Queueing systems with random volume customers and a sectorized unlimited memory buffer

• Autorzy: Tikhonenko Oleg , Ziółkowski Marcin , Kempa Wojciech M.

Identyfikatory

DOI

10.34768/amcs-2021-0032

• Języki publikacji: EN

Abstrakty

EN

In the present paper, we concentrate on basic concepts connected with the theory of queueing systems with random volume customers and a sectorized unlimited memory buffer. In such systems, the arriving customers are additionally characterized by a non-negative random volume vector. The vector’s indications can be understood as the sizes of portions of information of a different type that are located in the sectors of memory space of the system during customers’ sojourn in it. This information does not change while a customer is present in the system. After service termination, information immediately leaves the buffer, releasing its resources. In analyzed models, the service time of a customer is assumed to be dependent on his volume vector characteristics, which has influence on the total volume vector distribution. We investigate three types of such queueing systems: the Erlang queueing system, the single-server queueing system with unlimited queue and the egalitarian processor sharing system. For these models, we obtain a joint distribution function of the total volume vector in terms of Laplace (or Laplace-Stieltjes) transforms and formulae for steady-state initial mixed moments of the analyzed random vector, in the case when the memory buffer is composed of two sectors. We also calculate these characteristics for some practical case in which the service time of a customer is proportional to the customer’s length (understood as the sum of the volume vector’s indications). Moreover, we present some numerical computations illustrating theoretical results.

Słowa kluczowe

EN

queueing system; random volume customers; sectorized memory buffer; total volume vector; Laplace transform; Laplace–Stieltjes transform; multivariate L’Hospital rule

PL

system kolejkowania; wektor objętości; transformata Laplace'a; transformata Laplace'a-Stieltjesa

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2021
• Tom: Vol. 31, no. 3
• Strony: 471--486
• Opis fizyczny: Bibliogr. 40 poz., rys., tab., wykr.

Twórcy

autor

Tikhonenko Oleg

• Institute of Computer Science, Cardinal Stefan Wyszyński University in Warsaw, ul. Wóycickiego 1/3, 01-938 Warsaw, Poland

autor

Ziółkowski Marcin

• Institute of Information Technology, Warsaw University of Life Sciences (SGGW), ul. Nowoursynowska 159, 02-787 Warsaw, Poland

autor

Kempa Wojciech M.

• Department of Mathematics Applications and Methods for Artificial Intelligence, Silesian University of Technology, ul. Kaszubska 23, 44-100 Gliwice, Poland

