Automation and Remote Control

, Volume 80, Issue 6, pp 1069–1081 | Cite as

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

  • M. A. MatalytskiEmail author
  • D. Ya. KopatsEmail author
Stochastic Systems


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.


queueing network expected incomes requests of different classes method of successive approximations 


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  1. 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. 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. 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. 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. 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. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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.zbMATHGoogle Scholar
  8. 8.
    Gelenbe, E., Product-form Queueing Networks with Negative and Positive Customers, J. App. Probab., 1991, vol. 28, pp. 656–663.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jackson, J.R., Networks of Waiting Lines, Oper. Res., 1957, vol. 5, no. 4, pp. 518–521.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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.zbMATHGoogle Scholar
  11. 11.
    Ivnitskii, V.A., Teoriya setei massovogo obsluzhivaniya (Theory of Queueing Networks), Moscow: Fiz-matlit, 2004.Google Scholar
  12. 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.MathSciNetGoogle Scholar
  13. 13.
    Serfozo, R., Introduction to Stochastic Networks, New York: Springer-Verlag, 1999.CrossRefzbMATHGoogle Scholar
  14. 14.
    Gelenbe, E. and Schassberger, R., Stability of G-networks, Prob. Engin. Inform. Sci., 1992, vol. 6, no. 1, pp. 271–276.CrossRefzbMATHGoogle Scholar
  15. 15.
    Gelenbe, E., G-networks: A Unifying Model for Neural and Queueing Networks, Ann. Oper. Res., 1994, vol. 48, pp. 433–461.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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.CrossRefzbMATHGoogle Scholar
  19. 19.
    Matalytski, M., Analysis of G-network with Multiple Classes of Customers at Transient Behavior, Prob. Eng. Inform. Sci., 2018, pp. 1–14. DOI: Google Scholar
  20. 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.MathSciNetzbMATHGoogle Scholar
  21. 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. 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. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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. 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. 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. 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: Google Scholar
  28. 28.
    Rashevskii, P., Rimanova geometriya i tenzornyi analiz (Riemann’s Geometry and Tensor Analysis), Moscow: Nauka, 1967.Google Scholar
  29. 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. 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.MathSciNetGoogle Scholar
  31. 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.CrossRefGoogle Scholar
  32. 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

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Czestochowa University of TechnologyCzestochowaPoland
  2. 2.Yanka Kupala State University of GrodnoGrodnoBelarus

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