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Asymptotic analysis of an retrial queueing system \(M|M|1\) with collisions and impatient calls. (English. Russian original) Zbl 1411.90092
Autom. Remote Control 79, No. 12, 2136-2146 (2018); translation from Avtom. Telemekh. 2018, No. 12, 44-56 (2018).
Summary: We consider a single-line RQ-system with collisions with Poisson arrival process; the servicing time and time delay of calls on the orbit have exponential distribution laws. Each call in orbit has the “impatience” property, that is, it can leave the system after a random time. The problem is to find the stationary distribution of the number of calls on the orbit in the system under consideration. We construct Kolmogorov equations for the distribution of state probabilities in the system in steady-state mode. To find the final probabilities, we propose a numerical algorithm and an asymptotic analysis method under the assumption of a long delay and high patience of calls in orbit. We show that the number of calls in orbit is asymptotically normal. Based on this numerical analysis, we determine the range of applicability of our asymptotic results.

MSC:
90B22 Queues and service in operations research
60G15 Gaussian processes
Software:
MOSEL
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[1] Wilkinson, R. I., Theories for Toll Traffic Engineering in the USA, The Bell Syst. Techn. J., 35, 421-507, (1956)
[2] Cohen, J. W., Basic Problems of Telephone Trafic and the Influence of Repeated Calls, Philips Telecommun. Rev., 18, 49-100, (1957)
[3] Gosztony, G., Repeated Call Attempts and Their Effect on Trafic Engineering, Budavox Telecommun. Rev., 2, 16-26, (1976)
[4] Elldin, A. and Lind, G., Elementary Telephone Trafic Theory, Stockholm: Ericsson Public Telecommunications, 1971.
[5] Artalejo, J.R. and Gomez-Corral, A., Retrial Queueing Systems. A Computational Approach, Stockholm: Springer, 2008. · Zbl 1161.60033
[6] Falin, G.I. and Templeton, J.G.C., Retrial Queues, London: Chapman & Hall, 1997. · Zbl 0944.60005
[7] Artalejo, J. R.; Falin, G. I., Standard and Retrial Queueing Systems: A Comparative Analysis, Revista Mat. Complut., 15, 101-129, (2002) · Zbl 1009.60079
[8] Roszik, J.; Sztrik, J.; Kim, C., Retrial Queues in the Performance Modelling of Cellular Mobile Networks Using MOSEL, Int. J. Simulat., 6, 38-47, (2005)
[9] Kuznetsov, D. Yu.; Nazarov, A. A., Non-Markovian Models of Communication Networks with Adaptive Random Multiple Access Protocols, Autom. Remote Control, 62, 789-808, (2001) · Zbl 1066.90510
[10] Aguir, S.; Karaesmen, F.; Askin, O. Z.; Chauvet, F., The Impact of Retrials on Call Center Performance, OR Spektrum, 26, 353-376, (2004) · Zbl 1109.90019
[11] Sudyko, E. A.; Nazarov, A. A., A Study of a Markov RQ-System with Call Conflicts and Elementary Incoming Stream, Vestn. Tomsk. Gos. Univ., Upravlen., Vychisl. Tekh. Informat., 3, 97-106, (2010)
[12] Nazarov, A.; Sztrik, J.; Kvach, A., Comparative Analysis ofMethods of Residual and Elapsed Service Time in the Study of the Closed Retrial Queuing System M/GI/1//N with Collision of the Customers and Unreliable Server, Inform. Technol. Math. Model. Queueing Theory Appl. (ITMM 2017), Commun. Comp. Inform. Sci., 800, 97-110, (2017) · Zbl 1397.90128
[13] Berczes, T.; Sztrik, J.; Toth, A.; Nazarov, A., Performance Modeling of Finite-Source Retrial Queueing Systems with Collisions and Non-Reliable Server using MOSEL, Inform. Technol. Math. Model. Queueing Theory Appl. (ITMM 2017), Commun. Comp. Inform. Sci., 700, 248-258, (2017)
[14] Yang, T.; Posner, M.; Templeton, J., The M/G/1 Retrial Queue with Non-Persistent Customers, Queueing Syst., 7, 209-218, (1990) · Zbl 0745.60101
[15] Krishnamoorthy, A.; Deepak, T.; Joshua, V., An M/G/1 Retrial Queue with Non-Persistent Customers and Orbital Search, Stochast. Anal. Appl., 23, 975-997, (2005) · Zbl 1075.60577
[16] Kim, J., Retrial Queueing System with Collision and Impatience, Commun. Korean Math. Soc., 4, 647-653, (2010) · Zbl 1210.60102
[17] Fayolle, G., and Brun, M., On a System with Impatience and Repeated Calls, in Queueing Theory and Its Applications: Liber Amicorum for J.W. Cohen, Amsterdam: North Holland, 1988, pp. 283-305.
[18] Martin, M.; Artalejo, J., Analysis of an M/G/1 Queue with Two Types of Impatient units, Adv. Appl. Probab., 27, 647-653, (1995) · Zbl 0829.60085
[19] Aissani, A., Taleb, S., and Hamadouche, D., An Unreliable Retrial Queue with Impatience and Preventive Maintenance, Proc. 15 Appl. Stochast. Models Data Anal. (ASMDA2013), 2013, pp. 1-9.
[20] Kumar, M.; Arumuganathan, R., Performance Analysis of Single Server Retrial Queue with General Retrial Time, Impatient Subscribers, Two Phases of Service, and Bernoulli Schedule, Tamkang J. Sci. Eng., 13, 135-143, (2010)
[21] Fedorova, E.; Voytikov, K., Retrial Queue M/G/1 with Impatient Calls Under Heavy Load Condition, Inform. Technol. Math. Model. Queueing Theory Appl. (ITMM 2017), Commun. Comp. Inform. Sci., 800, 347-357, (2017) · Zbl 1397.90113
[22] Nazarov, A.A. and Fedorova, E.A., Asymptotic Analysis of the RQ-System MM1 with Impatient Calls under Long Patience, Proc. 19th Conf. Distrib. Comp. and Telecomm. Networks: Control, Computation, Communication (DCCN-2016), 2016, pp. 342-348.
[23] Dudin, A. N.; Klimenok, V. I., Queueing System BMAP/G/1 with Repeated Calls, (1999) · Zbl 1042.60535
[24] Stepanov, S. N., Algorithms Approximate Design Syst. Repeated Calls, Autom. Remote Control, 44, 63-71, (1983) · Zbl 0517.90032
[25] Nazarov, A. A.; Lyubina, T. V., The Non-Markov Dynamic RQ System with the Incoming MMP Flow of Requests, Autom. Remote Control, 74, 1132-1143, (2013) · Zbl 1297.93149
[26] 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
[27] Neuts, M. F.; Rao, B. M., Numerical Investigation of a Multiserver Retrial Model, Queueing Syst., 7, 169-189, (1990) · Zbl 0711.60094
[28] Nazarov, A.A. and Moiseeva, S.P., Metod asimptoticheskogo analiza v teorii massovogo obsluzhivaniya (The Asymptotic Analysis Method in Queueing Theory), Tomsk: Tomsk. Gos. Univ., 2006.
[29] Borovkov, A.A., Asymptotic Methods in Queueing Theory, New York: Wiley, 1984. · Zbl 0544.60085
[30] Zadorozhnii, V. N., Asymptotic Analysis of Systems with Intensive Interrupts, Autom. Remote Control, 69, 252-261, (2008) · Zbl 1178.90105
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