Multi–server loss queueing system with random volume customers, non–identical servers and a limited sectorized memory buffer

• Autorzy: Ziółkowski Marcin

Identyfikatory

DOI

10.24425/bpasts.2023.146764

• Języki publikacji: EN

Abstrakty

EN

In the present paper, the model of multi–server queueing system with random volume customers, non–identical (heterogeneous) servers and a sectorized memory buffer has been investigated. In such system, the arriving customers deliver some portions of information of a different type which means that they are additionally characterized by some random volume vector. This multidimensional information is stored in some specific sectors of a limited memory buffer until customer ends his service. In analyzed model, the arrival flow is assumed to be Poissonian, customers’ service times are independent of their volume vectors and exponentially distributed but the service parameters may be different for every server. Obtained results include general formulae for the steady–state number of customers distribution and loss probability. Special cases analysis and some numerical computations are attached as well.

Słowa kluczowe

EN

multi–server queueing system with heterogeneous servers; queueing systems with random volume customers; sectorized memory buffer; loss probability; Stieltjes convolution

PL

wieloserwerowy system kolejkowy z serwerami heterogenicznymi; prawdopodobieństwo straty; splot Stieltjesa; bufor pamięci sektorowany; system kolejkowy z losowymi klientami

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2023
• Tom: Vol. 71, nr 5
• Strony: art. no. e146764
• Opis fizyczny: Bibliogr. 28 poz., rys., tab.

Twórcy

autor

Ziółkowski Marcin

• Institute of Information Technology, Warsaw University of Life Sciences – SGGW, Poland

