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: The paper categorizes discrete multivariate distributions into classes according to the forms of their probability generating functions, putting especial emphasis on those with pgf's involving Lauricella functions.
12 citations
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TL;DR: This work first considers a 2-queue Markovian system with blocking at the second queue, analyzes it, and derives its stability condition, then studies a non-Markovian setting and derives the stability condition for an approximating diffusion, showing its similarity to the former condition.
Abstract: We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Markovian system with blocking at the second queue, analyze it, and derive its stability condition. We then study a non-Markovian setting and derive the stability condition for an approximating diffusion, showing its similarity to the former condition. In the Markovian setting, we use probability generating functions and matrix analytic techniques. In the diffusion setting, we consider expectations of the first hitting times of compact sets.
12 citations
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TL;DR: In this article, a resolvent analysis of the lattice Laplacian (the generator of a simple random walk on the -dimensional integer lattice) under large deviations of the random walk was carried out.
Abstract: We carry out a resolvent analysis of the lattice Laplacian (the generator of a simple random walk on the -dimensional integer lattice) under large deviations of the random walk. This enables us to obtain asymptotic representations for the transition probability of the simple random walk and the corresponding Green function. We explicitly describe the asymptotic behaviour of the transition probability as the spatial and temporal variables jointly tend to infinity. The resulting Cramer-type expansion for the transition probability is 'universal' in this sense. In particular, it enables us to construct a scale for measuring the transition probability as a function of the time assuming that the spatial variable is of order for various values of . We prove limit theorems on the asymptotic behaviour of the Green function of the transition probabilities under large deviations of the random walk.
12 citations
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TL;DR: In this article, the extinction probability of a branching process has been studied in the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment.
Abstract: In the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment the remainder term is proportional to another p.g.f. This leads to simple proofs of other power series expansions for p.g.f.'s, including an inversion formula giving the distribution in terms of the moments (when this can be done). Old and new inequalities for the extinction probability of a branching process are established. PROBABILITY GENERATING FUNCTIONS; FACTORIAL MOMENTS; BRANCHING PROCESS EXTINCTION PROBABILITY BOUNDS
12 citations
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20 Oct 2009TL;DR: This work analyzes a particular class of continuous-time multi-type branching processes, named Markovian binary trees (MBTs), and focuses on transient measures, such as the distribution of the population size at a finite time, the Distribution of the time until extinction, and the total family size until a given time.
Abstract: We analyze a particular class of continuous-time multi-type branching processes, named Markovian binary trees (MBTs), and we focus on transient measures, such as the distribution of the population size at a finite time, the distribution of the time until extinction, and the distribution of the total family size until a given time, as well as the total size until extinction. Our results mainly are formulated as differential equations for probability generating functions and expressions for the factorial moments. They are applied to a demographic comparison of three countries.
12 citations