The second moments of the absorption times in the Markov renewal branching process

1978 ◽  
Vol 15 (4) ◽  
pp. 707-714 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Busy Period ◽  
Branching Process ◽  
Queueing Models ◽  
Explicit Formulas ◽  
Absorption Time ◽  
Iterative Computation ◽  
Second Moments ◽  
Number Of Customers

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

The second moments of the absorption times in the Markov renewal branching process

1978 ◽  
Vol 15 (04) ◽  
pp. 707-714 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Busy Period ◽  
Branching Process ◽  
Queueing Models ◽  
Explicit Formulas ◽  
Absorption Time ◽  
Iterative Computation ◽  
Second Moments ◽  
Number Of Customers

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

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.

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.

Moments for a general branching process in a semi-Markovian environment

1980 ◽  
Vol 17 (02) ◽  
pp. 341-349 ◽  
Author(s):  
Craig Whittaker ◽  
Richard M. Feldman
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Markovian Environment ◽  
Iterative Computation ◽  
Second Moments ◽  
Environmental Process

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.

On finite exponential moments for branching processes and busy periods for queues

2004 ◽  
Vol 41 (A) ◽  
pp. 273-280 ◽  
Author(s):  
Marvin K. Nakayama ◽  
Perwez Shahabuddin ◽  
Karl Sigman
Keyword(s):  
Random Walk ◽  
Busy Period ◽  
Branching Process ◽  
Branching Processes ◽  
Exponential Moment ◽  
Exponential Moments ◽  
Simulation Based ◽  
Age Dependent ◽  
Semi Markov Processes ◽  
Number Of Customers

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

Moments for a general branching process in a semi-Markovian environment

1980 ◽  
Vol 17 (2) ◽  
pp. 341-349 ◽  
Author(s):  
Craig Whittaker ◽  
Richard M. Feldman
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Markovian Environment ◽  
Iterative Computation ◽  
Second Moments ◽  
Environmental Process

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.

On finite exponential moments for branching processes and busy periods for queues

2004 ◽  
Vol 41 (A) ◽  
pp. 273-280 ◽  
Author(s):  
Marvin K. Nakayama ◽  
Perwez Shahabuddin ◽  
Karl Sigman
Keyword(s):  
Random Walk ◽  
Busy Period ◽  
Branching Process ◽  
Branching Processes ◽  
Exponential Moment ◽  
Exponential Moments ◽  
Simulation Based ◽  
Age Dependent ◽  
Semi Markov Processes ◽  
Number Of Customers

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

Multitype processes with reproduction-dependent immigration

1998 ◽  
Vol 35 (02) ◽  
pp. 281-292
Author(s):  
Ibrahim Rahimov
Keyword(s):  
Limit Theorem ◽  
Discrete Time ◽  
Branching Process ◽  
Martingale Approach ◽  
Offspring Distribution ◽  
Second Moments ◽  
Branching Process With Immigration

The multitype discrete time indecomposable branching process with immigration is considered. Using a martingale approach a limit theorem is proved for such processes when the totality of immigrating individuals at a given time depends on evolution of the processes generating by previously immigrated individuals. Corollaries of the limit theorem are obtained for the cases of finite and infinite second moments of offspring distribution in critical processes.

The busy period of the E k/G/1 queue by finite waiting room

1975 ◽  
Vol 7 (02) ◽  
pp. 416-430
Author(s):  
A. L. Truslove
Keyword(s):  
Markov Chain ◽  
Busy Period ◽  
Waiting Room ◽  
Queueing Process ◽  
Number Of Customers

For the E k /G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of the busy period, together with the number of customers who arrive, and the number of customers served, during that period, is obtained. The limit as the size of the waiting room becomes infinite is found.

The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1

1967 ◽  
Vol 4 (01) ◽  
pp. 162-179 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Busy Period ◽  
Transition Probabilities ◽  
Queueing Systems ◽  
Entrance Time ◽  
Distribution Of The Maximum ◽  
Number Of Customers

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.

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