We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi(t), Gi(0+) = 0. At the end of its life, cell i is replaced by j1 new cells of type 1, j2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular non-linear component, initially proposed by Weiss (1965) for the spread of a carrier-borne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component.

A general age-dependent branching process is studied in which the population is distributed in N colonies. Migration from one colony to another is possible and no particular geometrical configuration is assumed for the colonies. Asymptotic expressions for the first two moments are given in several different cases.

The purpose of this paper is to show some statistical properties of a one-dimensional inhomogeneous flow of particles with their own constant velocities. The most interesting example of such flows is a low-density inhomogeneous road traffic flow; each vehicle drives at his own constant velocity (desired velocity), since passing is freely allowed. Therefore, to give a definite picture to the one-dimensional inhomogeneous flow of particles, the authors deal with a low-density inhomogeneous traffic flow.

A weir of capacity K is considered in which the water inflow is a process with stationary independent increments. Unless the weir is empty, there is a continuous release of water at unit rate; if K is finite the weir may become full in which case the excess water overflows instantaneously. A weir for which K is infinite will be referred to as infinite dam. For the latter the transient behaviour is well known if the input possesses a second moment (cf. e.g., Prabhu [7]) and serves as the starting point for the present paper. This result is first extended to yield the Laplace transform (L.T.) of the trivariate Laplace-Stieltjes transform (L.S.T.) of the content v(t) at time t, the input X(t) in (0, t) and the total time d(t) in the interval (0, t) during which the dam is dry. (Incidentally, the last two quantities, for relevant time intervals, will be carried throughout.) Then we use a relation between the latter and the L.S.T. of the expected number of downward level y crossings of the v(t) process established in Roes [9]. Since the dam processes considered are Markov processes, we have therewith the L.S.T. of the renewal function of the renewal process imbedded at level y. From this, one finds the L.S.T.'s of first entrance and taboo first entrance times (for their definition see introduction). Next we calculate the first skip times for the infinite dam from the first entrance times and the L.T. of the L.S.T. of v(t). It is then a routine matter to determine the taboo first skip times. From the (taboo) first entrance and skip times we derive the first entrance times for the finite dam, which in turn lead to the renewal functions of the renewal processes imbedded in the finite dam content process v*(t) and hence to the transient behaviour of the finite dam.

The advantage of the present approach over the one given in Roes [8] is that it is entirely probabilistic and avoids involved analytic arguments. As a result, the question of uniqueness of the solution does not arise, while more insight is obtained in the structure. The L.S.T. of several first entrance times and first skip times have been derived by Cohen [2] for compound Poisson input.

In a recent paper [7] the authors studied the optimal control policy of the following queueing system. Customers arrive at a service station according to a time homogeneous Poisson process with a known arrival intensity, λ. The service time at the station is a random variable having a negative exponential distribution with intensity μ, which is under control and can be varied over a certain range, according to the management policy.

For a GI/G/1 queue a limit theorem is obtained for the total input of work during (0, t]. This result is then used to obtain similar theorems for the waiting and idle times and for busy periods initiated by large service loads. Some of the results contained in the paper have recently been proved by other authors in more general settings. The intent of this work is to show how they may be obtained in lesser generality using simpler techniques.

A two-unit standby redundant system with repair and preventive maintenance is discussed. Assuming that the failure, repair, inspection, and preventive maintenance (preventive repair) time distributions of each unit for the system are all arbitrary, the Laplace-Stieltjes transform (L.S.T.) of the distribution of time to the first system down is derived and the mean time obtained from it. It is further shown that under suitable assumptions, the preventive maintenance policy for the system is effective in the sense that it increases the mean time to the first system down.

The semi-Markov decision model is considered under the criterion of long-run average cost. A new criterion, which for any policy considers the limit of the expected cost incurred during the first n transitions divided by the expected length of the first n transitions, is considered. Conditions guaranteeing that an optimal stationary (non-randomized) policy exist are then presented. It is also shown that the above criterion is equivalent to the usual one under certain conditions.

This paper is concerned with a continuous time parameter Markovian sequential decision process, and presents a method which transforms a given continuous parameter problem into a discrete one. It is proved that the optimal stationary policy for the resulting discrete time parameter Markovian decision process is also the optimal stationary policy for the original continuous one, and vice versa. The resulting discrete parameter problem may be more easily solved than the continuous one by applying the linear programming method. A simple numerical example is presented.

The random packing of geometric objects in one-, two- or three-dimensions may afford useful insights into the structure of crystals, liquids, absorbates on crystals, and in higher dimensions, into problems of pattern recognition. Random packing has accordingly received increasing attention in recent years. Two principal packing procedures have been formulated and each gives rise to different packing ratios. In one case, all possible configurations of a sphere-packed volume are assumed to be equally likely. In the other and most widely reported case, there is random sequential addition of spheres to the volume until it is packed. This is the situation we study in this paper. Most of the work to date has been limited to the theoretical study of the one-dimensional lattice or to continuous cases particularly in the limit for long lines. The higher dimensional cases have resisted theoretical attack but have been studied by computer simulation by Palasti [12] and Solomon [14] and by physical simulation by Bernal and Scott (see [14]).

Additive processes on finite Markov chains have been investigated by Miller ([8], [9]), Keilson and Wishart ([2], [3], [4]) and by Fukushima and Hitsuda [1]. These papers study a two-dimensional Markov Process {X(t), R(t)} whose state space is R1 × {1, 2, ···, R} characterized by the following properties:

1. (i) R(t) is an irreducible Markov chain on states 1,2, …,R governed by atransition probability matrix Bo = {brs}.

2. (ii) X(t) is a sum of random increments dependent on the chain, i.e., if the ith transition takes the chain from state r to state s, then the increment has the distribution

3. (iii) Nt, is t in discrete time while in the continuous time case Nt, might be an independent Poisson process.

Small sample and large sample properties of the bundle strength of parallel filaments studied earlier by Daniels (1945) and Sen, Bhattacharyya, Suh (1969) have been developed here by probabilistic argument. The statistics belong to a family or class of statistics, each of which forms a reverse semi-martingale sequence. Certain moment properties are also discussed.

Let X(t), t ≧ 0, be a stationary Gaussian process with zero mean, unit variance and continuous covariance function r(t). Suppose that, for some ε > 0 so that there is a version of the process whose sample functions are continuous [1].

Suppose F is a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such function F we shall write F{I} = F(b)– F(a) where I is the half-open interval (a, b]. Denote the k-fold convolution of F with itself by Fk* and let Now if z is a non-negative function we may form the convolution although Z may be infinite for some (and possibly all) points.

Let be a sequence of non-negative random variables and the associated point process defined by

Gnedenko's classical work [1] on the limit of the distribution of the maximum of a sequence of independent random variables is extended to the distribution of the maximum of a sequence of random variables defined on a finite Markov chain.

Assume the sequence of random variables x0, x1, x2, ··· forms a two-state, homogeneous Markov chain with transition probabilities and initial probabilities

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as opposed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3].

Let {Xk}, k = 1, 2, ··· be a sequence of random variables forming a homogeneous Markov chain on a finite state-space, S = {1, 2, ···, s}. Xk could be thought of as the state at time k of some physical system for which are the (one-step) transition probabilities. It is assumed that all the states are inter-communicating, so that the transition matrix P = ((pij)) is irreducible.