Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T20:38:50.520Z Has data issue: false hasContentIssue false
  • English
  • Français

PRODUCT-FORM MARKOVIAN QUEUEING SYSTEMS WITH MULTIPLE RESOURCES

Published online by Cambridge University Press:  14 June 2019

Valeriy Naumov  and
Konstantin Samouylov
Valeriy Naumov
Affiliation:
Service Innovation Research Institute, Annankatu 8 A, 00120 Helsinki, Finland E-mail: [email protected]
Konstantin Samouylov
Affiliation:
Peoples' Friendship University of Russia (RUDN University), Miklukho-Maklaya St. 6, 117198 Moscow, Russian Federation E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.

Keywords

loss systemMarkovian modelsmultiple resourcesrandom resource requirements
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Special Issue 1: Learning, Optimization, and Theory of G-Networks , January 2021 , pp. 180 - 188
Copyright
Copyright © Cambridge University Press 2019

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

1Artalejo, J.R. (2000). G-networks: A versatile approach for work removal in queueing networks. European Journal of Operational Research 126(2): 233–249.CrossRefGoogle Scholar
2Basharin, G.P. & Naumov, V.A. (1984). Simple matrix description of peaked and smooth traffic and its applications. Fundamentals of Teletraffic Theory. Proceedings of the Third International Seminar on Teletraffic Theory. Moscow: VINITI, 38–44.Google Scholar
3Basharin, G.P, Naumov, V.A., & Samuilov, K.E. (2018). On Markovian modelling of arrival processes. Statistical Papers 59(4): 1533–1540.CrossRefGoogle Scholar
4Breuer, L. (2003). From Markov jump processes to spatial queues. New York: Springer Science & Business Media.CrossRefGoogle Scholar
5Caglayan, M.U. (2017). G-networks and their applications to machine learning, energy packet networks and routing: Introduction to the special issue. Probability in the Engineering and Informational Sciences 31(4): 381–395.CrossRefGoogle Scholar
6Ezhov, I.I. & Skorokhod, A.V. (1969). Markov processes with homogeneous second component. Theory of Probability and its Applications Part I, 14(1): 1–13. Part II, 14(4): 652–667.CrossRefGoogle Scholar
7Fourneau, J.M. & Gelenbe, E. (2017). G-Networks with adders. Future Internet 9(3): 34.CrossRefGoogle Scholar
8Fourneau, J.M., Gelenbe, E., & Suros, R. (1996). G-networks with multiple classes of negative and positive customer. Theoretical Computer Science 155(1): 141–156.CrossRefGoogle Scholar
9Gelenbe, E. (1989). Random neural networks with negative and positive signals and product form solution. Neural Computation 1(4): 502–510.CrossRefGoogle Scholar
10Gelenbe, E. (1991). Product-form queuing-networks with negative and positive customers. Journal of Applied Probability 28(3): 656–663.CrossRefGoogle Scholar
11Gelenbe, E. (1993). G-networks with signals and batch removal. Probability in the Engineering and Informational Sciences 7: 335–342.CrossRefGoogle Scholar
12Gelenbe, E. & Ceran, E.T. (2016). Energy packet networks with energy harvesting. IEEE Access 4: 1321–1331.CrossRefGoogle Scholar
13Gelenbe, E. & Fourneau, J.M. (2002). G-networks with resets. Performance Evaluation 49(1): 179–191.CrossRefGoogle Scholar
14Gelenbe, E. & Labed, A. (1998). G-networks with multiple classes of signals and positive customers. European Journal of Operational Research 108(2): 293–305.CrossRefGoogle Scholar
15Gorbunova, A.V., Naumov, V.A., Gaidamaka, Yu.V., & Samouylov, K.E. (2019). Resource queuing systems with general service discipline. Informatics and Applications 13(1): 99–107.Google Scholar
16Naumov, V.A. (1976). On the independent work of the subsystems of a complex system. In Gnedenko, B.V., Gromak, Yu.I., & Chepurin, E.V., (eds.), Queueing Theory. Proceedings of the Third All-Union School on Queueing Theory, Puschtschino, 1974. Moscow: MSU Press, 169–177 (in Russian).Google Scholar
17Naumov, V. & Samouylov, K. (2017). Analysis of multi-resource loss system with state-dependent arrival and service rates. Probability in the Engineering and Informational Sciences 31(4): 413–419.CrossRefGoogle Scholar
18Naumov, V.A., Samouylov, K.E., & Samouylov, A.K. (2016). On the total amount of resources occupied by serviced customers. Automation and Remote Control 77(8): 1419–1427.CrossRefGoogle Scholar
19Romm, E.L. & Skitovitch, V.V. (1971). On certain generalization of problem of Erlang. Automation and Remote Control 32(6): 1000–1003.Google Scholar
20Tien, Van Do. (2011). Bibliography on G-networks, negative customers and applications. Mathematical and Computer Modelling 53(1–2): 205–212.Google Scholar