# zbMATH — the first resource for mathematics

A discrete-time $$\operatorname{Geo}/G/1$$ retrial queue with the server subject to starting failures. (English) Zbl 1101.90015
Summary: This paper studies a discrete-time $$\text{Geo}/G/1$$ retrial queue where the server is subject to starting failures. We analyse the Markov chain underlying the regarded queueing system and present some performance measures of the system in steady-state. Then, we give two stochastic decomposition laws and find a measure of the proximity between the system size distributions of our model and the corresponding model without retrials. We also develop a procedure for calculating the distributions of the orbit and system size as well as the marginal distributions of the orbit size when the server is idle, busy or down. Besides, we prove that the $$M/G/1$$ retrial queue with starting failures can be approximated by its discrete-time counterpart. Finally, some numerical examples show the influence of the parameters on several performance characteristics.

##### MSC:
 90B22 Queues and service in operations research 60K25 Queueing theory (aspects of probability theory)
Full Text:
##### References:
 [1] Aissani, A. and J.R. Artalejo. (1998). ”On the Single Server Retrial Queue Subject to Breakdowns.” Queueing Systems 30, 309–321. · Zbl 0918.90073 · doi:10.1023/A:1019125323347 [2] Artalejo, J.R. (1994). ”New Results in Retrial Queueing Systems with Breakdown of the Servers.” Statistica Neerlandica 48, 23–36. · Zbl 0829.60087 · doi:10.1111/j.1467-9574.1994.tb01429.x [3] Artalejo, J.R. (1999a). ”A Classified Bibliography of Research on Retrial Queues: Progress in 1990–1999.” Top 7, 187–211. · Zbl 1009.90001 · doi:10.1007/BF02564721 [4] Artalejo, J.R. (1999b). ”Accessible Bibliography on Retrial Queues.” Mathematical and Computer Modelling 30, 1–6. · Zbl 1009.90001 · doi:10.1016/S0895-7177(99)00128-4 [5] Artalejo, J.R. and G.I. Falin. (1994). ”Stochastic Decomposition for Retrial Queues.” Top 2, 329–342. · Zbl 0837.60084 · doi:10.1007/BF02574813 [6] Artalejo, J.R. and O. Hernández-Lerma. (2003). ”Performance Analysis and Optimal Control of the Geo/Geo/c Queue.” Performance Evaluation 52, 15–39. · doi:10.1016/S0166-5316(02)00161-X [7] Atencia, I. and P. Moreno. (2004a). ”A Discrete-Time Geo/G/1 Retrial Queue with General Retrial Times.” Queueing Systems 48, 5–21. · Zbl 1059.60092 · doi:10.1023/B:QUES.0000039885.12490.02 [8] Atencia, I. and P. Moreno. (2004b). ”Discrete-Time Geo[X]/GH/1 Retrial Queue with Bernoulli Feedback.” Computers and Mathematics with Applications 47, 1273–1294. · Zbl 1061.60092 · doi:10.1016/S0898-1221(04)90122-8 [9] Atencia, I. and P. Moreno. (2004c). ”The Discrete-Time Geo/Geo/1 Queue with Negative Customers and Disasters.” Computers and Operations Research 31, 1537–1548. · Zbl 1107.90330 · doi:10.1016/S0305-0548(03)00107-2 [10] Bruneel, H. and B.G. Kim. (1993). Discrete-Time Models for Communication Systems Including ATM. Kluwer Academic Publishers, Boston. [11] Chaudhry, M.L., J.G.C. Templeton, and U.C. Gupta. (1996). ”Analysis of the Discrete-Time GI/Geom(n)/1/N Queue.” Computers and Mathematics with Applications 31, 59–68. · Zbl 0844.90031 · doi:10.1016/0898-1221(95)00182-X [12] Choi, B.D. and J.W. Kim. (1997). ”Discrete-Time Geo1,Geo2/G/1 Retrial Queueing System with Two Types of Calls.” Computers and Mathematics with Applications 33, 79–88. · Zbl 0878.90041 · doi:10.1016/S0898-1221(97)00078-3 [13] Conway, J.B. (1973). Functions of One Complex Variable. Springer, New York. · Zbl 0277.30001 [14] Falin, G.I. (1990). ”A Survey of Retrial Queues.” Queueing Systems 7, 127–168. · Zbl 0709.60097 · doi:10.1007/BF01158472 [15] Falin, G.I. and J.G.C. Templeton. (1997). Retrial Queues. Chapman & Hall, London. · Zbl 0944.60005 [16] Fiems, D. and H. Bruneel. (2002). ”Analysis of a Discrete-Time Queueing System with Timed Vacations.” Queueing