Equations are derived for the distribution of the busy period of the GI/G/2 queue. The equations are analyzed for the M/G/2 queue, assuming that the service times have a density which is an arbitrary linear combination, with respect to both the number of stages and the rate parameter, of Erlang densities. The coefficients may be negative. Special cases and examples are studied.

It is proved that in a GI/G/s queue with the traffic intensity ρ < (S – m + l)/S, m ≧ 1, at least m servers are simultaneously idle infinitely often with probability one.

The paper establishes limit theorems for the total progeny of a simple branching process in which immigration is allowed whenever the population size reaches 0.

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

We study the asymptotics of the survival probability for the critical and decomposable branching processes in a random environment and prove Yaglom-type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in a constant environment.

This paper addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time, also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank $m\geq 1,$ under general covariance structures. These results are proven to hold, in particular, for a family of nonseparable covariance structures belonging to the Gneiting class. For $m=2,$ under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt-type limiting distribution for a suitable range of values of the long-memory parameter.

The first emptiness time of an infinite reservoir with unit release and discrete input forming a stationary. Markov chain is investigated. The exact distribution of the first emptiness time is derived without the use of moment generating functions. The first and second moments of this distribution are given explicity. The close relationship between the process with stationary independent input and Markov chain input is emphasized.

The first moment of the area beneath the sample path up to first emptiness is computed. This area is often used as a measure of total delay in traffic flow theory.

In a G/GI/c loss system with balking, reneging, or limited waiting space, deleting some of the arriving customers can either increase or decrease the fraction of the remaining arrivals who get served, depending on how customers are deleted. We present a model in which the random deletion of arrivals independently and with some fixed probability can never decrease the fraction of the remaining arrivals who get served.

A more general definition of MTP2 (multivariate total positivity of order 2) probability measure is given, without assuming the existence of a density. Under this definition the class of MTP2 measures is proved to be closed under weak convergence. Characterizations of the MTP2 property are proved under this more general definition. Then a precise definition of conditionally increasing measure is provided, and closure under weak convergence of the class of conditionally increasing measures is proved. As an application we investigate MTP2 properties of stationary distributions of Markov chains, which are of interest in actuarial science.

Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

In this paper we treat the general version of the semi-Markov storage model, introduced first by Senturia and Puri: transitions in the state of the system occur at a discrete sequence of time points, described by a two-state semi-Markov process. An input occurs at an instant of transition to state 1 and a demand for release occurs at an instant of transition to state 2.

Assuming general distributions for all the variables involved, we show that the dam contents just after the nth input converges properly in distribution as n →∞ under conditions of stability; likewise that after the nth demand. We also show that the demand lost due to shortage of stock, accumulated over instants of demand as well as over time, obeys a strong law and a central limit theorem.

Limit theorems for the thinning of renewal point processes according to two different schemes are studied. In the first scheme when a point is retained a random number of succeeding points are deleted. According to the second scheme a random number of points are deleted by an inhibitory Poisson process.

Using backward stochastic difference equations (BSDEs), this paper studies dynamic convex risk measures for risky positions in a simple discrete-time, binomial tree model. A relationship between BSDEs and dynamic convex risk measures is developed using nonlinear expectations. The time consistency of dynamic convex risk measures is discussed in the binomial tree framework. A relationship between prices and risks is also established. Two particular cases of dynamic convex risk measures, namely risk measures with stochastic distortions and entropic risk measures, and their mathematical properties are discussed.

The fluctuation of the counts of particles in m regions are used to yield an estimate of a fundamental motion parameter. The effect of varying m on the precision of the estimation method is considered. An exact analysis and simulation of the Fürth pedestrian vector process is presented.

The general stochastic epidemic model is used as a model for the spread of rumours. Recursive expressions are found for the mean of the final size of each generation of hearers. Simple expressions are found for the generation size and the asymptotic form of its final size in the deterministic model. An approximating process is presented.

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.

In this paper it is shown that the distribution of the instant of service of a customer is symmetric as between the distributions of service and interarrival time. Also U(t), the expected number of departures in (0, t), is a delayed renewal function for the GI/M/1 queue.

We establish an upper bound on the tails of a random variable that arises as a solution of a stochastic difference equation. In the nonnegative case our bound is similar to a lower bound obtained in Goldie and Grübel (1996).

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.