Topic
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.
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TL;DR: In this paper, the authors considered a bilateral birth-death process characterized by a constant transition rate from even states and a possibly different transition rate in odd states. And they derived the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state.
Abstract: We consider a bilateral birth-death process characterized by a constant transition rate $\lambda$ from even states and a possibly different transition rate $\mu$ from odd states. We determine the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state. Some features of the birth-death process confined to the non-negative integers by a reflecting boundary in the zero-state are also analyzed. In particular, making use of a Laplace transform approach we obtain a series form of the transition probability from state 1 to the zero-state.
14 Jun 2022
TL;DR: In this paper , a joint probability generating function for the message quene lengths in a slotted p-persistent CSMA/CD (Carrier-Sense Multiple Access with Collision Detection) system with a finite population of buffered users is developed.
Abstract: This dissertation develops a joint probability generating function for the message quene lengths in a slotted p-persistent CSMA/CD (Carrier-Sense Multiple-Access with Collision Detection) system with a finite population of buffered users. Each user is assumed to have an independent and identical process of packet generation and an infinite buffer for storing outstanding packets. A closed form formula is obtained for the generating function in case of a two user system. Analytic formulas for the stability condition of a p-persistent CSMA/CD system are derived using the generating function. The range of the transmission probability p that leads to stability is examined through numerical analysis over a wide variety of the user's communication demand, the packet length distribution, the number of users on the network, and the time to detect and abort collided packets.
01 Jan 2016
TL;DR: The behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution is studied.
Abstract: This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution. CONTINUOUS AND DISCRETE TIME QUEUEING SYSTEM; BATCH ARRIVALS OF
TL;DR: In this article, the convergence and analyticity of Maclaurin and Laurent random power series were investigated. But the convergence of the Laurent power series was not analyzed in this paper.
Abstract: We find the domains of convergence and analyticity of Maclaurin and Laurent random power series