Bibliografia

• [1] Alexandrov, A. and Kaz, B. (1973). Non-homogeneous demands flows service, Izvestiya AN SSSR: Tekhnicheskaya Kibernetika (2): 47–53, (in Russian).
• [2] Asmussen, S. (2003). Applied Probability and Queues, 2nd Edn, Springer, New York.
• [3] Bocharov, P., D’Apice, C., Pechinkin, A. and Salerno, S. (2004). Queueing Theory, VSP, Utrecht/Boston.
• [4] Boxma, O. (1989). Workloads and waiting times in single-server systems with multiple customer classes, Queueing Systems 5: 185–214, DOI: 10.1007/BF01149192.
• [5] Cascone, A., Manzo, R., Pechinkin, A. and Shorgin, S. (2010). A Geom/G/1/n queueing system with LIFO discipline, service interruptions and resumption, and restrictions on the total volume of demands, Proceedings of the World Congress on Engineering 2010, London, UK, Vol. III.
• [6] Chen, X., Stidwell, A. and Harris, M. (2009). Radiotelecommunications apparatus and method for communications internet data packets containing different types of data, Technical report, US Patent No. 7,558,240 B2, patents.google.com/patent/US7558240B2/en.
• [7] Dudin, S., Kim, C., Dudina, O. and Baek, J. (2013). Queueing system with heterogeneous customers as a model of a call center with a call–back for lost customers, Mathematical Problems in Engineering 2013, Article ID: 983723, DOI: 10.1155/2013/983723.
• [8] Fiems, D. and De Vuyst, S. (2018). From exhaustive vacation queues to preemptive priority queues with general interarrival times, International Journal of Applied Mathematics and Computer Science 28(4): 695–704, DOI: 10.2478/amcs-2018-0053.
• [9] Guo, P. and Zipkin, P. (2007). Analysis and comparison of queues with different levels of delay information, Management Science 53(6): 962–970, DOI: 10.1287/mnsc.1060.0686.
• [10] Ivlev, V. (2002). Indeterminate forms of many variables functions. Part I, Education of Mathematics 23(4): 90–100, (in Russian).
• [11] Ivlev, V. (2003). Indeterminate forms of many variables functions. Part II, Education of Mathematics 26(3): 77–85, (in Russian).
• [12] Juneja, S., Raheja, T. and Shimkin, N. (2012). The concert queueing game with a random volume of arrivals, in B. Gaujal et al. (Eds), Proceedings of the 6th International ICST Conference on Performance Evaluation Methodologies and Tools, IEEE, New York, pp. 317–325, DOI: 10.4108/valuetools.2012.250166.
• [13] Kerobyan, K., Covington, R., Kerobyan, R. and Enakoutsa, K. (2018). An infinite-server queueing MMAPk/Gk/∞ model in semi-Markov random environment subject to catastrophes, in A. Dudin, et al. (Eds), Communications in Computer and Information Science, Springer, Cham, pp. 195–212, DOI: 10.1007/978-3-319-97595-5_16.
• [14] Kim, H.-K. (2002). System and method for processing multimedia packets for a network, Technical report, US Patent No. 7,236,481 B2, patents.google.com/patent/US7236481B2/en.
• [15] Kim, J. and Ward, A. (2013). Dynamic scheduling of a GI/GI/1 + GI queue with multiple customer classes, Queueing Systems 75: 339–384, DOI: 10.1007/s11134-012-9325-7.
• [16] Lisovskaya, E., Moiseeva, S., Pagano, M. and Potatueva, V. (2017). Study of the MMPP/GI/∞ queueing system with random customers’ capacities, Informatics and Applications 11(4): 109–117, DOI: 10.14357/19922264170414.
• [17] Lisovskaya, E., Moiseeva, S. and Pagano, M. (2018). Multiclass GI/GI/∞ queueing systems with random resource requirements, in A. Dudin et al. (Eds), Communications in Computer and Information Science, Vol. 912, Springer, Cham, pp. 129–142, DOI: 10.1007/978-3-319-97595-5_11.
• [18] Matalytsky, M. and Zając, P. (2019). Application of HM-network with positive and negative claims for finding of memory volumes in information systems, Journal of Applied Mathematics and Computational Mechanics 18(1): 41–51, DOI: 10.17512/jamcm.2019.1.04.
• [19] Matveev, V.F. and Ushakov, V.G. (1984). Queueing Systems, Moscow University Edition, Moscow, (in Russian).
• [20] Naumov, V., Samouylov, K., Yarkina, N., Sopin, E., Andreev, S. and Samuylov, A. (2015). LTE performance analysis using queuing systems with finite resources and random requirements, 2015 7th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), Brno, Czech Republic, pp. 100–103, DOI: 10.1109/ICUMT.2015.7382412.
• [21] Naumov, V., Samuilov, K. and Samuilov, A. (2016). On the total amount of resources occupied by serviced customers, Automation and Remote Control 77: 1419–1427, DOI: 10.1134/S0005117916080087.
