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

top
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

top

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

top
  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).  
  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.  
  4. P.P. Bocharov and V.M. Vishnevskii, G-networks: development of the theory of multiplicative networks. Autom. Remote Control64 (2003) 714-739.  
  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.  
  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.  
  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.  
  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.  
  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).  
  21. E. Gelenbe, P. Glynn and K. Sigman, Queues with negative arrivals. J. Appl. Probab.28 (1991) 245-250.  
  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.  
  24. E. Gelenbe and I. Mitrani, Analysis and Synthesis of Computer Systems. New York, London Academic Press (1980).  
  25. E. Gelenbe and G. Pujolle, Introduction to Queueing Networks. New York, Wiley (1998).  
  26. E. Gelenbe and M. Schassberger, Stability of product form G-Networks. Probab. Eng. Inform. Sci.6 (1992) 271-276.  
  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.  
  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).  
  32. D. Kouvatsos, Entropy maximisation and queueing network models. Ann. Oper. Res.48 (1994) 63-126.  
  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).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.