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A discrete time $$\mathrm{Geo}/\mathrm{G}/1$$ retrial queue with general retrial times and balking customers. (English) Zbl 1293.60084
Summary: We consider a discrete time $$\mathrm{Geo}/\mathrm{G}/1$$ retrial queue with general retrial times and balking customers. If a new arriving customer finds the server busy, he may join the orbit to retry getting the required service again or depart completely from the system. Using the supplementary variable technique, this queueing system is modelled using a Markov chain. We derive the generating functions of the steady state distribution of this Markov chain. Hence, we establish the generating functions of the orbit size and the system size distributions. This set of generating functions is used to derive various performance measures. We prove a stochastic decomposition law and use it to a derive a measure of the proximity between the distributions of the system size in the present model and the corresponding one without retrials. A set of recursive formulae is built up to facilitate computing the orbit size and the system size distributions. Numerical results are presented with a focus on the effect of balking on the system performance.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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##### References:
 [1] Aboul-Hassan, A.; Rabia, S.I.; Kadry, A., Analytical study of a discrete-time retrial queue with balking customers and early arrival scheme, Alexandria engineering journal, 44, 911-917, (2005) [2] Aboul-Hassan, A.; Rabia, S.I.; Kadry, A., A recursive approach for analyzing a discrete-time retrial queue with balking customers and early arrival scheme, Alexandria engineering journal, 44, 919-925, (2005) [3] Aboul-Hassan, A., Rabia, S.I., & Mansy, A. (2006) Generating function analysis of a discrete-time stochastic dynamical system, in: International conference on mathematical analysis and its applications, Assuit, Journal of Computational and Applied Mathematics (in press) [4] Artalejo, J.R.; Falin, G.I., Standard and retrial queueing systems: A comparative analysis, Revista matemática complutense, 15, 101-129, (2002) · Zbl 1009.60079 [5] Artalejo, J.; Atencia, I.; Moreno, P., A discrete-time $$G e o^{[X]} / G / 1$$ retrial queue with control of admission, Applied mathematical modelling, 29, 1100-1120, (2005) · Zbl 1163.90413 [6] Artalejo, J.R.; Falin, G.I., Stochastic decomposition for retrial queues, Top, 2, 329-342, (1994) · Zbl 0837.60084 [7] Atencia, I.; Moreno, P., Discrete-time $$G e o^{[X]} / G_H / 1$$ retrial queue with Bernoulli feedback, Computers and mathematics with applications, 47, 1273-1294, (2004) · Zbl 1061.60092 [8] Atencia, I.; Moreno, P., A discrete-time $$G e o / G / 1$$ retrial queue with general retrial times, Queueing systems, 48, 5-21, (2004) · Zbl 1059.60092 [9] Atencia, I.; Moreno, P., A discrete-time $$G e o / G / 1$$ retrial queue with server breakdowns, Asia-Pacific journal of operational research, 23, 247-271, (2006) · Zbl 1113.90038 [10] Atencia, I.; Moreno, P., A discrete-time $$G e o / G / 1$$ retrial queue with the server subject to starting failures, Annals of operations research, 141, 85-107, (2006) · Zbl 1101.90015 [11] Bianchi, G., Performance analysis of the IEEE 802.11 distributed coordination function, IEEE journal on selected areas in communications, 18, 535-547, (2000) [12] Bruneel, H.; Kim, B.G., Discrete-time models for communication systems including ATM, (1993), Kluwer Academic Publishers Boston [13] Choi, B.D.; Kim, J.W., Discrete-time $$G e o_1, G e o_2 / G / 1$$ retrial queueing systems with two types of calls, Computers and mathematics with applications, 33, 79-88, (1997) · Zbl 0878.90041 [14] Fuhrmann, S.W.; Cooper, R.B., Stochastic decomposition in the $$M / G / 1$$ queue with generalized vocations, Operational research, 32, 1119-1129, (1985) [15] Falin, G.I.; Templeton, J.G.C., Retrial queues, (1997), Chapman & Hall London · Zbl 0944.60005 [16] Gravey, A.; Hébuterne, G., Simultaneity in discrete-time single server queues with Bernoulli inputs, Performance evaluation, 14, 123-131, (1992) · Zbl 0752.60079 [17] Hunter, J.J, Mathematical techniques of applied probability, () · Zbl 0539.60065 [18] Kulkarni, V.G.; Liang, H.M., Retrial queues revisited, (), 19-34 · Zbl 0871.60074 [19] Li, H.; Yang, T., $$G e o / G / 1$$ discrete time retrial queue with Bernoulli schedule, European journal of operational research, 111, 629-649, (1998) · Zbl 0948.90043 [20] Li, H.; Yang, T., Steady-state queue size distribution of discrete-time $$P H / G e o / 1$$ retrial queues, Mathematical and computer modelling, 30, 51-63, (1999) · Zbl 1042.60543 [21] Mansy, A.K. Stochastic dynamical systems, Master’s thesis, Alexandria university, Alexandria 2006 [22] Moreno, P., A discrete-time retrial queue with unreliable server and general server lifetime, Journal of mathematical sciences, 132, 643-655, (2006) · Zbl 1411.60138 [23] Takahashi, M.; Osawa, H.; Fujisawa, T., $$G e o^{[X]} / G / 1$$ retrial queue with non-preemptive priority, Asia-Pacific journal of operational research, 16, 215-234, (1999) · Zbl 1053.90505 [24] Wang, J.; Zhao, Q., Discrete-time $$G e o / G / 1$$ retrial queue with general retrial times and starting failures, Mathematical and computer modelling, 45, 853-863, (2007) · Zbl 1132.60322 [25] Wang, J.; Zhao, Q., A discrete-time $$G e o / G / 1$$ retrial queue with starting failures and second optional service, Computers and mathematics with applications, 53, 115-127, (2007) · Zbl 1172.90368 [26] Whitt, W., Improving service by informing customers about anticipated delays, Management science, 45, 192-207, (1999) · Zbl 1231.90285 [27] Yang, T.; Li, H., On the steady-state queue size distribution of the discrete-time $$G e o / G / 1$$ queue with repeated customers, Queueing systems, 21, 199-215, (1995) · Zbl 0840.60085
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