A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

A general age-dependent branching process is studied in which the population is distributed in N colonies. Migration from one colony to another is possible and no particular geometrical configuration is assumed for the colonies. Asymptotic expressions for the first two moments are given in several different cases.

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.

Mathematical models have been proposed for oil exploration and other kinds of search. They can be used to estimate the amount of undiscovered resources or to investigate optimal stopping times for the search. Here we consider a continuous search for hidden objects using a model which represents the number and values of the objects by mixtures of Poisson processes. The flexibility of the model and its complexity depend on the number of components in the mixture. In simple cases, optimal stopping rules can be found explicitly and more general qualitative results can sometimes be obtained.

Finiteness of mean extinction times for certain predator-prey models has been established by Hitchcock (1986) with the aid of a criterion of Reuter (1957). Using this criterion and a ‘minimisation' lemma this note shows that the mean extinction times tend to zero as the combined initial population of predators and prey becomes large.

Formulas for the asymptotic failure rate, long-term average availability, and the limiting distribution of the number of long ‘outages' are obtained for a general class of two-state reliability models for maintained systems. The results extend known formulas for alternating renewal processes to a wider class of point processes that includes sequences of dependent or non-identically distributed operating and repair times.

We discuss the limits of point processes which are generated by a triangular array of rare events. Such point processes are motivated by the exceedances of a high boundary by a random sequence since exceedances are rare events in this case. This application relates the problem to extreme value theory from where the method is used to treat the asymptotic approximation of these point processes. The presented general approach extends, unifies and clarifies some of the various conditions used in the extreme value theory.

Two competitors observe a Poisson stream of offers. The offers are independent and identically distributed random variables from some continuous distribution. Each of the competitors wishes to accept one offer in the interval [0, T] and each aims to select an offer larger than that of his competitor. Offers are observed sequentially and decisions to accept or reject must be made when the offers arrive. Optimal strategies and winning probabilities are obtained for the competitors under a priorized decision scheme. The time of first offer acceptance is also analyzed. In all cases the asymptotic results are obtained.

This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al. (2009) and Van Leeuwaarden and Winands (2006). The two methodologies are compared both in terms of applicability requirements and numerical complexity by analyzing their performance for the same classical queueing models considered in Van Leeuwaarden et al. (2009). The main findings are threefold. First, when both methods are applicable, the SL-based algorithms outperform the lattice path counting algorithm (LPCA). Second, there are important classes of problems (for example, models with (level) nonhomogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. Third, another main advantage of SL algorithms over lattice path counting is that the former includes a method to compute the steady state distribution using this rate matrix.

We consider a continuous, infinitely divisible random field in Rd given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields we compute the asymptotic probability that the supremum of the field exceeds the level x as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let Sn denote the partial sum Sn = θ(ξ1) + … + θ(ξn), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼SN = [limn→∞𝔼θ(ξn)]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξN). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξN) = o(𝔼N).

The optimality of the one step look-ahead stopping rule is shown to hold under conditions different from those discussed by Chow, Robbins and Seigmund [5]. These results are corollaries of the following theorem: Let {Xn, n = 0, 1, …}; X0 = x be a discrete-time homogeneous Markov process with state space (E, ℬ). For any ℬ-measurable function g and α in (0, 1], define Aαg(x) = αExg(X1) – g(x) to be the infinitesimal generator of g. If τ is any stopping time satisfying the conditions: Ex[αNg(XN)I(τ > N)]→0 as as N → ∞, then Applications of the results are considered.

Given a sequence (Mk, Qk)k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (Xn)n ≥ 0, defined by the random difference equation Xn = MnXn - 1 + Qn for n ≥ 1, where X0 is independent of (Mk, Qk)k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (Xn)n ≥ 0 is contractive in the sense that M1 ⋯ Mn → 0 almost surely, yet occasional large values of the Qn overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (Xn)n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.

We consider a multicomponent maintenance system controlled by an age replacement policy: when one of the components fails, it is immediately replaced; all components older than a threshold age θ are preventively replaced. Costs are associated with each maintenance action, such as replacement after failure or preventive replacement. We derive a weak derivative estimator for the derivative of the cost performance with respect to θ. The technique is quite general and can be applied to many other threshold optimization problems in maintenance. The estimator is easy to implement and considerably increases the efficiency of a Robbins-Monro type of stochastic approximation algorithm. The paper is self-contained in the sense that it includes a proof of the correctness of the weak derivative estimation algorithm.

Motivated by the need of studying a subset of components, ‘separate' from the other components, we introduce a new definition of ‘marginal distribution'. This is done by fixing the lives of the other components, but without the ‘knowledge' of the components of interest. Formally this is done by minimally repairing the components of no interest up to a predetermined time. Preservation properties of these ‘conditional marginal distributions', with respect to several stochastic orderings, are obtained. Also, inheritance of positive dependence properties, by the conditional marginal distributions, is shown. In addition, the preservations of dynamic multivariate aging properties, by the dynamic conditional marginal distributions, are obtained. The definitions and results are illustrated by a set of examples. Some applications for modelling ‘combinations' of sets of random lifetimes, and for bounding complex sets of random lifetimes, are described.

Assigning each arriving customer to the fastest idle server is shown to maximize throughput (equivalently, minimize blocking probability) in a queueing model with Poisson arrivals, heterogeneous exponential servers, and no waiting room. If a cost structure is imposed on this model, under specified conditions the same policy minimizes the expected discounted cost and the long-run average cost per unit time.

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.