We give a general construction of sequential games among multiple players, as well as a construction of the composition of sequential games. We obtain new properties of the optimal class of win-by-k games, including closure under composition and independence between the winner of the game and the number of points played. We obtain new results on the asymptotic efficiency of the n-point, win-by-k games.

Central limit theorems are given for the time taken for a population to increase by a factor k > 1 for a supercritical process, or to decrease by a factor k < 1 for a subcritical process. The size of errors is investigated so that confidence limits can be given for these times to O (m½) where m is the population size at t = 0.

Important parameters of particle size distributions in dispersed systems in engineering and related fields are ratios of moments and inverse powers of these ratios, known as mean sizes. The variation in these parameters is examined for the simplest growth model in which the size distribution is translated, and the results for this process considered in relation to the problems of models of other growth processes. For initial size distributions with monotone hazard rate, the results are particularly significant, and the properties of the normalised moments of other distributions are also considered.

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.

A trivariate equality of laws given by Doney and Yor is extended to a quadrivariate version. Some related explicit density calculations are carried out. It is shown that a bivariate case which has been used to establish the Dassios–Port–Wendel equality of laws is in fact equivalent to it.

The object of this paper is to study the actual waiting time of a customer in a GI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer than w in a given time interval (0, T) are calculated.

Branching processes are studied in random environments that are influenced by the population size and approach criticality as the population gets large. Results are applied to the polymerase chain reaction (PCR), which is empirically known to exhibit first exponential and then linear growth of molecule numbers.

Ebrahimi (2001) proposes an interesting model for assessing a system's reliability, expressing the failure time of the system in terms of a deterioration process and covariates. He also provides illustrative examples and gives some properties of the model. In this note, we give conditions for negative ageing of the system's lifetime, and we correct one of his statements. Moreover, for the lifetimes of two systems of the same kind, some stochastic comparisons are presented.

Motivated by a problem arising in the mining industry, we present a first study of the energy required to reduce a unit mass fragment by consecutively using several devices. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced in Bertoin and Martínez (2005), we compute the average energy required to attain a size η0 with this two-device procedure. We then asymptotically compare, as η0 goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that, for a certain range of parameters of the fragmentation processes and of their energy cost functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or vice versa.

This paper illustrates a simple method of deriving distributions for samples from cell populations in exponential growth. The distribution of the elapsed proportion of the DNA-synthetic phase of the cell cycle is derived and used to find the distribution of DNA content in a random sample of cells. Allowing for a normally-distributed measurement error, the resulting distribution agrees well with DNA distributions observed empirically. The statistical analysis of DNA distributions, and the interpretation of the distributional parameters in terms of cell population kinetics, are discussed and illustrated with an example.

Most generalizations of the classical occupancy model involve non-homogeneous shot assignment probabilities, but retain the independence of the individual shot assignments. Hence, these models are associated with non-homogeneous Poisson processes. The present article discusses a generalization in which the shot assignments are not independent, but which result in clustering of the shots. Conditions are given under which this clustered occupancy model converges to a Poisson cluster process. Limiting distributions for the number of empty cells are obtained for various allocation intensities when the total number of shots is deterministic as well as random. In particular, it is shown that when the allocation is sparse, then the limiting distribution of the number of empty cells is compound Poisson.

We consider the sum Sd of record values in a sequence of independent random variables that are uniformly distributed on 1,…,d. This sum can be interpreted as the total amount of time spent in record lifetimes in the standard renewal theoretic setup. We investigate the distributional limit of Sd and some related quantities as d→∞. Some explicit values are given for d=6, a case that can be interpreted as a simple game of chance.

This paper provides a direct approach to obtaining formulas for derivatives of functionals of point processes in rare perturbation analysis ([2], [6]). Results are obtained for arbitrary (not necessarily stationary) point processes in and d, d 2, under transparent conditions, close to minimal. Formulas for higher-order derivatives allow one to construct asymptotical expansions. The results can be useful in sensitivity analysis, in light traffic theory for queues and for computation by simulation of derivatives at positive intensity, while the computation of the derivatives via statistical estimation of the functional itself and its increments usually gives poor results.

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

We prove existence and uniqueness of a stationary regime for a model of a circuit-switching network with infinitely many nodes.

The random splitting model of Lloyd and Williams (1988) is generalised, to allow first beta splitting distributions and secondly, discrete splitting distributions. Analogues of some earlier results are developed. In particular, the equality of the average value of the largest split and the probability that this is achieved at the first split continues to hold.

Starting from a characterization of radiative transfer in terms of a collision rate λ and a single-collision transition probability Ψ, we study the distribution of the generalized state ζ(t) of a radiation particle at time t conditional on a specified initial state at time t= 0. The generalized state is a vector consisting of the state ω(t) at time t and the states ω1, ω2, …, ωn of the particle immediately after the collisions it experiences in the time interval (0, t]. The variable ζ(t) takes values in a population space and can be studied conveniently with the aid of a certain generating functional G. The first-collision integral equation and the backward integro-differential equation for G are derived. Simultaneous consideration of the first-collision and last-collision equations lead to a generalized reciprocity principle for G. First-passage problems are also considered. Finally a number of illustrative examples are given.

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.

Technology developed in a predecessor paper (Chen and Seneta (1996)) is applied to provide, in a unified manner, a sharpening of bivariate Bonferroni-type bounds on P(v1≥r, v2≥u) obtained by Galambos and Lee (1992; upper bound) and Chen and Seneta (1986; lower bound).

The characterization of the exponential distribution via the coefficient of the variation of the blocking time in a queueing system with an unreliable server, as given by Lin (1993), is improved by substantially weakening the conditions. Based on the coefficient of variation of certain random variables, including the blocking time, the normal service time and the minimum of the normal service and the server failure times, two new characterizations of the exponential distribution are obtained.