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A discrete-time Geo/G/1 retrial queue with $$J$$-vacation policy and general retrial times. (English) Zbl 1292.90094
Summary: The authors discuss a discrete-time Geo/G/1 retrial queue with $$J$$-vacation policy and general retrial times. As soon as the orbit is empty, the server takes a vacation. However, the server is allowed to take a maximum number $$J$$ of vacations, if the system remains empty after the end of a vacation. If there is at least one customer in the orbit at the end of a vacation, the server begins to serve the new arrivals or the arriving customers from the orbit. For this model, the authors focus on the steady-state analysis for the considered queueing system. Firstly, the authors obtain the generating functions of the number of customers in the orbit and in the system. Then, the authors obtain the closed-form expressions of some performance measures of the system and also give a stochastic decomposition result for the system size. Besides, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided.

##### MSC:
 90B22 Queues and service in operations research
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##### References:
 [1] Falin, G I, A survey of retrial queues, Queueing Systems, 1, 127-168, (1990) · Zbl 0709.60097 [2] Gomez-Corral, A, A bibliographical guide to the analysis of retrial queues through matrix analytic techniques, Annals of Operations Research, 141, 163-191, (2006) · Zbl 1100.60049 [3] Artalejo, J R, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 30, 1-6, (1999) [4] Artalejo, J R, A classified bibliography of research on retrial queues: progress in 1990-1999, Top, 7, 187-211, (1999) · Zbl 1009.90001 [5] Artalejo, J R, Accessible bibliography on retrial queues: progress in 2000-2009, Mathematical and Computer Modelling, 51, 1071-1081, (2010) · Zbl 1198.90011 [6] Yang, T; Li, H, On the steady-state queue size distribution of the discrete-time geo/G/1 queue with repeated customers, Queueing Systems, 21, 199-215, (1995) · Zbl 0840.60085 [7] Atencia, I; Moreno, P, A discrete-time geo/G/1 retrial queue with general retrial times, Queueing Systems, 48, 5-21, (2004) · Zbl 1059.60092 [8] Wang, J; Zhao, Q, Discrete-time geo/G/1 retrial queue with general retrial times and starting failures, Mathematical and Computer Modelling, 45, 853-863, (2007) · Zbl 1132.60322 [9] Atencia, I; Fortes, I; SĂˇnchez, S, A discrete-time retrial queueing system with starting failures, Bernoulli feedback and general retrial times, Computers and Industrial Engineering, 57, 1291-1299, (2009) [10] Aboul-Hassan, A K; Rabia, S I; Taboly, F A, A discrete time geo/G/1 retrial queue with general retrial times and balking customers, Journal of the Korean Statistical Society, 37, 335-348, (2008) · Zbl 1293.60084 [11] Aboul-Hassan, A K; Rabia, S I; Taboly, F A, Performance evaluation of a discrete-time geox$$/$$G$$/$$1 retrial queue with general retrial times, Computers and Mathematics with Applications, 58, 548-557, (2009) · Zbl 1189.90040 [12] Doshi, B, Single server queues with vacation: A survey, Queueing Systems, 1, 29-66, (1986) · Zbl 0655.60089 [13] Takagi H, Queueing Analysis, A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems, Elsevier, 1991. · Zbl 0744.60114 [14] Tian N and Zhang Z G, Vacation Queueing Models: Theory and Applications. Springer, 2006. · Zbl 1104.60004 [15] Ke, J C; Chu, Y K, A modified vacation model M\^{X}/G/1 system, Applied Stochastic Models in Business and Industry, 22, 1-16, (2006) · Zbl 1116.60053 [16] Ke, J C; Huang, K B; Pearn, W L, The randomized vacation policy for a batch arrival queue, Applied Mathematical Modelling, 34, 1524-1538, (2010) · Zbl 1193.90076 [17] Wang, T Y; Ke, J C; Chang, F M, On the discrete-time geo/G/1 queue with randomized vacations and at most $$J$$ vacations, Applied Mathematical Modelling, 35, 2297-2308, (2011) · Zbl 1217.90076 [18] Li, H; Yang, T, A single-server retrial queue with server vacations and a finite number of input sources, European Journal of Operational Research, 85, 149-160, (1995) · Zbl 0912.90139 [19] Artalejo, J R, Analysis of an M/G/1 queue with constant repeated attempts and server vacations, Computers and Operations Research, 24, 493-504, (1997) · Zbl 0882.90048 [20] Kumar, B K; Arivudainambi, D, The M/G/1 retrial queue with Bernoulli schedules and general retrial times, Computers and Mathematics with Applications, 43, 15-30, (2002) · Zbl 1008.90010 [21] Kumar, B K; Rukmani, R; Thangaraj, V, An M/M/c retrial queueing system with Bernoulli vacations, Journal of Systems Science and Systems Engineering, 18, 222-242, (2009) [22] Aissani, A, An M\^{X}/G/1 energetic retrial queue with vacations and its control, Electronic Notes in Theoretical Computer Science, 253, 33-44, (2009) [23] Chang, F M; Ke, J C, On a batch retrial model with $$J$$ vacations, Journal of Computational and Applied Mathematics, 232, 402-414, (2009) · Zbl 1175.60080 [24] Ke, J C; Chang, F M, Modified vacation policy for M/G/1 retrial queue with balking and feedback, Computers and Industrial Engineering, 57, 433-443, (2009) [25] Wang, J, Discrete-time geo/G/1 retrial queues with general retrial times and Bernoulli vacation, Journal of Systems Science and Complexity, 25, 504-513, (2012) · Zbl 1264.90065 [26] Hunter J J, Mathematical Techniques of Applied Probability, Discrete Time Models: Techniques and Applications, Volume 2, Academic Press, 1983. · Zbl 0539.60065 [27] Fuhrmann, S W; Cooper, R B, Stochastic decomposition in the M/G/1 queue with generalized vacations, Operation Research, 33, 1117-1129, (1985) · Zbl 0585.90033 [28] Artalejo, J R; Falin, G I, Stochastic decomposition for retrial queues, Top, 2, 329-342, (1994) · Zbl 0837.60084
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