scholarly journals Reservicing some customers inM/G/1queues under three disciplines

2004 ◽  
Vol 2004 (32) ◽  
pp. 1715-1723 ◽  
Author(s):  
M. R. Salehi-Rad ◽  
K. Mengersen ◽  
G. H. Shahkar
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Production Line ◽  
Busy Period ◽  
Probability Generating Function ◽  
Idle Period ◽  
The Mean ◽  
The Moment ◽  
State Size

Consider anM/G/1production line in which a production item is failed with some probability and is then repaired. We consider three repair disciplines depending on whether the failed item is repaired immediately or first stockpiled and repaired after all customers in the main queue are served or the stockpile reaches a specified threshold. For each discipline, we find the probability generating function (p.g.f.) of the steady-state size of the system at the moment of departure of the customer in the main queue, the mean busy period, and the probability of the idle period.

Moment formulas for the Markov renewal branching process

1976 ◽  
Vol 8 (4) ◽  
pp. 690-711 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Steady State ◽  
Numerical Computation ◽  
Busy Period ◽  
Branching Process ◽  
Queueing Models ◽  
Absorption Time ◽  
The Mean ◽  
The Moment

There are many queueing models in which there appears a semi-Markov matrix G(·), whose entries are absorption-time distributions in a Markov renewal branching process. The role of G(·) is similar to that of the busy period in the simple M/G/1 model. The computation of various quantities associated with G(·) is however much more complicated. The moment matrices, and particularly the mean matrix of G(·), are essential in the construction of general and mathematically well-justified algorithms for the steady-state distributions of such queues.This paper discusses the moment matrices of G(·) and algorithms for their numerical computation. Its contents are basic to the algorithmic solutions to several queueing models, which are to be presented in follow-up papers.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (2) ◽  
pp. 461-466 ◽  
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

Moment formulas for the Markov renewal branching process

1976 ◽  
Vol 8 (04) ◽  
pp. 690-711 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Steady State ◽  
Numerical Computation ◽  
Busy Period ◽  
Branching Process ◽  
Queueing Models ◽  
Absorption Time ◽  
The Mean ◽  
The Moment

There are many queueing models in which there appears a semi-Markov matrix G(·), whose entries are absorption-time distributions in a Markov renewal branching process. The role of G(·) is similar to that of the busy period in the simple M/G/1 model. The computation of various quantities associated with G(·) is however much more complicated. The moment matrices, and particularly the mean matrix of G(·), are essential in the construction of general and mathematically well-justified algorithms for the steady-state distributions of such queues. This paper discusses the moment matrices of G(·) and algorithms for their numerical computation. Its contents are basic to the algorithmic solutions to several queueing models, which are to be presented in follow-up papers.

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.

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (2) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

Theoretical solutions for degree distribution of decreasing random birth-and-death networks

2017 ◽  
Vol 31 (14) ◽  
pp. 1750161 ◽  
Author(s):  
Yin Long ◽  
Xiao-Jun Zhang ◽  
Kui Wang
Keyword(s):  
Steady State ◽  
Computer Simulations ◽  
Generating Function ◽  
Degree Distribution ◽  
Probability Generating Function ◽  
Function Approach ◽  
Poisson Summation

In this paper, theoretical solutions for degree distribution of decreasing random birth-and-death networks [Formula: see text] are provided. First, we prove that the degree distribution has the form of Poisson summation, for which degree distribution equations under steady state and probability generating function approach are employed. Then, based on the form of Poisson summation, we further confirm the tail characteristic of degree distribution is Poisson tail. Finally, simulations are carried out to verify these results by comparing the theoretical solutions with computer simulations.

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (02) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

How many random digits are required until given sequences are obtained?

1982 ◽  
Vol 19 (03) ◽  
pp. 518-531 ◽  
Author(s):  
Gunnar Blom ◽  
Daniel Thorburn
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Probability Generating Function ◽  
Recursive Formula ◽  
Fixed Number ◽  
Digit Sequence ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

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