Product form solution for g-networks with dependent service

Pavel Bocharov; Ciro D'Apice; Evgeny Gavrilov; Alexandre Pechinkin

RAIRO - Operations Research (2010)

  • Volume: 38, Issue: 2, page 105-119
  • ISSN: 0399-0559

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: 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 38.2 (2010): 105-119. <http://eudml.org/doc/105304>.

@article{Bocharov2010,
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: 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. },
author = {Bocharov, Pavel, D'Apice, Ciro, Gavrilov, Evgeny, Pechinkin, Alexandre},
journal = {RAIRO - Operations Research},
language = {eng},
month = {3},
number = {2},
pages = {105-119},
publisher = {EDP Sciences},
title = { Product form solution for g-networks with dependent service },
url = {http://eudml.org/doc/105304},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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: 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/105304
ER -

References

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  1. G.P. Basharin, P.P. Bocharov and Ya. A. Kogan, Analysis of Queues in Computer Networks. Theory and Design Methods, Moscow, Nauka (1989) (in Russian).  Zbl0708.68005
  2. G.P. Basharin and A.L. Tolmachev, The theory of queueing networks and its application to analysis of information-computer systems, in Itogi Nauki i Tekhniki. Teoria Veroyatnostei, Mat. Statistika, Teoret. Kibertetika21 3-120. Moscow, VINITI (1983).  
  3. F. Baskett, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed and mixed networks of queues with different classes of customers. J. ACM22 (1975) 248-260.  Zbl0313.68055
  4. P.P. Bocharov and V.M. Vishnevskii, G-networks: development of the theory of multiplicative networks. Autom. Remote Control64 (2003) 714-739.  Zbl1066.90009
  5. J. Bourrely and E. Gelenbe, Mémoires associatives: évaluation et architectures. C.R. Acad. Sci. ParisII309 (1989) 523-526.  
  6. C. Cramer and E. Gelenbe, Video quality and traffic QoS in learning-based subsampled and receiver-interpolated video sequences. IEEE J. on Selected Areas in Communications18 (2000) 150-167.  
  7. Y. Feng and E. Gelenbe, Adaptive object tracking and video compression. Network and Information Systems J.1 (1999) 371-400.  
  8. E. Gelenbe, Queuing networks with negative and positive customers. J. Appl. Prob.28 (1991) 656-663.  Zbl0741.60091
  9. J.-M. Fourneau, E. Gelenbe and R. Suros, G-networks with multiple classes of positive and negative customers. Theoret. Comp. Sci.155 (1996) 141-156.  Zbl0873.68010
  10. E. Gelenbe, Réseaux stochastiques ouverts avec clients négatifs et positifs, et réseaux neuronaux. C.R. Acad. Sci. ParisII309 (1989) 979-982.  
  11. E. Gelenbe, Random neural networks with positive and negative signals and product form solution. Neural Comput.1 (1989) 502-510.  
  12. E. Gelenbe, Réseaux neuronaux aléatoires stables. C.R. Acad. Sci.II310 (1990) 177-180.  
  13. E. Gelenbe, Stable random neural networks. Neural Comput.2 (1990) 239-247.  
  14. E. Gelenbe, G-networks with instantaneous customer movement. J. Appl. Probab.30 (1993) 742-748.  Zbl0781.60088
  15. E. Gelenbe, G-Networks with signals and batch removal. Probab. Eng. Inform. Sci.7 (1993) 335-342.  
  16. E. Gelenbe, Learning in the recurrent random network. Neural Comput.5 (1993) 154-164.  
  17. E. Gelenbe, G-networks: An unifying model for queuing networks and neural networks. Ann. Oper. Res.48 (1994) 433-461.  Zbl0803.90058
  18. E. Gelenbe, The first decade of G-networks. Eur. J. Oper. Res.126 (2000) 231-232.  
  19. E. Gelenbe and J.M. Fourneau, Random neural networks with multiple classes of signals. Neural Comput.11 (1999) 953-963.  
  20. E. Gelenbe and J.-M. Fourneau, G-Networks with resets. Perform. Eval.49 (2002) 179-192, also in Proc. IFIP WG 7.3/ACM-SIGMETRICS Performance '02 Conf., Rome, Italy (2002).  Zbl1043.68006
  21. E. Gelenbe, P. Glynn and K. Sigman, Queues with negative arrivals. J. Appl. Probab.28 (1991) 245-250.  Zbl0744.60110
  22. E. Gelenbe and K. Hussain, Learning in the multiple class random neural network. IEEE Trans. on Neural Networks13 (2002) 1257-1267.  
  23. E. Gelenbe and A. Labed, G-networks with multiple classes of signals and positive customers. Eur. J. Oper. Res.108 (1998) 293-305.  Zbl0954.90009
  24. E. Gelenbe and I. Mitrani, Analysis and Synthesis of Computer Systems. New York, London Academic Press (1980).  Zbl0484.68026
  25. E. Gelenbe and G. Pujolle, Introduction to Queueing Networks. New York, Wiley (1998).  Zbl0654.60079
  26. E. Gelenbe and M. Schassberger, Stability of product form G-Networks. Probab. Eng. Inform. Sci.6 (1992) 271-276.  Zbl1134.60396
  27. E. Gelenbe, E. Seref and Z. Xu, Simulation with learning agents. Proc. of the IEEE89 (2001) 148-157.  
  28. E. Gelenbe and H. Shachnai, On G-networks and resource allocation in multimedia systems. Eur. J. Oper. Res.126 (2000) 308-318.  Zbl0960.90007
  29. J.R. Jackson, Networks of waiting lines. Oper. Res.15 (1957) 234-265.  
  30. J.R. Jackson, Jobshop-like queueing systems. Manage. Sci.10 (1963) 131-142.  
  31. F.P. Kelly, Reversibility and Stochastic Networks. Chichester, Wiley (1979).  Zbl0422.60001
  32. D. Kouvatsos, Entropy maximisation and queueing network models. Ann. Oper. Res.48 (1994) 63-126.  Zbl0789.90032
  33. A.V. Pechinkin and V.V. Rykov, Product form for open queueing networks with dependent service times, in Proc. Distributed Computer Communication Networks. Theory and Applications, Moscow: Institute for Information Transmission Problems RAS (1977) 171-178.  
  34. M. Schwartz, Telecommunication Networks: Protocols, Modeling and Analysis. New York, Addison Wesley (1987).  
  35. N.M. Van Dijk, Queueing Networks and Product Forms. New York, Wiley (1993).  
  36. V.M. Vishnevskii, Theoretical Foundations of Computer Network Design. Moscow, Tekhnosfera 2003 (in Russian).  

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