The paper studies a single-server two-queue priority system with changeover times and switching threshold. The server serves queue 1 exhaustively and does not remain at an empty queue if the other one is non-empty. It immediately switches from queue 2 to queue 1 when the length of the latter reaches some level M. Whenever service is changed from one queue to the other a changeover time is required. Arrivals are Poisson, service times and changeover times are independent and exponentially distributed. Using an analytic method we obtain the steady-state joint probability generating function of the lengths of the two queues. By means of this probability generating function some performance measures of the system such as mean length of queue and mean delay can be calculated.

This paper investigates the probabilistic behaviour of the eigenvalue of the empirical transition matrix of a Markov chain which is of largest modulus other than 1, loosely called the second-largest eigenvalue. A central limit theorem is obtained for nonmultiple eigenvalues of the empirical transition matrix. When the Markov chain is actually a sequence of independent observations the distribution of the second-largest eigenvalue is determined and a test for independence is developed. The independence case is considered in more detail when the Markov chain has only two states, and some applications are given.

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

The Wright–Fisher model with varying population size is examined in the case where the selective advantage varies from generation to generation. Models are considered where the selective advantage is not always in favour of a particular genotype. Sufficient conditions in terms of the selection coefficients and the population growth are given to ensure ultimate homozygosity.

In a superfair red and black gambling house where the player must bet at least 2 units at each stage, a gambler wishes to maximize the probability of reaching a goal integer N before reaching zero. For win probability p > 1/2, when N is even, an optimal strategy is to bet 3 units when you have 3 units or N − 3 units and to bet 2 units otherwise. When N is odd, there are two strategies which are optimal depending on the value of the win probability p. When p is smaller than a certain value, p*, the above strategy is optimal, and when p is larger than this value, the timid strategy is optimal.

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.

This paper is concerned with the multidimensional Markov chain {X(n)} = {X(x, n)}, considered by Borovkov [1], [2], [3], with the form X(n + 1) = ?(n) + ?(?(n), n), where the distribution of depends only on x. Sufficient conditions on moments of |?(x)| are established for the Markov chain {X(n)} to have rate of convergence results (i.e. geometric ergodicity or sub-geometric rates for a variety of rate functions).

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

In many endemic areas the level of infection of malaria exhibits a regular cycle. A stochastic model is formulated to describe endemic malaria and is used to justify a set of deterministic equations. The model incorporates the properties peculiar to malaria, i.e., in endemic areas malaria is not usually fatal and an individual can be and usually is reinfected. The theory of small oscillations is applied to the deterministic equations, and under the assumption of stable populations of hosts and vectors an expression is obtained for the period of oscillation of the infection.

It is shown how to estimate centroids and other measures of location of an unknown embedded subset of a bounded specimen by random sectioning.

Let (Xt) and (Yt) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (Xt) and (Yt) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

We show in this paper that if a stationary traffic source is regulated by a leaky bucket with leak rate ρ and bucket size σ, then the amount of information generated in successive time intervals is dominated, in the increasing convex ordering sense, by that of a Poisson arrival process with rate ρ/σ, with each arrival bringing an amount of information equal to σ. By exploiting this property, we then show that the mean value in the stationary regime of the content of a buffer drained at constant rate and fed with the superposition of regulated flows is less than the mean value of the same buffer fed with an adequate Poisson process, whose characteristics depend upon the regulated input flows.

An alternative proof of a result of Blomqvist on the “heavy-traffic” problem for the two-barrier random walks using weak convergence methods is indicated.

The change with respect to time of the information in a population is studied, when the population is classified according to genetic type. The cases considered are (i) an infinite population classified at two loci, (ii) a large finite population classified at one locus, subject to sampling fluctuations and (iii) the same, subject to the effect of mutation.

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q1,Q2,…, Qn on a circle. If the counterclockwise way from Qi to Qj on the circle is shorter than the clockwise way, we say Qi dominates Qj. Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.

A basic process of simple queueing theory is S(t), an integer valued stochastic process which represents the “number of clients present, including the one in service” at epoch t. The queueing context physically limits S(t) to non-negative values, but if this impenetrable barrier is removed thereby permitting S(t) to assume negative values as well, we have a “randomized random walk” in which S(t) represents the position at time t of a mythical particle which moves on the x-axis according to the rules of the walk. The nomenclature “randomized random walk” is due to Feller (1966a). S(t) changes by unit positive or negative amounts, and our basic assumption is that the time intervals τ/σ separating successive positive/negative steps are i.i.d. with continuous d.f.'s. A(t)/B(t) (A(0)/B(0) = 0) possessing p.d.f.'s. a(t)/b(t): τ and σ are further assumed to be statistically independent. When A(t) = 1 – e–λt and B(t) = 1 – e–μt, we have a generalization of the M/M/1 queueing process which we shall denote by M/M. The walk generated by A(t) = 1 – e–λt and B(t) arbitrary may similarly be denoted by M/G. For G/M we clearly mean A(t) arbitrary and B(t) = 1 – e–μt. M/M has received extensive treatment elsewhere (cf. Feller (1966a), (1966b); Takács (1967); Gibson (1968); Conolly (1971)), and will not be considered here. Our interest centers on certain aspects of M/G and G/M, but since each is the dual of the other (loosely, G/M is M/G “upside down”) it is sufficient to restrict attention to one of them, as convenient; pointing out, where necessary, the interpretation in terms of the other.

The cumulative distribution of the finite sum of the binary sequence of order k is studied and some of its applications discussed. Certain properties of this sequence are investigated and uniformly superior bounds for the cumulative distribution under minimal information on the ‘success' probabilities are derived.

We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$-stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.