Probability-generating function

About: Probability-generating function is a research topic. Over the lifetime, 752 publications have been published within this topic receiving 9361 citations.

Quasi-Moment Functions in the Theory of Random Processes

Abstract: Instead of the system of generalized correlation functions, which statistically describe completely the stochastic process a new system of functions is introduced called quasi-moment functions, which also completely describe the process.The characteristic function of a multi-dimensional distribution (formula (1.17)) is as easily expressed through quasi-moment as through correlation functions. This characteristic function is represented as the product of two factors; the first factor gives the characteristic function for a given Gaussian process with about the same mean value and correlation function as for the initial process, while the second factor (represented as a series) accounts for the deviation of the charac-teristic function from the Gaussian. Moreover, probability densities of different multiplicities may be written through the introduced functions. This is important in solving problems connected with condition probability and others.Quasi-moment functions serve as the coefficients when expandin...

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

Abstract: Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions

Abstract: A probability distribution of fractional (or fractal) order is defined by the measure μ{ d x } = p ( x )(d x ) α , 0 α f ( x + h ) = E α ( D x α h α ) f ( x ) provided by the modified Riemann Liouville definition, one can expand a probability calculus parallel to the standard one. A Fourier’s transform of fractional order using the Mittag–Leffler function is introduced, together with its inversion formula; and it provides a suitable generalization of the characteristic function of fractal random variables. It appears that the state moments of fractional order are more especially relevant. The main properties of this fractional probability calculus are outlined, it is shown that it provides a sound approach to Fokker–Planck equation which are fractional in both space and time, and it provides new results in the information theory of non-random functions.

A retrial system with two input streams and two orbit queues

Abstract: Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i=1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a boundary value problem. Various performance measures are calculated and numerical results are presented.