On the Total Customers’ Capacity in Multi-server Queues

  • Ekaterina LisovskayaEmail author
  • Svetlana Moiseeva
  • Michele Pagano
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 800)


In this paper we consider a generalization of M/GI/N/\(\infty \) queues, in which customer capacity is an additional parameter of the system and it is independent of the service time. In more detail we focus on the distributions of the total capacity of customers in the different elements of the queue (waiting line, service and entire system) and provide approximate expressions for the corresponding characteristic functions. To verify the goodness of the proposed approximation, several sets of simulations have been carried out, considering discrete and continuous distributions of the customer capacity and using the Kolmogorov distance as a measure of similarity.


N-server queuing system Customer with random capacity Approximation of the probability distribution 


  1. 1.
    Apachidi, X.N., Katsman, Y.: Development of a queuing system with dynamic priorities. Key Eng. Mater. 685, 934–938 (2016)CrossRefGoogle Scholar
  2. 2.
    Efimushkina, T., Gabbouj, M., Samuylov, K.: Analytical model in discrete time for cross-layer video communication over LTE. Autom. Control Comput. Sci. 48(6), 345–357 (2014)CrossRefGoogle Scholar
  3. 3.
    Fedorova, E.: The second order asymptotic analysis under heavy load condition for retrial queueing system MMPP/M/1. In: Dudin, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2015. CCIS, vol. 564, pp. 344–357. Springer, Cham (2015). doi: 10.1007/978-3-319-25861-4_29 CrossRefGoogle Scholar
  4. 4.
    Lisovskaya, E., Moiseeva, S.: Study of the Queuing Systems M/GI/\(N\)/\(\infty \). Commun. Comput. Inf. Sci. 564, 175–184 (2015)Google Scholar
  5. 5.
    Lisovskaya, E., Pagano, M.: Imitacionnoe modelirovanie sistemy massovogo obsluzhivaniya trebovanij sluchajnogo ob”ema. Problemy optimizacii slozhnyh sistem: Trudy 12-j Mezhdunarodnoj Aziatskoj shkoly-seminara, 352–357 (in Russian)(2016)Google Scholar
  6. 6.
    Moiseev, A., Nazarov, A.: Queueing network MAP/(GI/\(\infty \))\(^K\) with high-rate arrivals. Eur. J. Oper. Res. 254(2), 161–168 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Moiseev, A., Sinyakov, M.: Razrabotka ob’ektno-orientirovannoj modeli sistemy imitacionnogo modelirovaniya processov massovogo obsluzhivaniya. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitel’naya tekhnika i informatika 1, 89–93 (In Russian)(2010)Google Scholar
  8. 8.
    Moiseev, A.: Asymptotic Analysis of the Queueing Network \(SM/(GI/\infty )^K\). Commun. Comput. Inf. Sci. 564, 73–84 (2015)Google Scholar
  9. 9.
    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
  10. 10.
    Naumov, V.A., Samuilov, K.E.: On Modeling Queueing Systems with Multiple Resources. Vestn. Ross. Univ. Druzhby Narodov, Ser. Mat. Informatika. Fiz. 3, 60–64 (2014)Google Scholar
  11. 11.
    Naumov, V.A., Samuilov, K.E., Samuilov, A.K.: On the total amount of resources occupied by serviced customers. Autom. Remote Control 77(8), 1419–1427 (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Nazarov, A., Broner, V.: Inventory management system with Erlang distribution of batch sizes. In: Dudin, A., Gortsev, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2016. CCIS, vol. 638, pp. 273–280. Springer, Cham (2016). doi: 10.1007/978-3-319-44615-8_24 CrossRefGoogle Scholar
  13. 13.
    Pankratova, E., Moiseeva, S.: Queueing system GI/GI/\(\infty \) with \(n\) types of customers. Commun. Comput. Inf. Sci. 564, 216–225 (2015)Google Scholar
  14. 14.
    Raspopov, A., Katsman, Y.Y.: Resource allocation algorithm modeling in queuing system based on quantization. Key Eng. Mater. 685, 886–891 (2016)CrossRefGoogle Scholar
  15. 15.
    Tikhonenko, O., Kawecka, M.: Busy period characteristics for single server queue with random capacity demands. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2012. CCIS, vol. 291, pp. 393–400. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31217-5_41 CrossRefGoogle Scholar
  16. 16.
    Tikhonenko, O., Kempa, W.M.: On the queue-size distribution in the multi-server system with bounded capacity and packet dropping. Kybernetika 49(6), 855–867 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Tikhonenko, O., Kempa, W.M.: Performance evaluation of an M/G/\(n\)-type queue with bounded capacity and packet dropping. Int. J. Appl. Math. Comput. Sci. 26(4), 841–854 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ekaterina Lisovskaya
    • 1
    Email author
  • Svetlana Moiseeva
    • 1
  • Michele Pagano
    • 2
  1. 1.Tomsk State UniversityTomskRussia
  2. 2.University of PisaPisaItaly

Personalised recommendations