We obtain corresponding matrices of derivatives, and measures of slope and normal curvature, and then discuss the assessment of local influence.

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.

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(logd–1n) for any polytope, the expected number of vertices is Ω(logd–1n) for any simple polytope, and the expected number of facets is O(logd–1n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.

The new concept of embedded level crossings is combined with the old principle of stationary set balance to produce an alternative approach for obtaining the steady-state distribution of the level in a dam with general release rule. The method yields the steady state distribution of the customer waiting time in the GI/G/1 queue as a special case. Results for a dam in which the instantaneous release rate is proportional to the level, and for the M/G/1, GI/M/1, Ek/M/1 and D/M/1 queues are derived using the new technique.

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

Let n be a positive integer. Denote by Mn the convex, closed, bounded, and n-dimensional set of all n-tuples (c1, c2,···, cn) such that for some probability measure a on the Borel subsets of the unit interval I = [0,1].

Extreme-value theory is considered in the context of independent but not identically distributed random variables. The departure from the case of identical distributions comes from a trend added to i.i.d. observations. Unlike previous authors we consider trends that do not completely destroy the asymptotic behaviour from the i.i.d. case.

In the framework of patterns in random texts, the Markov chain embedding techniques consist of turning the occurrences of a pattern over an order-m Markov sequence into those of a subset of states into an order-1 Markov chain. In this paper we use the theory of language and automata to provide space-optimal Markov chain embedding using the new notion of pattern Markov chains (PMCs), and we give explicit constructive algorithms to build the PMC associated to any given pattern problem. The interest of PMCs is then illustrated through the exact computation of P-values whose complexity is discussed and compared to other classical asymptotic approximations. Finally, we consider two illustrative examples of highly degenerated pattern problems (structured motifs and PROSITE signatures), which further illustrate the usefulness of our approach.

Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.

Stopping rules in the context of proofreading are studied. Previous results by Yang et al. [6] are corrected and a computational procedure to obtain an optimal rule is presented.

Let Xn be non-negative random variables, possessing the Markov property. We given criteria for deciding whether Pr(Xn →∞) is positive or 0. It turns out that essentially this depends on the magnitude of E(Xn+1 | Xn = x) compared to that of E(X2n+1 | Xn = x) for large x. The assumptions are chosen such that for example population-dependent branching processes can be treated by our results.

In this paper, we study a repair replacement model for a stochastically deteriorating system. For the expected discounted reward case, we show that the optimal replacement policy is of the form ‘replace at the time of the Nth failure'.

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

1. Recently Weiss [7] constructed a stochastic model for the spread of diseases e.g. typhoid, by carriers. He assumed that a carrier is eliminated in (t, t + dt) with probability βrdt, where r is the number of carriers extant at time t and that a susceptible is infected in (t, t + dt) with probability arsdt where s is the number of susceptibles at time t. Thus only the carriers are responsible for the spread of the disease.

A random mapping of a countable set N into itself is studied. An explicit formula is found for the probability that a subset of N will not contain cycles of the mapping. This formula is applied then for a detailed asymptotic analysis of two special mappings.

In genetics one frequently encounters variates which behave, to a close approximation, as diffusion variates with drift and diffusion coefficients m(x),v(x), respectively, where after a suitable rescaling of the time axis, m(x) and v(x) are of the form

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

A composite material is a parallel arrangement of stiff brittle fibers in a flexible matrix. Under load fibers fail, and the loads of failed fibers are locally redistributed onto nearby survivors through the matrix. In this paper we develop a new technique for computing the probability of failure under a previously studied model of the failure process. A recursion and limit theorem are obtained which apply separately to static strength and fatigue lifetime depending on the composite loading and the probability model for the failure of individual fibers under their own loads. The limit theorem yields an approximation for the distribution function for composite lifetime which is of the form 1 – [1 – W(t)]mn where W(t) is a characteristic distribution function and mn is the composite volume, reflecting a size effect. A similar result holds also for static strength. In both cases such a result was conjectured several years ago. This limit theorem is obtained from the recursion upon applying a key theorem in the theory of the renewal equation. In the proofs three technical conditions arise which must be verified in specific applications. In the case of static strength these conditions are quite easy to verify, but in the case of fatigue lifetime the verification is generally difficult, and entails considerable numerical computation.