Unbiased stereological estimators of d-dimensional volume in ℝn are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝn through O hits a fixed point in ℝn is given.

The maximum of a Gaussian random field was used by Worsley et al. (1992) to test for activation at an unknown point in positron emission tomography images of blood flow in the human brain. The Euler characteristic of excursion sets was used as an estimator of the number of regions of activation. The expected Euler characteristic of excursion sets of stationary Gaussian random fields has been derived by Adler and Hasofer (1976) and Adler (1981). In this paper we extend the results of Adler (1981) to χ2, F and t fields. The theory is applied to some three-dimensional images of cerebral blood flow from a study on pain perception.

This paper discusses a simple extension of the classical Voronoi tessellation. Instead of using the Euclidean distance to decide the domains corresponding to the cell centers, another translation-invariant distance is used. The resulting tessellation is a scaled version of the usual Voronoi tessellation. Formulas for the mean characteristics (e.g. mean perimeter, surface and volume) of the cells are provided in the case of cell centers from a homogeneous Poisson process. The resulting tessellation is stationary and ergodic but not isotropic.

Consider a forest of maple trees in autumn, with leaves falling on the ground. Those coming late cover the others below, so eventually the fallen leaves form a statistically homogeneous spatial pattern. In particular, the uncovered leaf boundaries form a mosaic. We formulate a mathematical model to describe this mosaic, firstly in the case where the leaves are polygonal and later for leaves with curved boundaries. Mean values of certain statistics of the mosaic are derived.

When a random electrical network has the structure of a rooted tree and the edge resistances are either inverse Gaussian or reciprocal inverse Gaussian random variables then, subject to some restrictions, the overall resistance of the network is shown to follow a reciprocal inverse Gaussian distribution.

We describe a computational procedure for evaluating the quasi-stationary distributions of a continuous-time Markov chain. Our method, which is an ‘iterative version' of Arnoldi's algorithm, is appropriate for dealing with cases where the matrix of transition rates is large and sparse, but does not exhibit a banded structure which might otherwise be usefully exploited. We illustrate the method with reference to an epidemic model and we compare the computed quasi-stationary distribution with an appropriate diffusion approximation.

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

We present two forms of weak majorization, namely, very weak majorization and p-weak majorization that can be used as sample path criteria in the analysis of queueing systems. We demonstrate how these two criteria can be used in making comparisons among the joint queue lengths of queueing systems with blocking and/or multiple classes, by capturing an interesting interaction between state and performance descriptors. As a result, stochastic orderings on performance measures such as the cumulative number of losses can be derived. We describe applications that involve the determination of optimal policies in the context of load-balancing and scheduling.

Consider the total service time of a job on an unreliable server under preemptive-repeat-different and preemptive-resume service disciplines. With identical initial conditions, for both cases, we notice that the distributions of the total service time under these two disciplines coincide, when the original service time (without interruptions due to server failures) is exponential and independent of the server reliability. We show that this fact under varying server reliability is a characterization of the exponential distribution. Further we show, under the same initial conditions, that the coincidence of the mean values also leads to the same characterization.

Obtaining good estimates for the distribution function of random variables like (‘perpetuity’) and (‘aggregate claim amount’), where the (Yi), (Zi) are independent i.i.d. sequences and (N(t)) is a general point process, is a key question in insurance mathematics. In this paper, we show how suitably chosen metrics provide a theoretical justification for bootstrap estimation in these cases. In the perpetuity case, we also give a detailed discussion of how the method works in practice.

Burn-in is a widely used method to improve quality of products after they have been produced. For a repairable component there are two common types of repair, complete repair and minimal repair. Preventive maintenance policies such as age replacement and block replacement are often employed in field operation. The present paper takes burn-in, maintenance and repair into consideration at the same time and considers related cost structures. The properties of the corresponding optimal burn-in times and optimal maintenance policies are discussed.

In this paper we study the extremal properties of several scheduling policies in an in-forest network consisting of multiserver queues. Each customer has a due date, and we assume that service times at the different queues form mutually independent sequences of independent and identically distributed random variables independent of the arrival times and due dates. Furthermore, the network is assumed to consist of a mixture of nodes, some of which permit only non-preemptive service policies whereas the others permit preemptive resume policies. In the case of non-preemptive queues, service times may be generally distributed if there is only one server; otherwise the service times are required to be increasing in likelihood ratio (ILR). In the case of preemptive queues, service times are restricted to exponential distributions. Using stochastic majorizations and partial orders on permutations, we establish that, within the class of work conserving service policies, the stochastically smallest due date (SSDD) and the stochastically largest due date (SLDD) policies minimize and maximize, respectively, the vector of the customer latenesses of the first n customers in the sense of the Schur-convex order and some weaker orders, provided the due dates are comparable in some stochastic sense. It then follows that the first come-first served (FCFS) and the last come-first served (LCFS) policies minimize and maximize, respectively, the vector of the response times of the first n customers in the sense of the Schur-convex order. We also show that the FCFS and LCFS policies minimize and maximize, respectively, the vector of customer end-to-end delays in the sense of the strong stochastic order. Extensions to the class of non-idling policies and to the stationary regime are also given.

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

A number of jobs on two interconnected queues are to be processed by m identical servers. The servers operate in parallel, so that every server can process any job. Jobs in queue i, i = 1, 2, incur an instantaneous holding cost Ci during the time they remain in the system. The service time for jobs in queue i, denoted by Xi, is a random variable with a general distribution. The interconnection process is independent of the service process. We establish sufficient conditions on the service times, the holding costs and the interconnection process under which the non-preemptive scheduling strategy that gives priority to queue 1 minimizes the total expected α -discounted cost. We call this strategy P1. We present counterexamples showing that if any of the sufficient conditions is not satisfied P1 may not be optimal, and that the optimal policy for the single-server problem is not necessarily optimal for the multiserver problem.