This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn, = j, then there is a random. ‘input’ Vn(j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn(j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn, Xn) and (Zn, OnLn), where Zn, is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

The growth rate of a cancerous tumour as a function of its age is a subject of intellectual and practical importance, as it influences both the effectiveness of proposed screening programs and strategy of treatment. Obtaining direct evidence on the growth rate is quite difficult owing to the ethical necessity to intervene when cancer is confirmed.

The linear compartmental model arises when ‘particles’ move independently between (or out of) a system of compartments in a stochastically similar way. With a given ‘initial’ particle count, the subsequent compartmental particle counts follow multinomial probability distributions (Faddy (1976)) for Markov or semi-Markov movement processes. One immediate consequence of this is that the variance of the compartmental particle count is always less than the mean, with the result that the coefficient of variation is very small for large mean counts. In applications (e.g, Faddy, Jones and Edwards (1976)) this underestimation of the variation can be a shortcoming of a compartmental analysis. Clustering, introduced by Matis and Wehrly (1981), where ‘clusters’ of particles may move together, is a way in which increased variability may be attained. Increased variability in general will result from relaxing the main assumption that gives rise to a linear model: independent behaviour of the particles. Such nonlinear compartmental models can be generally difficult to handle; in this paper some particular cases are discussed and illustrated with reference to one- and twocompartment systems.

The interpretation of mark-recapture data depends on a probabilistic model for the biological system. The assumptions of the general model must be verified against what is known about the particular system. This talk considers the application of markrecapture methods to the estimation of salmon smolt migrations from Babine Lake, British Columbia.

One of the more important parameters in the study of plant growth is the length of the root system.

The determination of the kinetics of early cancer growth has received considerable attention from cancer researchers interested in reducing the time required to assess carcinogenic risk. The three main difficulties in this study are: (1) tumors can only be identified after the animal has been sacrificed, (2) each animal exhibits considerable variability in tumor sizes and (3) tumors can only be identified once they have grown above a threshold size. Thus, the only data available to the researcher summarizes the sizes of observable tumors at the time of sacrifice. In this paper, we give techniques to help the researcher decide the type of tumor growth (e.g., exponential, logistic, Gompertz), the time at which the cancerous growth began, as well as the principal source of variability in tumor sizes (either the time of the initial growth, or the rate of growth). Additional techniques can be used to decide if the rate of growth increases with time (a consequence of the increased malignancy of the tumor) or decreases with time (due to the lack of oxygenation at the core of a large tumor). These techniques are applied to data on microscopic liver tumors (often called foci) from the lab of Dr Stanley Goldfarb (Department of Pathology, University of Wisconsin-Madison).

Offers become available from time to time. The offers constitute a sequence of i.i.d. positive random variables. Upon arrival of an offer a decision is taken whether to accept it or not. When an offer is accepted the process terminates. Otherwise, the offer is lost and the process continues until the next arrival or until a moment where it terminates by itself. The probability of self-termination depends on the age of the underlying process.

From the point of view of practical usefulness, the results of modelling exercises should not be too sensitive to the detailed assumptions built into the model. Consequently, it is a great advantage if structural assumptions are kept to a minimum.

Venereal disease is among one of the more active serious epidemics in the world today. Despite its prevalence and the ever-increasing literature on the mathematical treatment of epidemics very little progress has been made toward modelling the spread of this disease. Key features of the disease that help to account for this lack of progress are: (i) VD involves the cross-infection between two groups of individuals and (ii) VD is a recurrent disease. The heterosexual spread .of VD ties in the modelling of this disease with the yet unsolved problem in mathematical biology concerning the growth of two interacting species. In this paper we review the current state of the art in the stochastic modelling of this disease.

