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Retrial queue \(M/M/1\) with negative calls under heavy load condition. (English) Zbl 1459.60187
Vishnevskiy, Vladimir M. (ed.) et al., Distributed computer and communication networks. 20th international conference, DCCN 2017, Moscow, Russia, September 25–29, 2017. Proceedings. Cham: Springer. Commun. Comput. Inf. Sci. 700, 406-416 (2017).
Summary: In the paper, the retrial queueing system of \(M/M/1\) type with negative calls is considered. The system of Kolmogorov equations for the system states process is derived. The method of asymptotic analysis is proposed for the system solving under the heavy load condition. The theorem about the gamma form of the asymptotic characteristic function of the number of calls in the orbit is formulated and proved. During the study, the expression for the system throughput is obtained. Also the exact characteristic function is derived. Numerical examples of comparison asymptotic and exact distributions are presented. The conclusion about the asymptotic method application area is made.
For the entire collection see [Zbl 1393.68018].

MSC:
60K25 Queueing theory (aspects of probability theory)
Software:
MOSEL
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[1] Artalejo, J.R., Gómez-Corral, A.: Retrial Queueing Systems, A Computational Approach. Springer, Heidelberg (2008) · Zbl 1161.60033
[2] Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman & Hall, London (1997) · Zbl 0944.60005
[3] Artalejo, J.R., Falin, G.I.: Standard and retrial queueing systems: a comparative analysis. Rev. Mat. Complut. 15, 101-129 (2002) · Zbl 1009.60079
[4] Wilkinson, R.I.: Theories for toll traffic engineering in the USA. Bell Syst. Tech. J. 35(2), 421-507 (1956)
[5] Elldin, A., Lind, G.: Elementary Telephone Trafic Theory. Ericsson Public Telecommunications, Stockholm (1971)
[6] Gosztony, G.: Repeated call attempts and their efect on trafic engineering. Budavox Telecommun. Rev. 2, 16-26 (1976)
[7] Kuznetsov, D.Y., Nazarov, A.A.: Analysis of non-markovian models of communication networks with adaptive protocols of multiple random access. Avtomatika i Telemekhanika 5, 124-146 (2001)
[8] Roszik, J., Sztrik, J., Kim, C.S.: Retrial queues in the performance modelling of cellular mobile networks using MOSEL. Int. J. Simul. 6, 38-47 (2005)
[9] Tran-Gia, P., Mandjes, M.: Modeling of customer retrial phenomenon in cellular mobile networks. IEEE J. Sel. Areas Commun. 15, 1406-1414 (1997)
[10] Dudin, A.N., Klimenok, V.I.: Queueing system \(BMAP/G/1\) with repeated calls. Math. Comput. Modell. 30(3-4), 115-128 (1999) · Zbl 1042.60535
[11] Gómez-Corral, A.: A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Ann. Oper. Res. 141, 163-191 (2006) · Zbl 1100.60049
[12] Diamond, J.E., Alfa, A.S.: Matrix analytical methods for \(M/PH/1\) retrial queues. Stoch. Models 11, 447-470 (1995) · Zbl 0829.60091
[13] Artalejo, J.R., Gómez-Corral, A., Neuts, M.F.: Analysis of multiserver queues with constant retrial rate. Eur. J. Oper. Res. 135, 569-581 (2001) · Zbl 0989.90028
[14] Ridder, A.: Fast simulation of retrial queues. In: Third Workshop on Rare Event Simulation and Related Combinatorial Optimization Problems, pp. 1-5, Pisa, Italy (2000)
[15] Artalejo, J.R., Pozo, M.: Numerical calculation of the stationary distribution of the main multiserver retrial queue. Ann. Oper. Res. 116, 41-56 (2002) · Zbl 1013.90038
[16] Aissani, A.: Heavy loading approximation of the unreliable queue with repeated orders. In: Actes du Colloque Methodes et Outils d’Aide ’a la Decision (MOAD 1992), Bejaa, pp. 97-102 (1992)
[17] Anisimov, V.V.: Asymptotic analysis of reliability for switching systems in light and heavy traffic conditions. In: Recent Advances in Reliability Theory: Methodology, Practice, and Inference, pp. 119-133 (2000) · Zbl 0972.60085
[18] Gelenbe, E.: Random neural networks with positive and negative signals and product form solution. Neural Comput. 1(4), 502-510 (1989)
[19] Gelenbe, E.: Queueing networks with negative and positive customers. J. Appl. Probab. 28, 656-663 (1991) · Zbl 0741.60091
[20] Gelenbe, E.: The first decade of G-networks. Eur. J. Oper. Res. 126, 231-232 (2000) · Zbl 0955.00026
[21] Do, T.V.: Bibliography on G-networks, negative customers and applications. Math. Comput. Modell. 53(1-2), 205-212 (2011)
[22] Anisimov, V.V., Artalejo, J.R.: Analysis of Markov multiserver retrial queues with negative arrivals. Queueing Syst. Theo. Appl. 39(2/3), 157-182 (2001) · Zbl 0987.60097
[23] Berdjoudj, L., Aissani, D.: Martingale methods for analyzing the M/M/1 retrial queue with negative arrivals. J. Math. Sci. 131(3), 5595-5599 (2005) · Zbl 1416.60083
[24] Nazarov, A.A., Farkhadov, M.P., Gelenbe, E.: Markov and non-Markov probabilistic models of interacting flows of annihilating particles. In: Dudin, A., Gortsev, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2016. CCIS, vol. 638. Springer, Cham (2016). doi:10.1007/978-3-319-44615-8_25 · Zbl 1398.60095
[25] Moiseeva, E., Nazarov, A.: Asymptotic analysis of RQ-systems M/M/1 on heavy load condition. In: Proceedings of the IV International Conference Problems of Cybernetics and Informatics, Baku, Azerbaijan, pp. 164-166 (2012)
[26] Fedorova, E.
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