Mathematical Modeling of Bivariate Distributions of Polymer Properties Using 2D Probability Generating Functions. Part II: Transformation of Population Mass Balances of Polymer Processes

2013 ◽  
Vol 22 (5) ◽  
pp. 273-308 ◽  
Author(s):  
Adriana Brandolin ◽  
Mariano Asteasuain
Keyword(s):  
Mathematical Modeling ◽  
Generating Functions ◽  
Mass Balances ◽  
Bivariate Distributions ◽  
Probability Generating Functions ◽  
Polymer Properties

scholarly journals First Passage and Collective Marks

2019 ◽  
Vol 8 (6) ◽  
pp. 47
Author(s):  
Yiping Zhang ◽  
Myron Hlynka ◽  
Percy H. Brill
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
First Passage Times ◽  
System Of Equations ◽  
First Passage ◽  
Probability Generating Functions ◽  
Collective Marks ◽  
Passage Times

Probability generating functions for first passage times of Markov chains are found using the method of collective marks. A system of equations is found which can be used to obtain moments of the first passage times. Second passage probabilities are discussed.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

EXPLICIT SOLUTIONS FOR CONTINUOUS-TIME QBD PROCESSES BY USING RELATIONS BETWEEN MATRIX GEOMETRIC ANALYSIS AND THE PROBABILITY GENERATING FUNCTIONS METHOD

2020 ◽  
pp. 1-16 ◽  
Author(s):  
Gabi Hanukov ◽  
Uri Yechiali
Keyword(s):  
Generating Functions ◽  
Continuous Time ◽  
Geometric Analysis ◽  
Quadratic Equation ◽  
Unknown Boundary ◽  
Birth And Death Processes ◽  
Probability Generating Functions ◽  
Qbd Processes ◽  
The Matrix ◽  
The Stability

Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$ ; and (ii) the stability condition is readily extracted.

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