Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Calculating Expected Incomes in Open Markov Networks with Requests of Different Classes and Different Peculiarities

  • Stochastic Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

A system of difference-differential equations for the expected incomes of open Markov queueing networks with different peculiarities is considered. The number of network states and also the number of equations in this system are both infinite. The incoming flows of requests are elementary and independent while their service times have exponential distributions. The incomes from transitions between different states of the network are deterministic functions that depend on its states; the incomes gained by the queuing server systems per unit time under the invariable states also depend on these states only. The system of the difference-differential equations is solved using the modified method of successive approximations combined with the series method. An example of a Markov G-network with signals and the group elimination of positive requests is studied. As demonstrated below, the expected incomes can be increasing and decreasing time-varying functions; can take positive and negative values.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Crabil, T., Optimal Control of a Service Facility with Varible Exponential Service Times and Constant Arrival Rate, Manage. Sci., 1972, no. 18, pp. 560–566.

    Google Scholar 

  2. Foschini, G., On Heavy Traffic Diffusion Analysis and Dynamic Routing in Packet Switched Networks, Comput. Performance, 1977, no. 10, pp. 499–514.

    Google Scholar 

  3. Stidham, S. and Weber, R., A Survey of Markov Decision Models for Control of Networks of Queue, Queueing Syst., 1993, no. 3, pp. 291–314.

    Google Scholar 

  4. Matalytski, M. and Pankov, A., Analysis of the Stochastic Model of the Changing of Incomes in the Open Banking Network, Comput. Sci., 2003, vol. 3, no. 5, pp. 19–29.

    Google Scholar 

  5. Matalytski, M. and Pankov, A., Incomes Probabilistic Models of the Banking Network, Sci. Res. Inst. Math. Comput. Sci. Czestochowa Univ. Technol., 2003, vol. 1, no. 2, pp. 99–104.

    Google Scholar 

  6. Matalytski, M., On Some Results in Analysis and Optimization of Markov Networks with Incomes and Their Application, Autom. Remote Control, 2009, vol. 70, no. 10, pp. 1683–1697.

    Article  MathSciNet  MATH  Google Scholar 

  7. Howard, R.A., Dynamic Programming and Markov Processes, Boston: MIT Press, 1960. Translated under the title Dinamicheskoe programmirovanie i markovskie protsessy, Moscow: Sovetskoe Radio, 1964.

    MATH  Google Scholar 

  8. Gelenbe, E., Product-form Queueing Networks with Negative and Positive Customers, J. App. Probab., 1991, vol. 28, pp. 656–663.

    Article  MathSciNet  MATH  Google Scholar 

  9. Jackson, J.R., Networks of Waiting Lines, Oper. Res., 1957, vol. 5, no. 4, pp. 518–521.

    Article  MathSciNet  MATH  Google Scholar 

  10. Basharin, G.P., Bocharov, P.P., and Kogan, Ya.A., Analiz ocheredei v vychislitel’nykh setyakh (Queue Analysis in Computing Networks), Moscow: Nauka, 1989.

    MATH  Google Scholar 

  11. Ivnitskii, V.A., Teoriya setei massovogo obsluzhivaniya (Theory of Queueing Networks), Moscow: Fiz-matlit, 2004.

    Google Scholar 

  12. Malinkovsky, Yu.V., A Criterion for the Representability of the Stationary Distribution of the States of an Open Markov Queueing Network with Different Customer Classes in the Form of a Product, Autom. Remote Control, 1991, vol. 52, no. 4, pp. 503–509.

    MathSciNet  Google Scholar 

  13. Serfozo, R., Introduction to Stochastic Networks, New York: Springer-Verlag, 1999.

    Book  MATH  Google Scholar 

  14. Gelenbe, E. and Schassberger, R., Stability of G-networks, Prob. Engin. Inform. Sci., 1992, vol. 6, no. 1, pp. 271–276.

    Article  MATH  Google Scholar 

  15. Gelenbe, E., G-networks: A Unifying Model for Neural and Queueing Networks, Ann. Oper. Res., 1994, vol. 48, pp. 433–461.

    Article  MathSciNet  MATH  Google Scholar 

  16. Bocharov, P.P. and Vishnevski, B.M., G-networks: Development of the Theory of Multiplicative Networks, Autom. Remote Control, 2003, vol. 84, no. 5, pp. 714–739.

    Article  MathSciNet  MATH  Google Scholar 

  17. Fourneau, J.N., Gelenbe, E., and Suros, R., G-networks with Multiple Classes of Negative and Positive Customers, Theor. Comp. Sci., 1996, vol. 155, pp. 141–156.

