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Research of Heterogeneous Queueing System SM|M\(^{(n)}|\infty \)

  • Ekaterina PankratovaEmail author
  • Mais Farkhadov
  • Erol Gelenbe
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 800)

Abstract

One of the modifications of the mathematical models used to describe processes in multi-service communication networks and telecommunication systems is the queueing system with heterogeneous servers. As a rule, for simulation of such processes the system with non-Poisson input flows is used. We consider the queuing system with infinite number of servers of n different types and exponential service time. Incoming flow is a Semi Markovian Process (SM-flow). Investigation of n-dimensional stochastic process characterizing the number of occupied servers of different types is performed using the initial moments method.

Keywords

Queueing system Incoming sm-flow Heterogeneous servers Method of initial moments 

References

  1. 1.
    Artalejo, J.R., Gómes-Coral, A.: Retrial Queueing systems: A Computational Approach. Springer, Heidelberg (2008). 318 pCrossRefGoogle Scholar
  2. 2.
    Chechelnitsky, A.A., Kucherenko, O.V.: Stationary characteristics of parallel queueing systems with two-dimensional input flow. Collection of Scientific Articles, Minsk, vol. 2, pp. 262–268 (2009). (in Russian)Google Scholar
  3. 3.
    Down, D.G., Wu, R.: Multi-layered round robin routing for parallel servers. Queueing Syst. 53(4), 177–188 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Efrosinin, D., Farhadov, M., Kudubaeva, S.: Performance analysis and monotone control of a tandem queueing system. In: Vishnevsky, V., Kozyrev, D., Larionov, A. (eds.) DCCN 2013. CCIS, vol. 279, pp. 241–255. Springer, Cham (2014). doi: 10.1007/978-3-319-05209-0_21 CrossRefGoogle Scholar
  5. 5.
    Efrosinin, D., Sztrik, J.: Performance analysis of a two-server heterogeneous retrial queue with threshold policy. Qual. Technol. Quant. Manage. 8(3), 211–236 (2011)CrossRefGoogle Scholar
  6. 6.
    Iravani, S.M.R., Luangkesorn, K.L., Simchi-Levi, D.: A general decomposition algorithm for parallel queues with correlated arrivals. Queueing Syst. 47(4), 313–344 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kargahi, M., Movaghar, A.: Utility accrual dynamic routing in real-time parallel systems. IEEE Trans. Parallel Distrib Syst. (TDPS) 21(12), 1822–1835 (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    Lebedev, E.: On Asymptotic enlargement problem for stochastic networks. In: Mathematicals Methods and Optimization of Telecommunication Networks: Queues Flows, Systems, Networks, Minsk, pp. 108–109 (2011)Google Scholar
  9. 9.
    Lebedev, E., Chechelnitsky, A.: Limit diffusions for multi-channel networks with interdependent inputs. In: Modern Probabilistic Methods for Analysis and Optimization of Information and Telecommunication Networks, Minsk, pp. 133–136 (2011)Google Scholar
  10. 10.
    Lisovskaya, E., Moiseeva, S.: Study of the queuing systems \(M|GI|N|\infty \). In: Dudin, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2015. CCIS, vol. 564, pp. 175–184. Springer, Cham (2015). doi: 10.1007/978-3-319-25861-4_15 CrossRefGoogle Scholar
  11. 11.
    Lisovskaya, E., Moiseeva, S., Pagano, M.: The total capacity of customers in the infinite-server queue with MMPP arrivals. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) DCCN 2016. CCIS, vol. 678, pp. 110–120. Springer, Cham (2016). doi: 10.1007/978-3-319-51917-3_11 CrossRefGoogle Scholar
  12. 12.
    Lozhkhovsky, A.G., Kaptu, V.A., Verbanov, O.V., Kolchar, V.M.: A mathematical model of packet traffic. Bul. Tomsk Polytech. Univ. 9, 113–119 (2011). TPU, Tomsk (in Russian)Google Scholar
  13. 13.
    Moiseev, A.: Asymptotic analysis of the queueing network \(SM-(GI/\infty )^K\). In: Dudin, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2015. CCIS, vol. 564, pp. 73–84. Springer, Cham (2015). doi: 10.1007/978-3-319-25861-4_7 CrossRefGoogle Scholar
  14. 14.
    Moiseeva, S., Zadiranova, L.: Feedback in infinite-server queuing systems. In: Vishnevsky, V., Kozyrev, D. (eds.) DCCN 2015. CCIS, vol. 601, pp. 370–377. Springer, Cham (2016). doi: 10.1007/978-3-319-30843-2_38 CrossRefGoogle Scholar
  15. 15.
    Movaghar, A.: Analysis of a dynamic assignment of impatient customers to parallel queues. Queueing Syst. 67(3), 251–273 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pankratova, E., Moiseeva, S.: Queueing system with renewal arrival process and two types of customers. In: IEEE International Congress on Ultra Modern Telecommunications and Control Systems (ICUMT), pp. 514–517. IEEE, St. Petersburg (2015)Google Scholar
  17. 17.
    Pankratova, E., Moiseeva, S.: Queueing system \(GI|GI|\infty \) with n types of customers. In: Dudin, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2015. CCIS, vol. 564, pp. 216–225. Springer, Cham (2015). doi: 10.1007/978-3-319-25861-4_19 CrossRefGoogle Scholar
  18. 18.
    Rykov, V., Efrosinin, D.: On the slow server problem. Autom. Remote Control 70(12), 2013–2023 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sheu, R.S., Ziedins, I.: Asymptotically optimal control of parallel tandem queues with loss. Queueing Syst. 65(3), 211–227 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Taqqu, M.S., Leland, W.E., Willinger, W., Wilson, D.V.: Self-similarity in high-speed packet traffic: analysis and modeling of ethernet traffic measurements. Stat. Sci. 10, 67–85 (1995)CrossRefzbMATHGoogle Scholar
  21. 21.
    Willinger, W.: The discovery of self-similar traffic. In: Haring, G., Lindemann, C., Reiser, M. (eds.) Performance Evaluation: Origins and Directions. LNCS, vol. 1769, pp. 513–527. Springer, Heidelberg (2000). doi: 10.1007/3-540-46506-5_24 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ekaterina Pankratova
    • 1
    Email author
  • Mais Farkhadov
    • 1
  • Erol Gelenbe
    • 2
  1. 1.V. A. Trapeznikov Institute of Control Sciences of Russian Academy of SciencesMoscowRussia
  2. 2.Imperial CollegeLondonEngland

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