• https://orcid.org/0000-0001-8907-8813

Bibliografia

• [1] P.P. Bocharov, C. D’Apice, A.V. Pechinkin, and S. Salerno, Queueing Theory. Utrecht-Boston: VSP, 2004.
• [2] H. Gumbel, “Waiting lines with heterogeneous servers,” Oper. Res., vol. 8, no. 4, pp. 504–511, 1960.
• [3] V.P. Singh, “Two-server Markovian queues with balking: Heterogeneous vs. homogeneous servers,” Oper. Res., vol. 18, no. 1, pp. 145–159, 1970.
• [4] V.P. Singh, “Markovian queues with three heterogeneous servers,” AIIE Trans., vol. 3, no. 1, pp. 45–48, 1971.
• [5] D. Fakinos, “The generalized M/G/k blocking system with heterogeneous servers,” J. Oper. Res. Soc., vol. 33, no. 9, pp. 801–809, 1982.
• [6] M. Schwartz, Computer–communication Network Design and Analysis, Prentice-Hall, Englewood Cliffs, New York, 1977.
• [7] M. Schwartz, “Telecommunication networks: Protocols,” in Modeling and Analysis. New York: Addison-Wesley Publishing Company, 1987.
• [8] O. Tikhonenko, Computer Systems Probability Analysis. Warsaw: Akademicka Oficyna Wydawnicza EXIT, 2006, (in Polish).
• [9] A.M. Alexandrov and B.A. Kaz, “Non–homogeneous demands flows service,” Izvestiya AN SSSR. Tekhnicheskaya Kibernetika, vol. 2, pp. 47–53, 1973, (in Russian).
• [10] B. Sengupta, “The spatial requirements of an M/G/1 queue, or: How to design for buffer space,” in Modelling and Performance Evaluation Methodology. Lect. Notes in Contr. and Inf. Sci, vol. 60, F. Baccelli and G. Fayolle, Eds. Springer, Heidelberg, 1984, pp. 547–562.
• [11] O. Tikhonenko and M. Ziółkowski, “Queueing systems with random volume customers and their performance characteristics,” J. Inf. Organ. Sci., vol. 45, no. 1, pp. 21–38, 2021.
• [12] O.M. Tikhonenko, “Generalized Erlang problem for service systems with finite total capacity,” Probl. Inf. Transm., vol. 41, no. 3, pp. 243–253, 2005.
• [13] O.M. Tikhonenko, “Queuing systems with processor sharing and limited resources,” Autom. Remote Control, vol. 71, no. 5, pp. 803–815, 2010.
• [14] V. Naumov, K. Samuilov, and A. Samuilov, “On the total amount of resources occupied by serviced customers,” Autom. Remote Control, vol. 77, pp. 1419–1427, 2016.
• [15] E. Lisovskaya, S. Moiseeva, M. Pagano, and V. Potatueva, “Study of the MMPP/GI/∞ queueing system with random customers’ capacities,” Inform. Appl., vol. 11, no. 4, pp. 109–117, 2017.
• [16] K. Kerobyan, R. Covington, R. Kerobyan, and K. Enakoutsa, “An infinite–server queueing MMAPk/Gk/∞ model in semi–markov random environment subject to catastrophes,” in Information Technologies and Mathematical Modelling. Queueing Theory and Applications, A. Dudin, A. Nazarov, and A. Moiseev, Eds. Cham: Springer International Publishing, 2018, pp. 195–212.
• [17] E. Lisovskaya, S. Moiseeva, and M. Pagano, “Multi-class GI/GI/∞ queueing systems with random resource requirements,” in Information Technologies and Mathematical Modelling. Queueing Theory and Applications, A. Dudin, A. Nazarov, and A. Moiseev, Eds. Cham: Springer International Publishing, 2018, pp. 129–142.
• [18] K. Samouylov, Y. Gaidamaka, and E. Sopin, “Simplified analysis of queueing systems with random requirements,” in Statistics and Simulation. IWS 2015. Springer Proceedings in Mathematics and Statistics, J. Pilz et al., Ed. Cham: Springer, 2018, pp. 381–390.
• [19] M. Ziółkowski, “M/~G/n/0 Erlang queueing system with heterogeneous servers and non–homogeneous customers,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 66, no. 1, pp. 59–66, 2018.
• [20] O. Tikhonenko, M. Ziółkowski, and M. Kurkowski, “M/~G/n/(0,V ) Erlang queueing system with non-homogeneous customers, non-identical servers and limited memory space,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 67, no. 3, pp. 489–500, 2019.
• [21] H.-K. Kim, “System and method for processing multimedia packets for a network,” 26 June 2007, US Patent 7,236,481, https://patents.google.com/patent/US7236481B2/en.
• [22] X. Chen, A. Stidwell, and M. Harris, “Radiotelecommunications apparatus and method for communications internet data packets containing different types of data,” 2009, US Patent No. 7,558,240, https://patents.google.com/patent/US7558240B2/en.
• [23] M. Ziółkowski and O. Tikhonenko, “Multiserver queueing system with non-homogeneous customers and sectorized memory space,” in Computer Networks, P. Gaj, M. Sawicki, G. Suchacka, and A. Kwiecie´n, Eds. Cham: Springer International Publishing, 2018, pp. 272–285.
• [24] O. Tikhonenko and M. Ziółkowski, “Queueing systems with non-homogeneous customers and infinite sectorized memory space,” in Computer Networks, P. Gaj, M. Sawicki, and A. Kwiecień, Eds. Cham: Springer International Publishing, 2019, pp. 316–329.
• [25] O. Tikhonenko, M. Ziółkowski, and W.M. Kempa, “Queueing systems with random volume customers and a sectorized unlimited memory buffer,” Int. J. Appl. Math. Comput. Sci., vol. 31, no. 3, pp. 471–486, 2021.
• [26] M. Ziółkowski and O. Tikhonenko, “Single–server queueing system with limited queue, random volume customers and unlimited sectorized memory buffer,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 70, p. e143647, 2022.
• [27] J. Sztrik, Basic Queueing Theory. University of Debrecen, Faculty of Informatics, 2012.
• [28] M.L. Abell and J.P. Braselton, The Mathematica Handbook. Elsevier, 1992.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).