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 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].

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

Let ξn be the maximum of a set of n independent random variables with common distribution function F whose support consists of all sufficiently large positive integers. Some of the classical asymptotic results of extreme value theory fail to apply to ξn for such F and this paper attempts to find weaker ones which give some description of the behaviour of ξn as n → ∞. These are then applied to the extreme value theory of certain regenerative stochastic processes.

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.

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.

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.

Suppose that in a ballot candidate A scores a votes and candidate B scores b votes and all the possible voting records are equally probable. Denote by αr and βr the number of votes registered for A and B respectively among the first r votes counted. For j= 0, 1,···, a + b define Pj(a, b) as the probability that the inequality αr > aβr/b holds for exactly j subscripts r = 1, 2, .., a + b.

Let P= {p1,···, PN} be a finite discrete probability distribution. Then the entropy of the distribution P, introduced by Shannon [12], is defined as Throughout this paper ∑ will stand for and logarithms will be taken to the base D (D > 1).

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.

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]).

The fundamental principle underlying insurance is that the expected value of claims is equal to the premium. This was established by Bernoulli [4] in 1738. Subsequent work on the development of ‘risk theory’ led to research concerning the probability of ‘ruin’ of an insurance company. A critical study of these investigations was made by Borch [3] in a recent paper.

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.

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.

Consider an airplane which is at a distance t ≧ 0 from its destination (the distance being measured in time-units) and has n anti-aircraft missiles. The enemy sends its aircraft according to a Poisson process with known parameter, which by redefining the time-scale may, without loss of generality, be taken to be one. When meeting an enemy aircraft, the plane can use, simultaneously, some of its missiles, and it is assumed that the hits are independent random events with known probability 1 > 1 – q > 0. If no missile hits the enemy aircraft, the airplane is shot down with known probability 1 ≧ 1 – u > 0. The case u= 0 is of particular interest. (We have excluded the values q = 0, q = 1 and u= 1 since these yield trivial solutions.)

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.

Some inequalities are obtained which yield bounds for the mean life of series and of parallel systems in the case where component life distributions have properties such as a monotone failure rate, monotone failure rate average, or decreasing density. These bounds are based on comparisons with systems of exponential or uniform components. Similar comparisons are obtained when components have Weibull or Gamma distributions with different shape parameters. Some inequalities are also obtained for convolutions of life distributions helpful in the study of replacement policies.