(1.1) In this paper, acquaintance will be assumed with the basic mechanics of phage/bacterium, antibody/virus interactions; for background reading see, for example, Adams (1959), Bouanchaud (1970), Durham and King (1969) and Fraser (1967).

A general class of spatial-temporal Markov processes is defined leading to the standard spatial equilibrium distribution for nearest-neighbour models on a multi-dimensional lattice. Physical properties are obtainable from the marginal spatial spectral function. However, only the simplest one-dimensional case corresponds to a linear model with a readily derived spectrum. Non-linear models corresponding to two- and three-dimensional lattices are presented in their simplest terms, and a preliminary discussion of approximate solutions is included.

In this note we consider multi-type branching processes where certain terminal types of particles, once created, are subject neither to death nor to further reproduction.

The model considered in this paper describes a queueing system in which the station is dismantled at the end of a busy period and re-established on arrival of a new customer, in such a way that the closing-down process consists of N1 phases of random duration and that a customer 𝒞n who arrives while the station is being closed down must wait a random time idn(i = 1, ···, N1) if the ith phase is going on at the arrival instant. (For each fixed index i, the random variables idn are identically distributed.) A customer 𝒞n arriving when the closing-down of the station is already accomplished has to wait a random time (N1 + 1)dn corresponding to the set up time of the station. Besides, a customer 𝒞n who arrives when the station is busy has to wait an additional random time 0dn. We thus have (N1 + 2) types of “delay” (additional waiting time). Similarly, we consider (N2 + 2) types of service time and (N3 + 2) probabilities of joining the queue. This may be formulated as a model with (N + 2) types of triplets (delay, service time, probability of joining the queue). We consider the general case where the random variables defining the model all have an arbitrary distribution.

The process {wn}, where wn denotes the waiting time of customer 𝒞n if he joins the queue at all, is not necessarily Markovian, so that we first study (by algebraic considerations) the transient behaviour of a Markovian process {vn} related to {wn}, and then derive the distribution of the variables wn.

Equilibrium behavior of a stochastic system with two types of input of different statistical nature and with linear continuous output is investigated. The results have applications in queueing theory, storage theory and insurance-risk theory.

Queueing models have been used to describe the slowing or stopping of a stream of vehicles due to a traffic light, a merging disturbance, or an accident. The buildup of stopped vehicles, and their eventual dispersal after the delaying factor has been removed, has been likened to the busy period of a queue, e.g., Jewell (1964). Some studies have dealt with the total delay to all drivers involved in such a queue, Daley and Jacobs (1969), Gaver (1969); or the combination of this total delay and the distance of the last vehicle from the originating incident, McNeil (1969), Shaw (1970).

In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form where:

1. (i) ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P);

2. (ii) x(t; ω) is the unknown random variable for t ∊ R+, where R+ = [0, ∞);

3. (iii) y(t; ω) is the stochastic free term or free random variable for t ∊ R+;

4. (iv) k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and

5. (v) f(t, x) is a scalar function defined for t ∊ R+ and x ∊ R, where R is the real line.

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.

This is an extension of the paper [1] by Bather and Chernoff, where approximations are obtained to the solution of a sequential control problem involving the decision of when to use fuel to alter course during a space-ship's approach to a target. It is desired to come as close as possible to the target. If fuel is used too early, it may be used incorrectly due to imprecise information about the location of the target; however if the fuel is used too late it may be ineffective in reducing the miss distance.

The aim of this paper is to investigate the existence of a random solution and the stochastic absolute stability of the differential systems (1.0)–(1.1) and (1.2)–(1.3) with random parameters. These objectives are accomplished by reducing the differential systems into a stochastic integral equation of the convolution type of the form (1.4) and utilizing a generalized version of V. M. Popov's frequency response method.

1. Let T = (tij), i, j = 0, 1, ···, tij ≧ 0, denote an infinite non-negative matrix. We shall assume at all times that where are the matrix iterates of T; we take T0 = I = (δij).

Let ξ1, ξ2, ···, ξm be mutually independent random variables having a common distribution function P{ξr≦x} = F(x)(r = 1, 2, ···, m). Let Fm(x) be the empirical distribution function of the sample (ξ1, ξ2, ···, ξm), that is, Fm(x) is defined as the number of variables ≦x divided by m.

