Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-29T19:29:32.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Product form solution for g-networks with dependentservice

Published online by Cambridge University Press:  15 April 2004

Pavel Bocharov ,
Ciro D'Apice ,
Evgeny Gavrilov  and
Alexandre Pechinkin
Pavel Bocharov
Affiliation:
Department of Probability Theory and Mathematical Statistics Peoples' Friendship University of Russia, Moscow, Russia; pbocharov@sci.pfu.edu.ru.; tropic_mos@rambler.ru.
Ciro D'Apice
Affiliation:
Department of Information Engineering and Applied Mathematics, University of Salerno, Italy; dapice@diima.unisa.it.
Evgeny Gavrilov
Affiliation:
Department of Probability Theory and Mathematical Statistics Peoples' Friendship University of Russia, Moscow, Russia; pbocharov@sci.pfu.edu.ru.; tropic_mos@rambler.ru.
Alexandre Pechinkin
Affiliation:
Institute of Informatics Problems Russian Academy of Sciences Moscow, Russia; APechinkin@ipiran.ru.
Get access

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.

Type
Research Article
Information
RAIRO - Operations Research , Volume 38 , Issue 2: Advances in modelling of complex systems , April 2004 , pp. 105 - 119
Copyright
© EDP Sciences, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
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. Kibertetika 21 3-120. Moscow, VINITI (1983).
Baskett, F., Chandy, K.M., Muntz, R.R. and Palacios, F.G., Open, closed and mixed networks of queues with different classes of customers. J. ACM 22 (1975) 248-260. CrossRef
Bocharov, P.P. and Vishnevskii, V.M., G-networks: development of the theory of multiplicative networks. Autom. Remote Control 64 (2003) 714-739. CrossRef
J. Bourrely and E. Gelenbe, Mémoires associatives: évaluation et architectures. C.R. Acad. Sci. Paris II 309 (1989) 523-526.
Cramer, C. and Gelenbe, E., Video quality and traffic QoS in learning-based subsampled and receiver-interpolated video sequences. IEEE J. on Selected Areas in Communications 18 (2000) 150-167. CrossRef
Feng, Y. and Gelenbe, E., Adaptive object tracking and video compression. Network and Information Systems J. 1 (1999) 371-400.
Gelenbe, E., Queuing networks with negative and positive customers. J. Appl. Prob. 28 (1991) 656-663. CrossRef
Fourneau, J.-M., Gelenbe, E. and Suros, R., G-networks with multiple classes of positive and negative customers. Theoret. Comp. Sci. 155 (1996) 141-156. CrossRef
E. Gelenbe, Réseaux stochastiques ouverts avec clients négatifs et positifs, et réseaux neuronaux. C.R. Acad. Sci. Paris II 309 (1989) 979-982.
Gelenbe, E., Random neural networks with positive and negative signals and product form solution. Neural Comput. 1 (1989) 502-510. CrossRef
E. Gelenbe, Réseaux neuronaux aléatoires stables. C.R. Acad. Sci. II 310 (1990) 177-180.
Gelenbe, E., Stable random neural networks. Neural Comput. 2 (1990) 239-247. CrossRef
Gelenbe, E., G-networks with instantaneous customer movement. J. Appl. Probab. 30 (1993) 742-748.
Gelenbe, E., G-Networks with signals and batch removal. Probab. Eng. Inform. Sci. 7 (1993) 335-342. CrossRef
Gelenbe, E., Learning in the recurrent random network. Neural Comput. 5 (1993) 154-164. CrossRef
Gelenbe, E., G-networks: An unifying model for queuing networks and neural networks. Ann. Oper. Res. 48 (1994) 433-461. CrossRef
Gelenbe, E., The first decade of G-networks. Eur. J. Oper. Res. 126 (2000) 231-232. CrossRef
Gelenbe, E. and Fourneau, J.M., Random neural networks with multiple classes of signals. Neural Comput. 11 (1999) 953-963. CrossRef
Gelenbe, E. and Fourneau, J.-M., 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). CrossRef
Gelenbe, E., Glynn, P. and Sigman, K., Queues with negative arrivals. J. Appl. Probab. 28 (1991) 245-250. CrossRef
Gelenbe, E. and Hussain, K., Learning in the multiple class random neural network. IEEE Trans. on Neural Networks 13 (2002) 1257-1267. CrossRef
Gelenbe, E. and Labed, A., G-networks with multiple classes of signals and positive customers. Eur. J. Oper. Res. 108 (1998) 293-305. CrossRef
E. Gelenbe and I. Mitrani, Analysis and Synthesis of Computer Systems. New York, London Academic Press (1980).
E. Gelenbe and G. Pujolle, Introduction to Queueing Networks. New York, Wiley (1998).
Gelenbe, E. and Schassberger, M., Stability of product form G-Networks. Probab. Eng. Inform. Sci. 6 (1992) 271-276. CrossRef
Gelenbe, E., Seref, E. and Simulation, Z. Xu with learning agents. Proc. of the IEEE 89 (2001) 148-157. CrossRef
Gelenbe, E. and Shachnai, H., G-networks, On and resource allocation in multimedia systems. Eur. J. Oper. Res. 126 (2000) 308-318. CrossRef
Jackson, J.R., Networks of waiting lines. Oper. Res. 15 (1957) 234-265.
Jackson, J.R., Jobshop-like queueing systems. Manage. Sci. 10 (1963) 131-142. CrossRef
F.P. Kelly, Reversibility and Stochastic Networks. Chichester, Wiley (1979).
Kouvatsos, D., Entropy maximisation and queueing network models. Ann. Oper. Res. 48 (1994) 63-126. CrossRef
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.
M. Schwartz, Telecommunication Networks: Protocols, Modeling and Analysis. New York, Addison Wesley (1987).
N.M. Van Dijk, Queueing Networks and Product Forms. New York, Wiley (1993).
V.M. Vishnevskii, Theoretical Foundations of Computer Network Design. Moscow, Tekhnosfera 2003 (in Russian).