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.

Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates

Abstract: We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).

Limit laws of the coefficients of polynomials with only unit roots

Abstract: We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth normalized (by the standard deviation) central moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance goes unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms.

On fractional linear bounds for probability generating functions

Abstract: The best upper and lower bounds for any probability generating function with mean m and finite variance are derived within the family of fractional linear functions with mean m. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions. FRACTIONAL LINEAR FUNCTION; INFINITELY DIVISIBLE DISTRIBUTION; BOUNDS; BRANCHING PROCESS; S-I-S INFECTIOUS DISEASE

Certain classes of discrete distributions in modeling risk control operations and establishing a transformation for probability generating functions

Abstract: Stochastic interpretations in risk control operations for the class of discrete distributions corresponding to discrete random variables with values in the set of nonnegative integers and unique mode at the point zero, the class of discrete selfdecomposable distributions and the class of discrete distributions corresponding to integral part models are established by the paper. Moreover, the paper introduces a transformation which converts the probability generating function of an integral part model, based on a discrete random variable with values in the set of nonnegative integers and probability function having a unique mode at the point zero, into the probability generating function of a discrete random variable with values in the set of nonnegative integers and selfdecomposable distribution.

Analysis of a queueing system in random environment with an unreliable server and geometric abandonments

Abstract: This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e. , an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phase i with probability q i , i = 1, 2, …, d . Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.