The paper considers a random sample of r chromosomes, each having n genes subject to intrachromosomal gene conversion, and mutation. The probability distribution and moments for the number of alleles present is investigated, when the number, k, of possible alleles at each locus, is either finite or infinite. Explicit formulas are given for the mean numbers of alleles on r = 1, 2, or 3 chromosomes, which simplify previously known results. For fixed r, in the infinitely-many-alleles case, the mean number increases asymptotically like r θ log (n) as n→∞, where θ is a mutation parameter. But results for large samples remain elusive.

Let Y0, Y1, Y2, … be an i.i.d. sequence of random variables with continuous distribution function, and let P be a simple point process on 0≦t≦∞, independent of the Yj's. We assume that P has a point at t = 0; we associate Yj with the jth point of j≧0, and we say that the Yj's occur at the arrival times of P. Y0 is considered a ‘reference value'. The first Yj (j≧1) to exceed all previous ones is called the first ‘record value', and the time of its occurrence is the first ‘record time'. Subsequent record values and times are defined analogously. We give an infinite series representation for the joint characteristic function of the first n record times, for general P; in some cases the series can be summed. We find the intensity of the record process when P is a general birth process, and when P is a linear birth process with m immigration sources we find the distribution of the number of records in (0, t]. For m = 0 (the Yule process) we give moments of record times and a compact form for the record process intensity. We show that the records occur according to a homogeneous Poisson process when m = 1, and we display a different model with the same behavior, leading to statistical non-identifiability if only the record times are observed. For m = 2, the records occur according to a semi-Markov process; again we display a different model with the same behavior. Finally we give a new derivation of the joint distribution of the interrecord times when P is an arbitrary Poisson process. We relate this result to existing work and to the classical record model. We also obtain a new characterization of the exponential distribution.

The Voronoi tessellation generated by a Gibbs point process is considered. Using the algebraic formalism of polymer expansion, the limit theorem and the large deviation principle for the number of Voronoi vertices are proved.

An efficient algorithm to compute upper and lower bounds for the first-passage time in the presence of a second absorbing barrier by means of a continuously differentiable decomposable process, e.g. a smooth function of a continuously differentiable Gaussian vector field, is given. The method is used to obtain accurate approximations for the joint density of the zero-crossing wavelength and amplitude and the distribution of the rainflow cycle amplitude. Numerical examples illustrating the results are also given.

The necessary and sufficient condition for unilateral characterization of Gaussian Markov fields and the Besag-Moran positivity condition for second-order autonormal bilateral models define the same tetrahedral domain of achievable regression parameters. A bijective function maps this domain to a different tetrahedral domain of parameters in the Pickard model. These two domains are identical to the corresponding ones in the Welberry-Carroll model. We obtain series solutions for correlation coefficients and study their limits near the boundaries of the first domain.

Let s1, …, sn be generated governed by an r-state irreducible aperiodic Markov chain. The partial sum process is determined by a realization of states with s0 = α and the real-valued i.i.d. bounded variables Xαß associated with the transitions si = α, si+1 = β. Assume Χ αβ has negative stationary mean. The explicit limit distribution of the maximal segmental sum is derived. Computational methods with potential applications to the analysis of random Markov-dependent letter sequences (e.g. DNA and protein sequences) are presented.

Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

This paper studies computer simulation methods for estimating the sensitivities (gradient, Hessian etc.) of the expected steady-state performance of a queueing model with respect to the vector of parameters of the underlying distribution (an example is the gradient of the expected steady-state waiting time of a customer at a particular node in a queueing network with respect to its service rate). It is shown that such a sensitivity can be represented as the covariance between two processes, the standard output process (say the waiting time process) and what we call the score function process which is based on the score function. Simulation procedures based upon such representations are discussed, and in particular a control variate method is presented. The estimators and the score function process are then studied under heavy traffic conditions. The score function process, when properly normalized, is shown to have a heavy traffic limit involving a certain variant of two-dimensional Brownian motion for which we describe the stationary distribution. From this, heavy traffic (diffusion) approximations for the variance constants in the large sample theory can be computed and are used as a basis for comparing different simulation estimators. Finally, the theory is supported by numerical results.

This paper complements two previous studies (Daley and Rolski (1984), (1991)) by investigating limit properties of the waiting time in k-server queues with renewal arrival process under ‘light traffic' conditions. Formulae for the limits of the probability of waiting and the waiting time moments are derived for the two approaches of dilation and thinning of the arrival process. Asmussen's (1991) approach to light traffic limits applies to the cases considered, of which the Poisson arrival process (i.e. M/G/k) is a special case and for which formulae are given.

We have furnished further examples on the connection between some standard one-dimensional chaotic deterministic models and stochastic time series models via time reversal.

We give a finite form for the probability mass function of the wrapped Poisson distribution, together with a probabilistic proof. We also describe briefly its connection with existing results.

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk) and H(t) is established.

Two types of minimal repair are discussed. After statistical minimal repair, the state of a system is statistically identical to what it was just before the failure. Physical minimal repair restores the failed unit to its exact physical condition before the failure. Several examples are given.

This note is concerned with the continuing misconception that a discrete-time network of queues, with independent customer routing and the number of arrivals and services in a time interval following geometric and truncated geometric distributions respectively, has a product-form equilibrium distribution.

In this note we give a counterexample which shows that the µc-rule is not optimal in the second node of the tandem queue. This counterexample contradicts the interchange argument in Nain [1] and Nain et al. [2].

A result relating the probability that kth customer finds the system empty to the distribution of the number of customers served in a busy period, for an M/M/1 queue, has been obtained. This relationship is similar to the relationship between the probability that the queue is empty at time t and the distribution of the length of the busy period.