    Article  MathSciNet  MATH  Google Scholar 

  18. Gelenbe, E. and Labed, A., G-networks with Multiple Classes of Signals and Positive Customers, Eur. J. Oper. Res., 1998, vol. 108, no. 2, pp. 293–305.

    Article  MATH  Google Scholar 

  19. Matalytski, M., Analysis of G-network with Multiple Classes of Customers at Transient Behavior, Prob. Eng. Inform. Sci., 2018, pp. 1–14. DOI:https://doi.org/10.1017/S0269964818000086

    Google Scholar 

  20. Matalytski, M., Finding Non-Stationary State Probability of G-networks with Signal and Customers Batch Removal, Prob. Eng. Inform. Sci., 2017, vol. 31, no. 4, pp. 346–412.

    MathSciNet  MATH  Google Scholar 

  21. Matalytski, M., Analysis and Forecasting of Expected Incomes in Markov Networks with Bounded Waiting Time for Claims, Autom. Remote Control, 2015, no. 6, pp. 1005–1017.

    Google Scholar 

  22. Matalytski, M., Analysis and Forecasting of Expected Incomes in Markov Networks with Unreliable Servicing Systems, Autom. Remote Control, 2015, no. 12, pp. 2179–2189.

    Google Scholar 

  23. Matalytski, M., Forecasting Anticipated Incomes in the Markov Networks with Positive and Negative Customers, Autom. Remote Control, 2017, vol. 78, no. 5, pp. 815–825.

    Article  MathSciNet  MATH  Google Scholar 

  24. Kopats, D.Ya. and Matalytskii, M.A., Calculating Expected Incomes in a Network with Positive and Negative Requests of Different Types, Vestn. Grodn. Gos. Univ. Ser. 2, 2018, vol. 8, no. 1, pp. 132–144.

    Google Scholar 

  25. Kopats, D.Ya., Naumenko, V.V., and Matalytskii, M.A., Calculating Expected Incomes in a Queueing Network with Random Waiting Times of Positive and Negative Requests, Vestn. Grodn. Gos. Univ. Ser. 2, 2017, vol. 7, no. 1, pp. 147–153.

    Google Scholar 

  26. Matalytskii, M.A. and Kopats, D.Ya., Calculating Expected Incomes in a Network with Heterogeneous Positive and Negative Requests, Vestn. Grodn. Gos. Univ. Ser. 2, 2018, vol. 8, no. 2, pp. 129–140.

    Google Scholar 

  27. Matalytski, M. and Kopats, D., Analysis of the Network with Multiple Classes of Positive Customers and Signals at a Non-Stationary Regime, Prob. Eng. Inform. Sci., 2018, pp. 1–13. DOI: https://doi.org/10.1017/S0269964818000219

    Google Scholar 

  28. Rashevskii, P., Rimanova geometriya i tenzornyi analiz (Riemann’s Geometry and Tensor Analysis), Moscow: Nauka, 1967.

    Google Scholar 

  29. Valeev, K.G. and Zhautykov, O.A., Beskonechnye sistemy differentsial’nykh uravnenii (Infinite Systems of Differential Equations), Alma-Ata: Nauka, 1974.

    Google Scholar 

  30. Korobeinik, Yu.F., Differential Equations of Infinite Order and Infinite Systems of Differential Equations, Izv. Akad. Nauk SSSR, Ser. Mat., 1970, vol. 34, no. 4, pp. 881–922.

    MathSciNet  Google Scholar 

  31. Matalytski, M. and Naumenko, V., Simulation Modeling of HM-networks with Consideration of Positive and Negative Messages, J. Appl. Math. Comput. Mechan., 2015, vol. 14, no. 2, pp. 49–60.

    Article  Google Scholar 

  32. Matalytskii, M.A. and Naumenko, V.V., Stokhasticheskie seti s nestandartnym peremeshcheniem zayavok (Stochastic Networks with Nonstandard Transfer of Requests), Grodno: Grodn. Gos. Univ., 2016.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    M. A. Matalytski

  2. Yanka Kupala State University of Grodno, Grodno, Belarus

    D. Ya. Kopats

Authors
  1. M. A. Matalytski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Ya. Kopats
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to M. A. Matalytski or D. Ya. Kopats.

Additional information

Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 6, pp. 104–120.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matalytski, M.A., Kopats, D.Y. Calculating Expected Incomes in Open Markov Networks with Requests of Different Classes and Different Peculiarities. Autom Remote Control 80, 1069–1081 (2019). https://doi.org/10.1134/S0005117919060067

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117919060067

Keywords

Search

Navigation