Polymeric materials are encountered everywhere: rubber, plastics, paint, soft contact lenses; the list is endless. There are two basic types of polymer, linear and non-linear. The linear polymers can be envisioned as a mass of long spaghetti-like strands. The molecules of non-linear polymers have a branched tree-like structure and can form ‘infinite’ networks. The basic building blocks of polymers are monomers. Monomers have reactive sites: chemical bonds can form between sites on different monomers leading to the formation of many monomers into a large molecule (polymer). If all monomers have one or two reactive sites, linear polymers result. If some of the monomers have three or more reactive sites, a non-linear (branched) molecule results. For many materials the reaction between monomers can be modelled as a random process; thus probability theory is the natural tool for studying polymers.

This paper reviews a variety of results for genealogical (or line-of-descent) processes that arise in connection with the theory of some classical selectively neutral haploid population genetics models. While some new results and derivations are included, the principal aim of the paper is to demonstrate the central importance and simplicity of genealogical Markov chains in this theory. Considerable attention is given to ‘diffusion time scale’ approximations of such genealogical processes. A wide variety of results pertinent to (diffusion approximations of) the classical multi-allele single-locus WrightFisher model and its relatives are unified by this approach. Other examples where the genealogical process plays an explicit role (for example, the infinite-sites models) are discussed.

Plasmids are subcellular particles somewhat similar to, but sirnpler than, viruses. A small part of the host cell's genome, they multiply independently within host cells and are apparently inherited in a random manner. They code important genetic traits not normally encoded and are responsible for several plant and animal diseases; they can also confer the ability to resist a wide variety of toxic agents.

The M/M/∞ queue can be considered as a compartmental system which may be modified by allowing its arrival and service rates to depend on an exogenous ‘environment process’. The states of the environment process are known as environments. We present some examples of this, chosen for their analytic or algorithmic tractability. We discuss (i) models with arrivals in only one environment, (ii) models with service in only one environment, (iii) reversible models and (iv) a model in which the environment process is also a queue.

Conservative mutations (those which do not change the amino acid coded for a codon) are discussed. In particular, formulas are given for the computation of the probability that a mutation is conservative and the expected value of this probability if the genetic code were randomly permuted, assuming wobble and non-wobble conditions.

A predominance of certain ‘sequence systemmetries’ in DNA/RNA has led to various conjectures about the possible structural/functional role these symmetries might play in nucleic acid sequences. De Wachter employed a binomial probability model to compare the observed number of ‘direct repeats’ with those expected in a random sequence. Counting of direct repeats essentially leads to a sequence of m-dependent trials. We develop a stochastic model for studying various types of symmetries. Expressions for means and variances of the statistics employed are derived. The asymptotic distributions are obtained using the central limit theorem for m-dependent random variables. It is proposed that each sequence pattern be examined separately for its chance occurrence as opposed to what de Wachter suggests, i.e., clumping of all patterns together. It is also shown how our model can be used to detect various gene-amplification events, if any, in nucleic acid sequences. Finally, for certain types of patterns, it is indicated how the theory of recurrent events can be used to get a better handle on the analysis of direct repeats.

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when

The critical time tc when diverges corresponds to a critical branching process, while post-critical times t> tc correspond to supercritical branching processes.

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.

Let be a double sequence of random variables such that there exists a ‘dual' sequence satisfying , and that can be expressed as a sum of independent random variables. If (suitably centered and rescaled) is approximately normally distributed as k and N → ∞ in some fashion, we can use this fact to obtain a remainder-term estimate for the asymptotic normality of as n and N → ∞ in some prescribed manner. The result in the general theorem is used in two specific situations: (i) classical occupancy where the balls can fall through the boxes, (ii) a capture-recapture problem where tagging affects catchability.

From one point of view this paper adds to a previous formulation of the best-choice problem (Petruccelli (1981)) the possibility that the number of available observations, rather than being known, is a bounded random variable N with known distribution. From another perspective, it expands the formulations of Presman and Sonin (1972) and Rasmussen and Robbins (1975) to include recall and uncertainty of selection of observations. The behaviour of optimal stopping rules is examined under various assumptions on the general model. For optimal stopping rules and their probabilities of best choice relations are obtained between the bounded and unbounded N cases. Two particular classes of stopping rules which generalize the s(r) rules of Rasmussen and Robbins (1975) are considered in detail.