In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α 1 ≤α 0 < + ∞ and 0 < C 1, C 2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C 1 k -α 0 and above by C 2 k -α 1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.

In this paper we present some necessary conditions for the uniqueness, recurrence, and ergodicity of a class of multidimensional Q-processes, using the dual Yan-Chen comparison method. Then the coupling method is used to study the multidimensional processes in a specific space. As applications, three models of particle systems are illustrated.

Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.

It is proved that the strong Doeblin condition (i.e., ps(x,y) ≥ asπ(y) for all x,y in the state space) implies convergence in the relative supremum norm for a general Markov chain. The convergence is geometric with rate (1 - as)1/s. If the detailed balance condition and a weak continuity condition are satisfied, then the strong Doeblin condition is equivalent to convergence in the relative supremum norm. Convergence in other norms under weaker assumptions is proved. The results give qualitative understanding of the convergence.

We study discrete-time population models where the nearest future of an individual may depend on the individual's life-stage (age and reproduction history) and the current population size. A criterion is given for whether there is a positive probability that the population survives forever. We identify the cases when population size grows exponentially and linearly and show that in the latter population size scaled by time is asymptotically Γ-distributed.

The set of transient states of a Markov chain is considered as a system. If numbers of arrivals to the system at discrete time points have constant mean and covariance matrix then there is a limiting distribution of numbers in the states. Necessary and sufficient conditions are given for this distribution to yield zero correlations between states.

We propose a stochastic model describing a process of awareness, evaluation, and decision making by agents on the d-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0, 1, 2. In this model 0 stands for ignorants, 1 for aware, and 2 for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate λ. At rate α an agent in aware state becomes an adopter due to the influence of adopters' neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant, at rate 1. Our purpose is to analyze the influence of the parameters on the qualitative behavior of the process. We obtain sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability.

A certain single server queueing system with negative exponential service with mean rate nμ, when the system contains n customers, and Poisson arrivals, is formally equivalent to the infinite capacity system M/M/∞. This equivalence is exploited to yield in a very simple manner results for the single server system which were previously obtained by difficult analysis (see Hadidi (1969)).

In this article we provide new results for the asymptotic behavior of a time-fractional birth and death process $N_{\alpha}(t)$ , whose transition probabilities $\mathbb{P}[N_{\alpha}(t)=\,j\mid N_{\alpha}(0)=i]$ are governed by a time-fractional system of differential equations, under the condition that it is not killed. More specifically, we prove that the concepts of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. In addition, exact formulas for the quasi-limiting distribution and its rate convergence are presented. In the first sections, we revisit the two equivalent characterizations for this process: the first one is a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Finally, we apply our main theorems to the linear model originally introduced by Orsingher and Polito [23].

The coupon-collector's and sampling tagging problems are considered, under the model that each tagged element is γ > 0 times as likely to be caught at the next stage as every untagged element. Let WN(γ, k) and SN(γ, j) denote, respectively, the waiting time until k + 1 distinct elements are obtained, and the number of distinct elements in a sample of size j, when the population size is N. Complete characterisations are obtained for the limiting distributions of WN (γ, k) and SN(γ, j), in terms of the rates at which k and j tend to infinity with N.

An anticipation process is a continuous-time Markov process, having an element of deterministic drift in its transition structure. The theory of such processes, developed here from an elementary (Chapman-Kolmogorov) point of view, draws together a number of threads of theoretical and applied probability.

The busy period of the GI/D/∞ queue is determined as the time when at least one customer is served. Let v be the number of customers served during this period. The busy period of order n is defined as a busy period for which v ≧ n. In this paper we derive the exact distributions, integral equations, characteristic functions and all moments of those periods. Finally, some properties of the idle periods of order n are established.

We consider a semimartingale X which is reflected at an upper barrier T and a lower barrier S, where S and T are also semimartingales such that T is bounded away from S. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective barriers to X, S, and T. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions on X, S, and T, e.g. if we take X, S, and T to be independent martingales (which satisfy some extra technical conditions).

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

Two classes of finite and infinite moving-average sequences of bivariate random vectors are considered. The first class has bivariate exponential marginals while the second class has bivariate geometric marginals. The theory of positive dependence is used to show that in various cases the two classes consist of associated random variables. Association is then applied to establish moment inequalities and to obtain approximations to some joint probabilities of the bivariate processes.

Consider a situation where a known number, n, of objects appear sequentially in a random order. At each stage, the present object is presented to d ≥ 2 different selectors, who must jointly decide whether to select or reject it, irrevocably. Exactly one object must be chosen. The observation at stage j is a d-dimensional vector R(j) = (R1(j),…, Rd(j)), where Ri(j) is the relative rank of the jth object, by the criterion of the ith selector. The decision whether to stop or not at time j is based on the d-dimensional random vectors R(1),…, R(j). The criteria according to which each selector ranks the objects can either be dependent or independent. Although the goal of each selector is to maximize the probability of choosing the best object from his/her point of view, all d selectors must cooperate and chose the same object. The objective studied here is the maximization of the minimum over the d individual probabilities of choosing the best object. We exhibit the structure of the optimal rule. For independent criteria we give a full description of the rule and show that the optimal value tends to d-d/(d-1), as n → ∞. Furthermore, we show that as n → ∞, the liminf of the values under negatively associated criteria is bounded below by d-d/(d-1).

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.

The utility of the moments of the binomial distribution in finding the n-stage transition probabilities was indicated by Khazanie and McKean (1966). In this paper, maintaining the same breeding structure as in the two-allele case, it is proposed to extend the technique to cover the multiallelic case in general, with specific emphasis on the triallelic case. The existing literature on the multiple allelic case is at best scanty (see for example Kimura (1955)). Section 6 will briefly touch on a further extension to multiple loci which segregate independently.

The purpose of this paper is to analyse the real-time trading of electricity. We address a model for an auction-like trading which captures key features of real-world electricity markets. Our main result establishes that, under certain conditions, the expected total payment for electricity is independent of the particular auction type. This result is analogous to the revenue-equivalence theorem known for classical auctions and could contribute to an improved understanding of different electricity market designs and their comparison.

Motivated by applications to wireless networks, cloud computing, data centers, etc., stochastic processing networks have been studied in the literature under various asymptotic regimes. In the heavy traffic regime, the steady-state mean queue length is proved to be $\Theta({1}/{\epsilon})$ , where $\epsilon$ is the heavy traffic parameter (which goes to zero in the limit). The focus of this paper is on obtaining queue length bounds on pre-limit systems, thus establishing the rate of convergence to heavy traffic. For the generalized switch, operating under the MaxWeight algorithm, we show that the mean queue length is within $\textrm{O}({\log}({1}/{\epsilon}))$ of its heavy traffic limit. This result holds regardless of the complete resource pooling (CRP) condition being satisfied. Furthermore, when the CRP condition is satisfied, we show that the mean queue length under the MaxWeight algorithm is within $\textrm{O}({\log}({1}/{\epsilon}))$ of the optimal scheduling policy. Finally, we obtain similar results for the rate of convergence to heavy traffic of the total queue length in load balancing systems operating under the ‘join the shortest queue’ routeing algorithm.