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Discrete-time Geo/G/1 retrial queue with general retrial times and starting failures. (English) Zbl 1132.60322
Summary: This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to starting failures. It is assumed that the server, after each service completion, begins a process of search in order to find the following customer to be served. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or down. Then, we give two stochastic decomposition laws and as an application we give bounds for the proximity between the system size distributions of our model and the corresponding model without retrials. Finally, some numerical examples show the influence of the parameters on several performance characteristics.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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##### References:
 [1] Bruneel, H.; Kim, B.G., Discrete-time models for communication systems including ATM, (1993), Kluwer Boston [2] Hunter, J.J., Mathematical techniques of applied probability, vol. 2, discrete-time models: techniques and applications, (1983), Academic Press New York · Zbl 0539.60065 [3] Takagi, H., Queueing analysis: A foundation of performance evaluation, vol. 3, discrete-time systems, (1993), North-Holland Amsterdam [4] Woodward, M.E., Communication and computer networks: modelling with discrete-time queues, (1994), IEEE Computer Soc. Press Los Alamitos, CA [5] Artalejo, J.R., A classified bibliography of research on retrial queues: progress in 1990-1999, Top, 7, 2, 187-211, (1999) · Zbl 1009.90001 [6] Artalejo, J.R., Accessible bibliography on retrial queues, Mathematical and computer modelling, 30, 1-6, (1999) · Zbl 1009.90001 [7] Falin, G.I.; Templeton, J.G.C., Retrial queues, (1997), Chapman & Hall London · Zbl 0944.60005 [8] 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 [9] Atencia, I.; Moreno, P., Discrete-time $$G e o^{[X]} / G_H / 1$$ retrial queue with Bernoulli feedback, Computers and mathematics with applications, 47, 8-9, 1273-1294, (2004) · Zbl 1061.60092 [10] Choi, B.D.; Kim, J.W., Discrete-time $$G e o_1, G e o_2 / G / 1$$ retrial queueing system with two types of calls, Computers and mathematics with applications, 33, 10, 79-88, (1997) · Zbl 0878.90041 [11] Li, H.; Yang, T., $$G e o / G / 1$$ discrete-time retrial queue with Bernoulli schedule, European journal of operational research, 111, 3, 629-649, (1998) · Zbl 0948.90043 [12] 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 [13] 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 [14] 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 [15] Kulkarni, V.G.; Choi, B.D., Retrial queue with server subject to breakdown and repairs, Queueing systems, 7, 2, 191-208, (1990) · Zbl 0727.60110 [16] Wang, J.; Cao, J.; Li, Q., Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing systems, 38, 363-380, (2001) · Zbl 1028.90014 [17] Artalejo, J.R.; Falin, G.I., Stochastic decomposition for retrial queues, Top, 2, 329-342, (1994) · Zbl 0837.60084
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