In this paper we aim to apply simple actuarial methods to build an insurance plan protecting against an epidemic risk in a population. The studied model is an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals. The costs due to the epidemic are measured through the expected epidemic size and infectivity time. The premiums received during the epidemic outbreak are measured through the expected susceptibility time. Using martingale arguments, a method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model. Some numerical examples illustrate the effect of removals and the premium calculation in an insurance plan.

A characterization of the exponential distribution was obtained by Grosswald et al. (1980) using the relevation transform introduced by Krakowski (1973). Here we obtain an improved version of the result in Grosswald et al. (1980).

We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk and of the finiteness of both primal and dual value functions.

Processes that exhibit repeatability in their kth-order moments are frequently studied in signal analysis. Such repeatability can be conveniently expressed with the help of almost periodicity. In particular, almost periodically correlated (APC) processes play an important role in the analysis of repeatable signals. This paper presents a study of asymptotic distributions of the estimator of the spectral covariance function for APC processes. It is demonstrated that, for a large class of APC processes, the functional central limit theorem holds.

Some inequalities are obtained which yield bounds for the mean life of series and of parallel systems in the case where component life distributions have properties such as a monotone failure rate, monotone failure rate average, or decreasing density. These bounds are based on comparisons with systems of exponential or uniform components. Similar comparisons are obtained when components have Weibull or Gamma distributions with different shape parameters. Some inequalities are also obtained for convolutions of life distributions helpful in the study of replacement policies.

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which yi → ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk < yo, · · ·, yN–1 for some k ≧ N, is necessary and sufficient for transience.

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin (t) be the state probabilities and p∗in the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn = λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p∗i0, p∗in (n ≥ 1) being determined recursively. An exact expression for p10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

Design engineers are well aware that a system where active spare allocation is made at the component level has a lifetime stochastically larger than the corresponding system where active spare allocation is made at the system level. In view of the importance of hazard rate ordering in reliability and survival analysis, Boland and El-Neweihi (1995) recently investigated this principle in hazard rate ordering and demonstrated that it does not hold in general. They showed that for a 2-out-of-n system with independent and identical components and spares, active spare allocation at the component level is superior to active spare allocation at the system level. They conjectured that such a principle holds in general for a k-out-of-n system when components and spares are independent and identical. We prove that for a k-out-of-n system where components and spares have independent and identical life distributions active spare allocation at the component level is superior to active spare allocation at the system level in likelihood ratio ordering. This is stronger than hazard rate ordering, thus establishing the conjecture of Boland and El-Neweihi (1995).

We analyze certain self-organizing filing techniques when accesses are assumed to be dependent on each other. The stream of requests for accessing records in a file is modelled as a Markov chain. A general framework is introduced to obtain the asymptotic search cost of a memory-free self-organizing heuristic. The move-to-front heuristic is studied in detail. A formula for the asymptotic search cost, which generalizes that in the case of independent accesses, is obtained. Numerical examples on the performance of the transposition heuristic are provided, and compared with that of the move-to-front heuristic.

In this paper we review some recent work on limit results on realised power variation, that is, sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, quadratic variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.

This paper describes a set of stochastic processes that is useful for modeling and analyzing a new genetic mapping method. Some of the processes are Markov chains, and some are best described as functions of Markov chains. The central issue is boundary-crossing probabilities, which correspond to p-values for the existence of genes for particular traits. The methods elaborated by Aldous (1989) provide very accurate approximate p-values, as spot-checked against simulations.

The relationship between consistency of model selection and that of parameter estimation is investigated. It is shown that the consistency of model selection is achieved at the cost of a lower order of consistency of the resulting estimate of parameters in some domain. The situation is different when selecting autoregressive moving average models, since the information matrix becomes singular when overfitted. Some detailed analyses of the consistency are given in this case.

Laha and Lukacs (1960) have studied distributions with the property that a quadratic statistic has quadratic regression on the sample mean. In doing this, they have arrived at some interesting characterizations for the normal, Poisson, gamma, binomial and negative binomial distributions. Starting with an exponential-type probability density function, the present paper investigates all the distributions for which the 3 × 3 Bhattacharya matrix is diagonal. It is found that the normal, Poisson, gamma, binomial and negative binomial distributions can be characterized by this property. Further, it is observed that for these distributions an s × s Bhattacharya matrix is defined for all s and is also diagonal.

A model for road traffic delays at intersections is considered where vehicles arriving, possibly in bunches, in a Poisson process in a one way minor road yield right of way to traffic, which forms alternate bunches and gaps, in a major road. The gap acceptance times are random variables, and depend on whether or not a minor road vehicle is immediately following another minor road vehicle into the intersection or not.

The transforms of the stationary waiting time and queue size distributions, and the mean stationary delay, for minor road vehicles are obtained by substitution of determined service time distributions into results for a generalisation of the M/G/1 queueing system. Some numerical results are given to illustrate the increase in the mean delay for variable gap acceptance times for a Borel-Tanner distribution of major road traffic, and a partial solution is given for a two way major road.

We consider the problem of routing jobs to parallel queues with identical exponential servers and unequal finite buffer capacities. Service rates are state-dependent and non-decreasing with respect to queue lengths. We establish the extremal properties of the shortest non-full queue (SNQ) and the longest non-full queue (LNQ) policies, in systems with concave/convex service rates. Our analysis is based on the weak majorization of joint queue lengths which leads to stochastic orderings of critical performance indices. Moreover, we solve the buffer allocation problem, i.e. the problem of how to distribute a number of buffers among the queues. The two optimal allocation schemes are also ‘extreme', in the sense of capacity balancing. Some extensions are also discussed.

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues under standard heavy-traffic scalings. The expressions require the solution of a set of only M—N linear equations, where M is the length of the polling table and N is the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routeing schemes commonly used in practice, such as the star and elevator visit order, in a general parameter setting. The results reveal a number of asymptotic properties of the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routeing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))i∊S letters from a finite alphabet S and v=(v(i))i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.

Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.