The subject of this paper is the study of the distribution of integrals of the type where {X(t); t ≧ 0} is some appropriately defined continuous-time parameter stochastic process, and f is a suitable non-negative function of its arguments. This subject has also sometimes been labelled as “the occupation time or the sojourn time problem” in literature. These integrals arise in several domains of applications such as in the theory of inventories and storage (see Moran [14], Naddor [15]), in the study of the cost of the flow-stopping incident involved in the automobile traffic jams (see Gaver [8], Daley [3], Daley and Jacobs [4]). The author encountered such integrals while studying certain stochastic models suitable for the study of response time distributions arising in various live situations. In fact in [19], it was shown that such a distribution is equivalent to the study of an integral of the type (1). Again, in the study of response of host to injection of virulent bacteria, Y(t) with f(X(t), t) = bX(t), with b > 0, could be regarded as a measure of the total amount of toxins produced by the bacteria during (0, t), assuming a constant toxin-excretion rate per bacterium. Here X(t) denotes the number of live bacteria at time t, the growth of which is governed by a birth and death process (see Puri [16], [17] and [18]).

In [3] this author gave conditions under which a sequence of jump Markov processes Xn(t) will converge to the solution X(t) of a system of first order ordinary differential equations, in the sense that for every δ > 0.

A replacement of type (jk) in a Markov process with rewards means an instantaneous shift of the trajectory from state j into state k. A formula is obtained for the characteristic function of the total reward from the process under an arbitrary non-anticipative replacement policy. From the formula results are deduced concerning Markov policies, optimal policies and policies minimizing the variance of the reward.

Suppose there are k kinds of balls identical except for their colours. The number of balls of the ith colour is ri (i = 1, 2, ···, k) and Put down the R balls in a sequence (a line), then an acceptable sequence is one which does not have two adjacent balls of the same colour. The problem is to generate a random sample of the set of acceptable sequences S(r1, r2, ···, rk).

A basic process of simple queueing theory is S(t), an integer valued stochastic process which represents the “number of clients present, including the one in service” at epoch t. The queueing context physically limits S(t) to non-negative values, but if this impenetrable barrier is removed thereby permitting S(t) to assume negative values as well, we have a “randomized random walk” in which S(t) represents the position at time t of a mythical particle which moves on the x-axis according to the rules of the walk. The nomenclature “randomized random walk” is due to Feller (1966a). S(t) changes by unit positive or negative amounts, and our basic assumption is that the time intervals τ/σ separating successive positive/negative steps are i.i.d. with continuous d.f.'s. A(t)/B(t) (A(0)/B(0) = 0) possessing p.d.f.'s. a(t)/b(t): τ and σ are further assumed to be statistically independent. When A(t) = 1 – e–λt and B(t) = 1 – e–μt, we have a generalization of the M/M/1 queueing process which we shall denote by M/M. The walk generated by A(t) = 1 – e–λt and B(t) arbitrary may similarly be denoted by M/G. For G/M we clearly mean A(t) arbitrary and B(t) = 1 – e–μt. M/M has received extensive treatment elsewhere (cf. Feller (1966a), (1966b); Takács (1967); Gibson (1968); Conolly (1971)), and will not be considered here. Our interest centers on certain aspects of M/G and G/M, but since each is the dual of the other (loosely, G/M is M/G “upside down”) it is sufficient to restrict attention to one of them, as convenient; pointing out, where necessary, the interpretation in terms of the other.

Consider first a Markov chain with two ergodic states E1 and E2, and discrete time parameter set {0, 1, 2, ···, n}. Define the random variables Z0, Z1, Z2, ···, Znby then the conditional probabilities for k = 1,2,···, n, are independent of k. Thus the matrix of transition probabilities is

Diffusion and birth-death processes have basic theoretical and practical importance for statistics. Insight into the structure of transition distributions and passage time distributions for such processes has been given in recent years by Feller. Karlin, Kemperman, D. G. Kendall, Reuter and many others. An elementary account of this work and partial bibliography has been given elsewhere ([9], [10)]. Certain key passage time densities and sojourn time densities for such processes have a simple property of log-concavity or log-convexity and associated unimodality. Such properties provide information on the character of distributions unavailable from the spectral representations, Laplace transforms and series of convolutions at hand. These properties may also have value for purposes of estimation and optimization.

In a short article McKean [2] inaugurated the study of a class of time-inhomogeneous Markov processes which arise from the reaction of a single particle (the “tagged” particle) with an evolving environment of other particles. Other more recent references on these processes are [1], [4] and [5], in which some existence and uniqueness theorems for the solution of the resulting non-linear equation are obtained. We confine attention here to a very particular type of these processes which arises when two interacting particles simply coalesce, and the mass is taken as the process variable.