Showing papers on "Probability-generating function published in 1978"

Galerkin Finite Element Procedure for analyzing flow through random media

Abstract: A method is presented for analyzing flow through a porous medium whose parameters are random functions. Such a medium is conceptualized as an ensemble of media with an associated probability mass function. The flow problem in each member of this ensemble is deterministic in the usual sense. All the solutions belong to a particular Hilbert space, and hence they can be written in terms of linear combinations of its basis functions. This is similar to the Galerkin formulation except that the coefficients in the linear combination are no longer deterministic quantities but random functions. The finite element method in conjunction with a Taylor series expansion is used to get the first two moments of the solution approximately. The method does not require specification of full probability mass functions of the parameters but only their first two moments, and spatial correlations can be easily accounted for. However, it is assumed that the probability mass functions are peaked at the expected value and are smooth in its vicinity. A sample problem is solved to illustrate the procedure. It is observed that the result is sensitive to the element size in the numerical scheme and the variances and spatial correlations of parameters. The expected value of the hydraulic head is found to differ significantly from the results that would have been obtained if the problem had been solved deterministically.

Multiplicity distributions of created bosons: the method of combinants

Abstract: The number of bosons that are created by an excitation process, in a system which initially has none, is a random variable having a discrete probability distribution, defined on the non-negative integers, which satisfies P(0)>0. The logarithm of the resulting probability generating function is therefore analytic at the origin, and the series expansion coefficients thereby generated can each be expressed as a finite combination of ratios of the P(n), giving them an interestingly close kinship to experimental data. These 'combinants' are additive for sums of boson multiplicity random variables which are independent, and they all vanish except for the first-order one in the important Poisson case. The combinants are readily calculated in a number of theoretical models for created boson multiplicities, including the thermal model some of the related chaotic radiation models, and in some models described by master rate equations.

A lofistic rlgression analysis of response-time data where the hazard function is time dependent

Abstract: A logistic-exponential model for analyzing response-time data involving regressor variables is modified to allow for non-consrarey of the hazard function. For the discrete observation case illustrated the logit of the probability of responding in a time interval cf arbitrary length is taken as the sum of a function of resressor variables and a function of the time variable. The particular functions chosen in the two medical examples analyzed are linear in the parameters involved. A polynomial function of time is employed in the absence of knowledge as to a more appropriate form. Various issues arising in the analysis made are discussed.

Uniform bounding of probability generating functions and the evolution of reproduction rates in birds

Abstract: Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. In this paper we show, using a multitype Galton-Watson process to model a bird population, that such behaviour can correspond to maximization of the probability of survival of the species to time t for each finite t. This is also a conclusion which might be drawn from the theory of natural selection and hence provides some mathematical evidence of the force of evolution. The results of the paper rest on a bounding of probability generating functions.

A generating-function analysis of multiprogramming queues

Abstract: A simple MP system consisting of an input-output facility and a central processor is modeled as a two-parameter Markov chain. The conditions for stability are demonstrated, and the steady-state joint probabilities are calculated explicitly. Various priority and capacity assignments result in radically different analytical situations, some of which have been considered in the literature. The present work treats a version that was considered for a time intractable. This paper emphasizes the analytical properties of the probability-generating functions and a method to solve a resultant functional equation. The numerical results display the importance of dependence between variables in the model.

Escape probability of a random walker on a lattice doped with absorbers

Abstract: We present a method for calculating the escape probability of a particle performing a random walk on a lattice doped with one or more absorbers. This method utilizes generating functions. We apply this method to several cases of practical interest, including the escape probability from a pair of absorbers. Comparison is made between the present discrete approach and the continuous one based on the Smoluchowski equation.

Marriage between the Supplementary Variable Technique and the Imbedded Markov Chain Technique-I

Abstract: Although the main purpose of this paper is to derive relations between the probability generating functions of the number in the system for the queueing process M/GK/I in which the size of service batch is fixed at three epochs of time — random, just after departure and just before arrival, the method discussed is general and can be applied to several other queueing processes in which the size of arrival groups or of service groups is variable. In addition, usefulness of the mathematical expressions is illustrated by indicating as to how some of the parameters of the process under consideration may be evaluated.

Polynomial bounds for probability generating functions. II

Abstract: Consider the set of proper probability distributions on the nonnegative integers having the first k moments (fixed) in common. It is desired to find the element of this set whose corresponding probability generating function (p.g.f.) lies entirely above or below the others. Using convexity arguments, it is shown that the extremal distribution exists, is unique, and is necessarily an element of a certain subclass of the class of all (k + 1)-point distributions. This subclass is entirely characterized by the geometrical properties of its set of support. The alternation of upper and lower bounds with the parity of k is also explained. There is mention of how the techniques developed here apply to more general discrete optimization problems, and the connection with the theory of cyclic polytopes is also discussed.