scholarly journals Investigation of Probability Generating Function in an Interdependent <i>M/M/</i>1:(∞; GD) Queueing Model with Controllable Arrival Rates Using Rouche’s Theorem

2012 ◽  
Vol 01 (02) ◽  
pp. 34-38
Author(s):  
Vishwa Nath Maurya
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Queueing Model ◽  
Controllable Arrival Rates ◽  
Rouche’S Theorem ◽  
Rouché's Theorem

scholarly journals Transient Solution of 𝑀𝑋/𝑀/𝐶 Queueing Model for Homogenous Servers, with Catastrophes, Balking & Vacation

2020 ◽  
Vol 5 (4) ◽  
pp. 65-69
Author(s):  
G. Kavitha ◽  
◽  
K.Julia Rose Mary ◽  
Keyword(s):  
Exponential Distribution ◽  
Generating Function ◽  
Probability Generating Function ◽  
Service Rate ◽  
Queueing Model ◽  
Transient Solution

In this paper we analyze 𝑴𝑿/𝑴/𝑪 Queueing model of homogenous service rate with catastrophes, balking and vacation. Here we consider the customers, where arrival follow a poisson and the service follows an exponential distribution. Based on the above considerations, under catastrophes, balking and vacation by using probability generating function along with the Bessel properties we obtain the transient solution of the model in a simple way.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 868
Author(s):  
Khrystyna Prysyazhnyk ◽  
Iryna Bazylevych ◽  
Ludmila Mitkova ◽  
Iryna Ivanochko
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Probability Generating Function ◽  
Time Interval ◽  
The Moment ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

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.

A Remark on Rouché's Theorem

1976 ◽  
Vol 83 (3) ◽  
pp. 186-187 ◽  
Author(s):  
I. Glicksberg
Keyword(s):  
Rouche’S Theorem ◽  
Rouché's Theorem

Interconnected population processes

1973 ◽  
Vol 10 (01) ◽  
pp. 1-14 ◽  
Author(s):  
E. Renshaw
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Approximate Solutions ◽  
Death Process ◽  
Birth And Death Process ◽  
Population Processes

This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.

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