Skip to main content
SpringerLink
Log in

Invariance of the stationary state distribution for mass service networks with multi-regime strategies, different demands, and a “generalized processor sharing” discipline

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

Mass service networks with multiregime strategies and several types of demands are studied. Single-line nodes can operate in several regimes corresponding to different degrees of performance. The time of existence of each regime has an exponential distribution. The discipline of the demand servicing by the device is the “generalized processor sharing” (GPS) with random channel choice in the node. The amount of work on servicing a demand is a random quantity with an arbitrary distribution function. The invariance of the stationary probability distribution of network states with respect to the functional form of the distribution of the amount of work required for the demand servicing is established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Sevast’yanov, B.A., Limit Theorem for Markov Processes and Its Application to Telephone Systems with Rejections, Teor. Veroyat. Prim., 1957, vol. 2, no. 1, pp. 106–116.

    MATH  MathSciNet  Google Scholar 

  2. Kovalenko, I.N., About One Method in the Theory of Mass Service, Trudy VI Vses. soveshchaniya po teorii veroyatnostej i mat. statistike, (Proc. VI All-Un. Conf. on Theory of Probabilities and Math. Statistics), Vilnus, 1962, pp. 357–358.

  3. Kovalenko, I.N., Issledovaniya po analizu nadezhnosti slozhnykh sistem (Studies of Analysis of Complex System Reliability), Kiev: Naukova Dumka, 1976.

    Google Scholar 

  4. Kenig, D., Rykov, V.V., and Shmidt, F., Stationary Systems of Mass Service with Dependences, in Itogi nauki i tekhniki. Teoriya veroyatnostei. Mat. statistika. Teoretich. kibernetika. T. 18 (Resume of Science and Technics. Theory of Probabilities. Mathematical Statistics. Theoretical Cybernetics. Vol. 18), Moscow: VINITI, 1981.

    Google Scholar 

  5. 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. Assoc. Comput. Mach., 1975, vol. 22, no. 2, pp. 248–260.

    MATH  MathSciNet  Google Scholar 

  6. Stoyan, D., Queuing Networks Insensivity and Heuristic Approximation, Electron. Information. Kibern., 1978, vol. 14, no. 1, pp. 135–143.

    MATH  MathSciNet  Google Scholar 

  7. Jansen, U. and Konig, D., Insensivity and Steady-State Probabilities in Product Form for Queuing Networks, Electron. Information. Kibern., 1980, vol. 16, no. 3, pp. 385–397.

    MATH  MathSciNet  Google Scholar 

  8. Ivnitskii, V.A., On the Condition of Invariance of Stationary Probabilities for Mass Service Networks, Teoriya Veroyat. Primen., 1982, vol. 27, no. 1, pp. 188–192.

    MathSciNet  Google Scholar 

  9. Ivnitskii, V.A., Teoriya setei massovogo obsluzhivaniya (Theory of Mass Service Networks), Moscow: Fizmatlit, 2004.

    Google Scholar 

  10. Kovalev, E.A., Networks with Unreliable Channels and Reserve, Materialy VI Bel. shkoly-seminara po TMO “Matematicheskie metody issledovaniya setei svyazi i setei EVM” (Proc. VI Belgor. School-Semin. on Theory of Mass Service, Mathematical Methods of Connection Networks and Computer Networks), Minsk, 1990, pp. 70–71.

  11. Malinkovskii, Yu.V. and Nueman, A.Yu., Closed Information Networks with Multimode Strategies of Service, Materialy I mezhdunarodnoi konferentsii “Informatsionnye sistemy i tekhnologii” (ISTKh2002) (Proc. I Int. Conf. Information Systems and Technologies), Minsk, 2002, part 1.

  12. Malinkovskii, Yu.V. and Nueman, A.Yu., Multiplikativity of Stationary Distribution in Open Networks with Multimode Strategies of Service, Vesti NANB. Ser. Fiz.-Mat. Navuk, 2001, no. 3, pp. 129–134.

  13. Starovoitov, A.N., Invariance of Stationary Distribution in Networks with Multimode Strategies of Service and PS Service Discipline, Materialy IX Respubl. Nauchn. Konf. Stud. i Aspir. “Novye Mat. Metody i Komp’yuternye Tekhnologii v Proektirovanii, Proizvodstvennykh i Nauchnykh Issledovaniyakh” (Proc. IX Rep. Sci. Conf. St. Postgrad. St. Mew Math. Methods Comp. Technol. in Design, Industry and Sci. Studies), Gomel’, 2006, pp. 199–200.

  14. Malinkovskii, Yu.V., Starovoitov, A.N., and Eremina, A.R., Starovoitov, A.N., Invariance of Stationary Distribution of Probabilities of State of Networks with Multimode Strategies of Service, Claims of Different Types and LSFS RR Service Discipline, Vestnik Tomsk. Gos. Univ. Upravl., Vych. Tekhn. Inform., 2009, no. 3(8), pp. 33–39.

  15. Letunovich, Yu.E., Stationary Distribution of States of Open Inhomogeneous Network with Multimode Strategies and Immediate Service, Sb. Nauch. St. Konf. “Sovremennye Informatsionnye Komp’yuternye Tekhnologii” (Proc. Conf. Contemp. Inform. Comp. Techn.), Grodno, 2008, pp. 97–99.

Download references

Author information

Authors and Affiliations

  1. Chair of Stochastic Analysis and Econometric Simulation, Y. Kupala State University of Grodno, ul. Ozheshko 22, Grodno, 220023, Belarus

    A. R. Eremina

Authors
  1. A. R. Eremina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. R. Eremina.

Additional information

Original Russian Text © A.R. Eremina, 2011, published in Avtomatika i Vychislitel’naya Tekhnika, 2011, No. 1, pp. 43–54.

About this article

Cite this article

Eremina, A.R. Invariance of the stationary state distribution for mass service networks with multi-regime strategies, different demands, and a “generalized processor sharing” discipline. Aut. Conrol Comp. Sci. 45, 29–38 (2011). https://doi.org/10.3103/S0146411611010044

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411611010044

Keywords

Search

Navigation