• [22] Naumov, V. and Samuilov, K. (2018). Analysis of networks of the resource queuing systems, Automation and Remote Control 79: 822–829, DOI: 10.1134/S0005117918050041.
• [23] Nowak, B., Piechowiak, M., Stasiak, M. and Zwierzykowski, P. (2020). An analytical model of a system with priorities servicing a mixture of different elastic traffic streams, Bulletin of the Polish Academy of Sciences: Technical Sciences 68(2): 263–270, DOI: 10.24425/bpasts.2020.133111.
• [24] Rumyantsev, A. and Morozov, E. (2017). Stability criterion of a multiserver model with simultaneous service, Annals of Operation Research 252: 29–39, DOI: 10.1007/s10479-015-1917-2.
• [25] Samouylov, K., Sopin, E. and Vikhrova, O. (2015). Analyzing blocking probability in LTE wireless network via queuing system with finite amount of resources, in A. Dudin et al. (Eds), Communications in Computer and Information Science, Springer, Cham, pp. 393–403, DOI: 10.1007/978-3-319-25861-4_33.
• [26] Samouylov, K., Sopin, E., Vikhrova, O. and Shorgin, S. (2017). Convolution algorithm for normalization constant evaluation in queuing system with random requirements, AIP Conference Proceedings 1863(090004): 090004-1–090004-3, DOI: 10.1063/1.4992269.
• [27] Samouylov, K., Gaidamaka, Y. and Sopin, E. (2018). Simplified analysis of queueing systems with random requirements, in J. Pilz et al. (Eds), Statistics and Simulation. IWS 2015. Springer Proceedings in Mathematics and Statistics, Springer, Cham, pp. 381–390, DOI: 10.1007/978-3-319-76035-3_27.
• [28] Schwartz, M. (1977). Computer-Communication Network Design and Analysis, Prentice-Hall, Englewood Cliffs.
• [29] Schwartz, M. (1987). Telecommunication Networks: Protocols, Modeling and Analysis, Addison-Wesley Publishing Company, New York.
• [30] Sengupta, B. (1984). The spatial requirements of an M/G/1 queue, or: How to design for buffer space, in F. Baccelli and G. Fayolle (Eds), Modelling and Performance Evaluation Methodology, Springer, Heidelberg, pp. 547–562.
• [31] Shun-Chen, N. (1980). A single server queueing loss model with heterogeneous arrival and service, Operations Research 28(3): 584–593, DOI: 10.1287/opre.28.3.584.
• [32] Tikhonenko, O. (2005). Generalized Erlang problem for service systems with finite total capacity, Problems of Information Transmission 41(3): 243–253, DOI: 10.1007/s11122-005-0029-z.
• [33] Tikhonenko, O. (2015). Determination of loss characteristics in queueing systems with demands of random space requirement, in A. Kwiecień et al. (Eds), Communications in Computer and Information Science, Springer, Cham, pp. 209–215, DOI: 10.1007/978-3-319-25861-4_18.
• [34] Tikhonenko, O. and Kempa, W.M. (2016). Performance evaluation of an M/G/N-type queue with bounded capacity and packet dropping, International Journal of Applied Mathematics and Computer Science 26(4): 841–854, DOI: 10.1515/amcs-2016-0060.
• [35] Tikhonenko, O. and Ziółkowski, M. (2018). Single-server queueing system with external and internal customers, Bulletin of the Polish Academy of Sciences: Technical Sciences 66(4): 539–551, DOI: 10.24425/124270.
• [36] Tikhonenko, O. and Ziółkowski, M. (2019). Queueing systems with non–homogeneous customers and infinite sectorized memory space, in A. Kwiecień et al. (Eds), Communications in Computer and Information Science, Springer, Cham, pp. 316–329, DOI: 10.1007/978-3-030-21952-9_24.
• [37] Tikhonenko, O., Ziółkowski, M. and Kurkowski, M. (2019). M/_G/n/(0, V ) Erlang queueing system with non-homogeneous customers, non-identical servers and limited memory space, Bulletin of the Polish Academy of Sciences: Technical Sciences 67(3): 489–500, DOI: 10.24425/bpasts.2019.129648.
• [38] Yashkov, S. and Yashkova, A. (2007). Processor sharing: A survey of the mathematical theory, Automation and Remote Control 68(9): 1662–1731, DOI: 10.1134/S0005117907090202.
• [39] Zhernovyi, Y. and Kopytko, B. (2016). The potentials method for the M/G/1/m queue with customer dropping and hysteretic strategy of the service time change, Journal of Applied Mathematics and Computational Mechanics 15(1): 197–210, DOI: 10.17512/jamcm.2016.1.20.
• [40] Ziółkowski, M. and Tikhonenko, O. (2018). Multiserver queueing system with non-homogeneous customers and sectorized memory space, in A. Kwiecień et al. (Eds), Communications in Computer and Information Science, Vol. 860, Springer, Cham, pp. 272–285, DOI: 10.1007/978-3-319-92459-5_22.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).