Abstract
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.
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References
Wilkinson, R.I., Theories for Toll Traffic Engineering in the USA, The Bell Syst. Techn. J., 1956, vol. 35, no. 2, pp. 421–507.
Cohen, J.W., Basic Problems of Telephone Trafic and the Influence of Repeated Calls, Philips Telecommun. Rev., 1957, vol. 18, no. 2, pp. 49–100.
Gosztony, G., Repeated Call Attempts and Their Effect on Trafic Engineering, Budavox Telecommun. Rev., 1976, vol. 2, pp. 16–26.
Elldin, A. and Lind, G., Elementary Telephone Trafic Theory, Stockholm: Ericsson Public Telecommunications, 1971.
Artalejo, J.R. and Gomez-Corral, A., Retrial Queueing Systems. A Computational Approach, Stockholm: Springer, 2008.
Falin, G.I. and Templeton, J.G.C., Retrial Queues, London: Chapman & Hall, 1997.
Artalejo, J.R. and Falin, G.I., Standard and Retrial Queueing Systems: A Comparative Analysis, Revista Mat. Complut., 2002, vol. 15, pp. 101–129.
Roszik, J., Sztrik, J., and Kim, C., Retrial Queues in the Performance Modelling of Cellular Mobile Networks Using MOSEL, Int. J. Simulat., 2005, no. 6, pp. 38–47.
Kuznetsov, D.Yu. and Nazarov, A.A., Non-Markovian Models of Communication Networks with Adaptive Random Multiple Access Protocols, Autom. Remote Control, 2001, vol. 62, no. 5, pp. 789–808.
Aguir, S., Karaesmen, F., Askin, O.Z., and Chauvet, F., The Impact of Retrials on Call Center Performance, OR Spektrum, 2004, no. 26, pp. 353–376.
Sudyko, E.A. and 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., 2010, no. 3(12), pp. 97–106.
Nazarov, A., Sztrik, J., and 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., 2017, vol. 800, pp. 97–110.
Berczes, T., Sztrik, J., Toth, A., and 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., 2017, vol. 700, pp. 248–258.
Yang, T., Posner, M., and Templeton, J., The M/G/1 Retrial Queue with Non-Persistent Customers, Queueing Syst., 1990, no. 7(2), pp. 209–218.
Krishnamoorthy, A., Deepak, T., and Joshua, V., An M/G/1 Retrial Queue with Non-Persistent Customers and Orbital Search, Stochast. Anal. Appl., 2005, no. 23, pp. 975–997.
Kim, J., Retrial Queueing System with Collision and Impatience, Commun. Korean Math. Soc., 2010, no. 4, pp. 647–653.
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.
Martin, M. and Artalejo, J., Analysis of an M/G/1 Queue with Two Types of Impatient units, Adv. Appl. Probab., 1995, no. 27, pp. 647–653.
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.
Kumar, M. and 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., 2010, no. 13(2), pp. 135–143.
Fedorova, E. and 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., 2017, vol. 800, pp. 347–357.
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.
Dudin, A.N. and Klimenok, V.I., Queueing System BMAP/G/1 with Repeated Calls, Math. Comp. Model., 1999, vol. 30, no. 3–4, pp. 115–128.
Stepanov, S.N., Algorithms Approximate Design Syst. Repeated Calls, Autom. Remote Control, 1983, vol. 44, no. 1, pp. 63–71.
Nazarov, A.A. and Lyubina, T.V., The Non-Markov Dynamic RQ System with the Incoming MMP Flow of Requests, Autom. Remote Control, 2013, vol. 74, no. 7, pp. 1132–1143.
Artalejo, J.R. and Pozo, M., Numerical Calculation of the Stationary Distribution of the Main Multiserver Retrial Queue, Ann. Oper. Res., 2002, no. 116, pp. 41–56.
Neuts, M.F. and Rao, B.M., Numerical Investigation of a Multiserver Retrial Model, Queueing Syst., 1990, vol. 7, no. 2, pp. 169–189.
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.
Borovkov, A.A., Asymptotic Methods in Queueing Theory, New York: Wiley, 1984.
Zadorozhnii, V.N., Asymptotic Analysis of Systems with Intensive Interrupts, Autom. Remote Control, 2008, vol. 69, no. 2, pp. 252–261.
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Original Russian Text © E.Yu. Danilyuk, E.A. Fedorova, S.P. Moiseeva, 2018, published in Avtomatika i Telemekhanika, 2018, No. 12, pp. 44–56.
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Danilyuk, E.Y., Fedorova, E.A. & Moiseeva, S.P. Asymptotic Analysis of an Retrial Queueing System M|M|1 with Collisions and Impatient Calls. Autom Remote Control 79, 2136–2146 (2018). https://doi.org/10.1134/S0005117918120044
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DOI: https://doi.org/10.1134/S0005117918120044