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Asymptotic analysis of the \(MMPP|M|1\) retrial queue with negative calls under the heavy load condition. (Russian. English summary) Zbl 1455.90040
Summary: In the paper, a single-server retrial queueing system with \(MMPP\) arrivals and an exponential law of the service time is studied. Unserviced calls go to an orbit and stay there during random time distributed exponentially, they access to the server according to a random multiple access protocol. In the system, a Poisson process of negative calls arrives, which delete servicing positive calls. The method of the asymptotic analysis under the heavy load condition for the system studying is proposed. It is proved that the asymptotic characteristic function of a number of calls on the orbit has the gamma distribution with the obtained parameters. The value of the system capacity is obtained, so, the condition of the system stationary mode is found. The results of a numerical comparison of the asymptotic distribution and the distribution obtained by simulation are presented. Conclusions about the method applicability area are made.
90B22 Queues and service in operations research
60K25 Queueing theory (aspects of probability theory)
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
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[1] Kuznetsov D. Y., Nazarov A. A., “Analysis of a communication network governed by an adaptive random multiple access protocol in critical load”, Problems of Information Transmission, 40:3 (2004), 243-253 · Zbl 1088.94502
[2] Tran-Gia P., Mandjes M., “Modeling of customer retrial phenomenon in cellular mobile networks”, IEEE Journal on Selected Areas in Communications, 15 (1997), 1406-1414
[3] Roszik J., Sztrik J., Kim C. S., “Retrial queues in the performance modelling of cellular mobile networks using MOSEL”, I. J. of Simulation, 6:1-2 (2005), 38-47
[4] Kim C. S., Klimenok V., Dudin A., “Analysis and optimization of guard channel policy in cellular mobile networks with account of retrials”, Computers and Operation Research, 43 (2014), 181-190 · Zbl 1348.90175
[5] Artalejo J. R., Gómez-Corral A., Retrial Queueing Systems. A Computational Approach, Springer, Berlin, 2008, 267 pp. · Zbl 1161.60033
[6] Falin G. I., Templeton J. G. C., Retrial queues, Chapman & Hall, L., 1997, 328 pp. · Zbl 0944.60005
[7] Gelenbe E., “Random neural networks with positive and negative signals and product form solution”, Neural Computation, 1:4 (1989), 502-511
[8] Gelenbe E., “Product-form queueing networks with negative and positive customers”, Journal of Applied Probability, 28 (1991), 656-663 · Zbl 0741.60091
[9] Do T. V., “Bibliography on G-networks, negative customers and applications”, Mathematical and Computer Modelling, 53:1-2 (2011), 205-212
[10] Shin Y. W., “Multi-server retrial queue with negative customers and disasters”, Queueing Systems, 55:4 (2007), 223-237 · Zbl 1115.60094
[11] Anisimov V. V., Artalejo J. R., “Analysis of Markov multiserver retrial queues with negative arrivals”, Queueing Systems, 39:2/3 (2001), 157-182 · Zbl 0987.60097
[12] Berdjoudj L., Aissani D., “Martingale methods for analyzing the M/M/1 retrial queue with negative arrivals”, Journal of Mathematical Sciences, 131:3 (2005), 5595-5599 · Zbl 1416.60083
[13] Wu J., Lian Z., “A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule”, Computers & Industrial Engineering, 64:1 (2013), 84-93
[14] Kirupa K., Udaya Chandrika K., “Batch arrival retrial queue with negative customers, multi-optional service and feedback”, Communications on Applied Electronics, 2:4 (2015), 14-18
[15] Klimenok V. I., Dudin A. N., “A BMAP/PH/N queue with negative customers and partial protection of service”, Communications in Statistics — Simulation and Computation, 41:7 (2012), 1062-1082 · Zbl 1267.60105
[16] Dimitriou I., “A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations”, Applied Mathematical Modelling, 37:3 (2013), 1295-1309 · Zbl 1351.90075
[17] Rajadurai P., “A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy”, International Journal of Mathematical Modelling and Numerical Optimisation (IJMMNO), 8:4 (2018), 393-417
[18] Zidani N., Djellab N., “On the multiserver retrial queues with negative arrivals”, International Journal of Mathematics in Operational Research (IJMOR), 13:2 (2018), 219-242 · Zbl 1452.90124
[19] Bertsekas D., Gallager R., Information transmission networks, Mir, M., 1989, 544 pp. (in Russian)
[20] Bljek Ju., Networks: protocols, standards, interfaces, Mir, M., 1990, 510 pp. (in Russian)
[21] Gómez-Corral A., “A bibliographical guide to the analysis of retrial queues through matrix analytic techniques”, Annals of Operations Research, 141:1 (2006), 163-191 · Zbl 1100.60049
[22] Kim C. S., Mushko V. V., Dudin A. N., “Computation of the steady state distribution for multi-server retrial queues with phase type service process”, Annals of Operations Research, 201:1 (2012), 307-323 · Zbl 1260.90067
[23] Artalejo J. R., Pozo M., “Numerical calculation of the stationary distribution of the main multiserver retrial queue”, Annals of Operations Research, 116:1-4 (2002), 41-56 · Zbl 1013.90038
[24] Ridder A., “Fast simulation of retrial queues”, Third Workshop on Rare Event Simulation and Related Combinatorial Optimization Problems (Piza, 2000), 1-5 · Zbl 1418.91390
[25] Moiseev A., Nazarov A., “Queueing network MAP \(-(GI/\infty)^K\) with high-rate arrivals”, European Journal of Operational Research, 254:1 (2016), 161-168 · Zbl 1346.90259
[26] Nazarov A. A., Fedorova E. A., “Retrial queuing system MMPP|GI|1 researching by means of the second-order asymptotic analysis method under a heavy load condition”, Bulletin of the Tomsk Polytechnic University. Geo Assets Engineering, 325:5 (2014), 6-15 (in Russian)
[27] Lisovskaja E. Ju., Moiseeva S. P., “Asymptotical analysis of a non-Markovian queueing system with renewal input process and random capacity of customers”, Tomsk State University Journal of Control and Computer Science, 2017, no. 39, 30-38 (in Russian)
[28] Moiseev A., Demin A., Dorofeev V., Sorokin V., “Discrete-event approach to simulation of queueing networks”, Key Engineering Materials, 685 (2016), 939-942
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