We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

In this paper we explore the potential of the pairwise-type modelling approach to beextended to weighted networks where nodal degree and weights are not independent. As abaseline or null model for weighted networks, we consider undirected, heterogenousnetworks where edge weights are randomly distributed. We show that the pairwise modelsuccessfully captures the extra complexity of the network, but does this at the cost oflimited analytical tractability due the high number of equations. To circumvent thisproblem, we employ the edge-based modelling approach to derive models corresponding to twodifferent cases, namely for degree-dependent and randomly distributed weights. Thesemodels are more amenable to compute important epidemic descriptors, such as early growthrate and final epidemic size, and produce similarly excellent agreement with simulation.Using a branching process approach we compute the basic reproductive ratio for both modelsand discuss the implication of random and correlated weight distributions on this as wellas on the time evolution and final outcome of epidemics. Finally, we illustrate that thetwo seemingly different modelling approaches, pairwise and edge-based, operate on similarassumptions and it is possible to formally link the two.

In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes over time, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

The distributions that result from zero-truncating mixed Poisson (ZTMP) distributions and those obtained from mixing zero-truncated Poisson (MZTP) distributions are characterised based on their probability generating functions. One consequence is that every ZTMP distribution is an MZTP distribution, but not vice versa. These characterisations also indicate that the size-biased version of a Poisson mixture and, under certain regularity conditions, the shifted version of a Poisson mixture are neither ZTMP distributions nor MZTP distributions.

We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries Xk,j and independently distributed rows. We study the distribution of which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of Sn for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

The mathematical model we consider here is a decomposable Galton-Watson process with individuals of two types, 0 and 1. Individuals of type 0 are supercritical and can only produce individuals of type 0, whereas individuals of type 1 are subcritical and can produce individuals of both types. The aim of this paper is to study the properties of the waiting time to escape, i.e. the time it takes to produce a type-0 individual that escapes extinction when the process starts with a type-1 individual. With a view towards applications, we provide examples of populations in biological and medical contexts that can be suitably modeled by such processes.

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.

Let pk, k ≧ 0, be a probability distribution having finite binomial moments Br, r ≧ 0, and the probability generating function U(z) with a radius of convergence α (≧ 1). In this note explicit and recursive formulae are derived allowing computation of the pk in terms of the Br if α > 1. Ch. Jordan's formula, which holds if α > 2, turns out to be a special case.

Motivated by the problem of solid catalytic particle attrition during chemical reactions we formulate a continuous-time Markov model to describe the shattering of a particle when it is assumed that particles can be classified into a small number of types by size. We then obtain a recursive expression for the joint probability generating function of the count of the different types of particle at time t and derive a system of differential equations for the mean number of particle counts and a system of matrix differential equations for the covariance matrix of the particle counts. Solutions to these differential equations are presented in an important special case.

It is proved that, for a large class of stable stationary queueing systems with renewal arrival processes and without losses, a necessary condition for the departure process also to be a renewal process is that its interval distribution be the same as that of the arrival process. This result is then applied to the classical GI/G/s queueing systems. In particular, alternative proofs of known characterizations of the M/G/1 and GI/M/1 systems are given, as well as a characterization of the GI/G/∞ system. In the course of the proofs, sufficient conditions for the existence of all the moments of the stationary queue-size distributions of both the GI/G/1 and GI/G/∞ systems are derived.

We show that the Harris-Sevast'yanov transformation for supercritical Galton-Watson processes with positive extinction probability q can be modified in such a way that the extinction probability of the new process takes any value between 0 and q. We give a probabilistic interpretation for the new process. This note is closely related to Athreya and Ney (1972), Chapter 1.12.

Let X1, X2, · ··, Xn, · ·· be independent and identically distributed non-negative integer-valued random variables with finite mean and variance. For any positive integer n and m we consider the random vector i.e., L has the same distribution as the conditional distribution of (X1, · ··, Xm) given the condition It is easy to see that our model includes the classical urn model, the Bose–Einstein urn model and the Pólya urn model as special cases. For any non-negative integer s define G(s) = the number of Li′s such that Li = s, and U = the number of Li′s such that Li is an even number; in this paper we study the asymptotic behaviour of the random variables considered above. Some central limit theorems and a multinormal local limit theorem are proved.

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. In this paper we show, using a multitype Galton–Watson process to model a bird population, that such behaviour can correspond to maximization of the probability of survival of the species to time t for each finite t. This is also a conclusion which might be drawn from the theory of natural selection and hence provides some mathematical evidence of the force of evolution. The results of the paper rest on a bounding of probability generating functions.

In this paper we derive a successive method of solution to a general class of models pertaining to the investigation of the migration process. A specific application of the results to a problem arising in a study of the epidemiology of leprosy has been considered. The application to explorative studies in biological modelling using an interactive computing mode has been discussed.