# Product form solution for g-networks with dependent service

• [1] Institute of Informatics Problems Russian Academy of Sciences Moscow, Russia
• Volume: 38, Issue: 2, page 105-119
• ISSN: 0399-0559

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## Abstract

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We consider a G-network with Poisson flow of positive customers. Each positive customer entering the network is characterized by a set of stochastic parameters: customer route, the length of customer route, customer volume and his service length at each route stage as well. The following node types are considered: (0) an exponential node with ${c}_{n}$ servers, infinite buffer and FIFO discipline; (1) an infinite-server node; (2) a single-server node with infinite buffer and LIFO PR discipline; (3) a single-server node with infinite buffer and PS discipline. Negative customers arriving at each node also form a Poisson flow. A negative customer entering a node with $k$ customers in service, with probability $1/k$ chooses one of served positive customer as a “target”. Then, if the node is of a type 0 the negative customer immediately “kills” (displaces from the network) the target customer, and if the node is of types 1–3 the negative customer with given probability depending on parameters of the target customer route kills this customer and with complementary probability he quits the network with no service. A product form for the stationary probabilities of underlying Markov process is obtained.

## How to cite

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Bocharov, Pavel, et al. "Product form solution for g-networks with dependent service." RAIRO - Operations Research - Recherche Opérationnelle 38.2 (2004): 105-119. <http://eudml.org/doc/245053>.

@article{Bocharov2004,
abstract = {We consider a G-network with Poisson flow of positive customers. Each positive customer entering the network is characterized by a set of stochastic parameters: customer route, the length of customer route, customer volume and his service length at each route stage as well. The following node types are considered: (0) an exponential node with $c_n$ servers, infinite buffer and FIFO discipline; (1) an infinite-server node; (2) a single-server node with infinite buffer and LIFO PR discipline; (3) a single-server node with infinite buffer and PS discipline. Negative customers arriving at each node also form a Poisson flow. A negative customer entering a node with $k$ customers in service, with probability $1/k$ chooses one of served positive customer as a “target”. Then, if the node is of a type 0 the negative customer immediately “kills” (displaces from the network) the target customer, and if the node is of types 1–3 the negative customer with given probability depending on parameters of the target customer route kills this customer and with complementary probability he quits the network with no service. A product form for the stationary probabilities of underlying Markov process is obtained.},
affiliation = {Institute of Informatics Problems Russian Academy of Sciences Moscow, Russia},
author = {Bocharov, Pavel, D'Apice, Ciro, Gavrilov, Evgeny, Pechinkin, Alexandre},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
language = {eng},
number = {2},
pages = {105-119},
publisher = {EDP-Sciences},
title = {Product form solution for g-networks with dependent service},
url = {http://eudml.org/doc/245053},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Bocharov, Pavel
AU - D'Apice, Ciro
AU - Gavrilov, Evgeny
AU - Pechinkin, Alexandre
TI - Product form solution for g-networks with dependent service
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 2
SP - 105
EP - 119
AB - We consider a G-network with Poisson flow of positive customers. Each positive customer entering the network is characterized by a set of stochastic parameters: customer route, the length of customer route, customer volume and his service length at each route stage as well. The following node types are considered: (0) an exponential node with $c_n$ servers, infinite buffer and FIFO discipline; (1) an infinite-server node; (2) a single-server node with infinite buffer and LIFO PR discipline; (3) a single-server node with infinite buffer and PS discipline. Negative customers arriving at each node also form a Poisson flow. A negative customer entering a node with $k$ customers in service, with probability $1/k$ chooses one of served positive customer as a “target”. Then, if the node is of a type 0 the negative customer immediately “kills” (displaces from the network) the target customer, and if the node is of types 1–3 the negative customer with given probability depending on parameters of the target customer route kills this customer and with complementary probability he quits the network with no service. A product form for the stationary probabilities of underlying Markov process is obtained.
LA - eng
UR - http://eudml.org/doc/245053
ER -

## References

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