Systems 42, 243–254. · Zbl 1011.60073 · doi:10.1023/A:1020571814186 [17] Fiems, D., B. Steyaert, and H. Bruneel. (2002). ”Randomly Interrupted GI-G-1 Queues: Service Strategies and Stability Issues.” Annals of Operations Research 112, 171–183. · Zbl 1013.90028 · doi:10.1023/A:1020937324199 [18] Fiems, D., B. Steyaert, and H. Bruneel. (2003). ”Analysis of a Discrete-Time GI-G-1 Queueing Model Subjected to Bursty Interruptions.” Computers and Operations Research 30, 139–153. · Zbl 1029.90021 · doi:10.1016/S0305-0548(01)00086-7 [19] Fuhrmann, S.W. and R.B. Cooper. (1985). ”Stochastic Decompositions in the M/G/1 Queue with Generalized Vacations.” Operations Research 33, 1117–1129. · Zbl 0585.90033 · doi:10.1287/opre.33.5.1117 [20] Gravey, A. and G. Hébuterne. (1992). ”Simultaneity in Discrete-Time Single Server Queues with Bernoulli Inputs.” Performance Evaluation 14, 123–131. · Zbl 0752.60079 · doi:10.1016/0166-5316(92)90014-8 [21] Gupta, U.C. and V. Goswami. (2002). ”Performance Analysis of Finite Buffer Discrete-Time Queue with Bulk Service.” Computers and Operations Research 29, 1331–1341. · Zbl 0994.90047 · doi:10.1016/S0305-0548(01)00034-X [22] Hunter, J.J. (1983). Mathematical Techniques of Applied Probability, Vol. 2, Discrete-Time Models: Techniques and Applications. Academic Press, New York. · Zbl 0539.60065 [23] Krishna Kumar, B., S. Pavai Madheswari, and A. Vijayakumar. (2002). ”The M/G/1 Retrial Queue with Feedback and Starting Failures.” Applied Mathematical Modelling 26, 1057–1075. · Zbl 1018.60088 · doi:10.1016/S0307-904X(02)00061-6 [24] Kulkarni, V.G. and B.D. Choi. (1990). ”Retrial Queues with Server Subject to Breakdowns and Repairs.” Queueing Systems 7, 191–208. · Zbl 0727.60110 · doi:10.1007/BF01158474 [25] Li, H. and T. Yang. (1998). ”Geo/G/1 Discrete Time Retrial Queue with Bernoulli Schedule.” European Journal of Operational Research 111, 629–649. · Zbl 0948.90043 · doi:10.1016/S0377-2217(97)90357-X [26] Li, H. and T. Yang. (1999). ”Steady-State Queue Size Distribution of Discrete-Time PH/Geo/1 Retrial Queues.” Mathematical and Computer Modelling 30, 51–63. · Zbl 1042.60543 · doi:10.1016/S0895-7177(99)00131-4 [27] Meisling, T. (1958). ”Discrete Time Queueing Theory.” Operations Research 6, 96–105. · doi:10.1287/opre.6.1.96 [28] Takagi, H. (1993). Queueing Analysis: A Foundation of Performance Evaluation. Discrete-Time Systems, Vol. 3, North-Holland, Amsterdam. [29] Takahashi, M., H. Osawa, and T. Fujisawa. (1999). ”Geo[X]/G/1 Retrial Queue with Non-Preemptive Priority.” Asia-Pacific Journal of Operational Research 16, 215–234. · Zbl 1053.90505 [30] Tian, N. and Z.G. Zhang. (2002). ”The Discrete-Time GI/Geo/1 Queue with Multiple Vacations.” Queueing Systems 40, 283–294. · Zbl 0993.90029 · doi:10.1023/A:1014711529740 [31] Wang, J., J. Cao, and Q. Li. (2001). ”Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs.” Queueing Systems 38, 363–380. · Zbl 1028.90014 · doi:10.1023/A:1010918926884 [32] Woodward, M.E. (1994). Communication and Computer Networks: Modelling with Discrete-Time Queues. IEEE Computer Society Press, Los Alamitos, California. [33] Yang, T. and H Li. (1994). ”The M/G/1 Retrial Queue with the Server Subject to Starting Failures.” Queueing Systems 16, 83–96. · Zbl 0810.90046 · doi:10.1007/BF01158950 [34] Yang, T. and H. Li. (1995). ”On the Steady-State Queue Size Distribution of the Discrete-Time Geo/G/1 Queue with Repeated Customers.” Queueing Systems 21, 199–215. · Zbl 0840.60085 · doi:10.1007/BF01158581 [35] Yang, T. and J.G.C. Templeton. (1987). ”A Survey on Retrial Queues.” Queueing Systems 2, 201–233. · Zbl 0658.60124 · doi:10.1007/BF01158899 [36] Zhang, Z.G. and N. Tian. (2001). ”Discrete Time Geo/G/1 Queue with Multiple Adaptive Vacations.” Queueing Systems 38, 419–429. · Zbl 1079.90525 · doi:10.1023/A:1010947911863
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.