The purpose of this paper is to give some very simple applications of martingales to epidemics. The results are all connected with stopping times T (for instance the classical end of epidemic) and include the expression of the joint generating function Laplace transform of and and simple relations between moments of these three variables. (Here Xt and Yt respectively denote the numbers of susceptibles and carriers.) We also give several relations between different types of epidemics. Although this paper only deals with Downton's model, some of the methods are still valid for more general models.

In this article, we assume that the state of a system forms a continuous-time Markov chain or a higher-dimensional Markov process after introducing some supplementary variables. A formula for evaluating the rate of occurrence of failures for the system is derived. As an application of the theory, a maintenance model for a two-component system is also studied.

A system is subject to a sequence of shocks occurring randomly at times n = 1, 2, ···; each shock causes a random amount of damage. The system might fail at any point in time n, and the probability of a failure depends on the history of the system. Upon failure the system is replaced by a new and identical system and a cost is incurred. If the system is replaced before failure a smaller cost is incurred. We study the problem of specifying a replacement rule which minimizes the long-run (expected) average cost per unit time. A special case, in which the system fails when the total damage first exceeds a fixed threshold, is analysed in detail.

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

In the present paper we study the number of occurrences of non-overlapping success runs of length in a sequence of (not necessarily identical) Bernoulli trials arranged on a circle. An exact formula is given for the probability function, along with some sharp bounds which turn out to be very useful in establishing limiting (Poisson convergence) results. Certain applications to statistical run tests and reliability theory are also discussed.

In many applications of queueing theory assumptions of either Poisson arrivals or exponential service times are made. The implicit assumption is that if the actual arrival process approximates a Poisson process and the service times are close to exponential, then the quantities of interest in the real queueing system (viz. the virtual waiting time, queue length, idle times, etc.), will approximate those of the idealized model. The continuity of the single server queue acting as functionals of the arrival and service processes is established. The proof involves an application of the theory of weak convergence of probability measures on metric spaces.

The problem is what to do with a slow server in a service facility which has fast and slow servers. Should the slow server be used to render service, or is it better not to use it at all? Simple models for answering this question are formulated and studied.

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.

Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.

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.

Given n events A1, A2, · ··, An, bounds are obtained for the probability of the occurrence of at least m out of the n events. The bounds are linear in terms of S1 and S2 where S1 = Σi P(Ai) and S2 = Σi<jΡ (Α i,Α j). The bounds are illustrated with an example. For m = 1 the bounds are the same as the best known linear bounds in terms of S1 and S2.

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.

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.

Recently, Rao et al. (2004) introduced an alternative measure of uncertainty known as the cumulative residual entropy (CRE). It is based on the survival (reliability) function F̅ instead of the probability density function f used in classical Shannon entropy. In reliability based system design, the performance characteristics of the coherent systems are of great importance. Accordingly, in this paper, we study the CRE for coherent and mixed systems when the component lifetimes are identically distributed. Bounds for the CRE of the system lifetime are obtained. We use these results to propose a measure to study if a system is close to series and parallel systems of the same size. Our results suggest that the CRE can be viewed as an alternative entropy (dispersion) measure to classical Shannon entropy.

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].

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.

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.

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}.

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.