The problem of age specific bacteriophage growth is dealt with, using the product density approach. Expressions for the product densities, taking into account the intermediate stages of transformations from the vegetative phage to mature phage, are also given.

Consider a deterministic diploid population and a character determined by two loci with alleles A, a at the first and B, b at the second. When the effects of the alleles at each of the two loci on the relative fitnesses of the genotypes are additive, the selection matrix may be written

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.

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

To current workers in the mathematical theory of road traffic, one of the most challenging and important problems is to describe and analyze the congestion suffered by motorists on freeways. While there is, as yet, no satisfactory unified theory, several different models have been suggested, each of which is reasonable for a limited class of situations. These approaches fall largely into two categories. The first may be described as looking at traffic “in the large” and its principal exponents were Lighthill and Whitham (1955) and Prigogine (1961). The theory of Lighthill and Whitham involves treating the system of vehicles as a fluid subject to a rule of conservation of mass (that is, vehicles) and an assumed relation between the flow and concentration at each point. This theory is completely deterministic. Prigogine's approach is to regard the system of vehicles as behaving like the particles in a gas and subject to the laws of statistical mechanics.

Denote by X(t) the total quantity of water flowing into a weir of capacity K in the interval (0, t). Unless the weir is empty, there is a continuous release of water at unit rate; if it becomes full, the excess water overflows instantaneously.

In many situations, it is reasonable to assume that a priority rule does not affect the total time spent in service of any job. Rules with this property are said to be work-conserving. This concept unifies and simplifies the analysis of a variety of priority queues. Some results are obtained for rules applied to the GI/G/1 queue. Some special properties of Poisson arrivals are discussed, and a new proof of the equivalence of averaging over all time with averaging over arrival epochs is presented. In this case, explicit results for particular rules are obtained in examples. In another example, the optimal rule (from a very restrictive class) is determined without specializing the arrival stream.

The most widely used probabilistic model of a series of “events” occurring randomly in a time or space continuum is the Poisson process. It is the proper basis for statistical analysis when general information about the rate of occurrence of events is the primary purpose of an investigation, or when the assumption of independent occurrences is reasonable and valid. A unified treatment of the statistical methods for estimating parameters in stationary and non-stationary Poisson models, and tests of the Poisson assumption against various alternatives, is contained in the lucid monograph by Cox and Lewis [6].

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

A Monte Carlo method is proposed and demonstrated for obtaining an approximate algebraic solution to linear equations with algebraic coefficients arising from first order master equations at steady state. Exact solutions are hypothetically obtainable from the spanning trees of an associated graph, each tree contributing an algebraic term. The number of trees can be enormous. However, because of a high degeneracy, many trees yield the same algebraic term. Thus an approximate algebraic solution may be obtained by taking a Monte Carlo sampling of the trees, which yields an estimate of the frequency of each algebraic term. The accuracy of such solutions is discussed and algorithms are given for picking spanning trees of a graph with uniform probability. The argument is developed in terms of a lattice model for membrane transport, but should be generally applicable to problems in unimolecular kinetics and network analysis. The solution of partition functions and multivariable problems by analogous methods is discussed.

Our main concern in this paper is the convergence, as t → ∞, of the quantities i, j ∈ E; where Pij(t) is the transition probability of a semi-Markov process whose state space E is irreducible but not closed (i.e., escape from E is possible), and rj is the probability of eventual escape from E conditional on the initial state being i. The theorems proved here generalize some results of Seneta and Vere-Jones ([8] and [11]) for Markov processes.

Many phenomena studied in the social sciences and elsewhere are complexes of more or less independent characteristics which develop simultaneously. Such phenomena may often be realistically described by time-continuous finite Markov processes. In order to define such a model which will take care of all the relevant a priori information, there ought to be a way of defining a Markov process as a vector of components representing the various characteristics constituting the phenomenon such that the dependences between the characteristics are represented by explicit requirements on the Markov process, preferably on its infinitesimal generator.

We consider the limiting behaviour of a k-type (k < ∞) Galton-Watson process which is augmented at each generation by a stochastic immigration component. In Section 2, conditions for ergodicity are found for a subclass of such processes. In Section 3, expressions are derived for the first two moments of the nth generation (by way of a recurrence relation) and for the first two asymptotic moments, in a manner which to some extent generalises previous results.

Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.

Let X1, X2, X3, ··· be independent, identically distributed random variables on a probability space (Ω, F, P); and with a continuous distribution function. Let the sequence of indices {Vr} be defined as Also define The following theorem is due to Renyi [5].

The concept of cost has not, until recently, been investigated in any detail within the framework of epidemic theory. In the case of minor illnesses a pertinent estimate of cost may be the man-hours lost in industry during an outbreak. For the general epidemic, in which the removals are immune recoveries (or deaths), this is conveniently represented by the area generated by the trajectory in time t of infectives I(t) until the epidemic ceases. A small element of area under the trajectory is I(t)δt and represents I(t) infectives withdrawn from the labour force for a small time δt; the total work-time lost is thus represented by where T is the duration of the epidemic.

Denote by SN,r the set of r-tuples n having non-negative integer components summing to N, and by ei the r-tuple having a one in the ith position and zeros elsewhere. For m and n belonging to SN r, we let P(n, t|m, s) represent the transition probability of moving from state m at time s to state n at time t. Bartlett's conservative process [1] assumes that for n–ei + ej and n ∈ SN,r and Δt > 0 where the λij(t) are non-negative continuous functions of t.

We consider a population which evolves at discrete points in time by branching and immigration, and in which each member reproduces independently of all others. Let Fn(x) denote the probability generating function (P.G.F.) of the number of offspring produced by each member of the nth generation, Bn–1(x) the P.G.F. of the number of immigrants joining the nth generation and Zn the population size in the nth generation. We write

Let Z0 = 1, Z1, Z2, ··· denote a super-critical Galton-Watson process whose non-degenerate offspring distribution has probability generating function where 1 < m = EZ1 < ∞. The Galton-Watson process evolves in such a way that the generating function Fn(s) of Znis the nth functional iterate of F(s). The convergence problem for Zn, when appropriately normed, has been studied by quite a number of authors; for an ultimate form see Heyde [2]. However, no information has previously been obtained on the rate of such convergence. We shall here suppose that in which case Wn = m –nZn converges almost surely to a non-degenerate random variable W as n → ∞ (Harris [1], p. 13). It is our object to establish the following result on the rate of convergence of Wn to W.

Let Zn be the number of descendents in the nth generation of a simple Galton-Watson branching process, initiated by a single progenitor, Z0 = 1. If E(Z1) < ∞ the limiting distribution of Zn is known in some detail, and a comprehensive account is given in Seneta [1]. If E{Z1) = ∞ (the “explosive” case) the behavior of the distribution of Zn for large n seems not to be known. As shown by Seneta [1], there are no constants cn such that cnZn has a non-degenerate limiting distribution but it turns out that, under conditions given below, log(Zn + 1) has a limiting distribution.