×

A discrete-time \(Geo^{[x]}/g/1\) retrial queue with control of admission. (English) Zbl 1163.90413

Summary: A discrete-time \(Geo/G/1\) retrial queue with batch arrivals in which individual arriving customers have a control of admission. We study the underlying Markov chain at the epochs immediately after the slot boundaries making emphasis on the computation of its steady-state distribution. To this end we employ numerical inversion and maximum entropy techniques. We also establish a stochastic decomposition property and prove that the continuous-time \(M/G/1\) retrial queue with batch arrivals and control of admission can be approximated by our discrete-time system. The outcomes agree with known results for special cases.

MSC:

90B22 Queues and service in operations research

Software:

MCQueue
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Artalejo, J. R., A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7, 187-211 (1999) · Zbl 1009.90001
[2] Artalejo, J. R., Accessible bibliography on retrial queues, Math. Comput. Modell., 30, 1-6 (1999) · Zbl 1009.90001
[3] Falin, G. I., A survey of retrial queues, Queue. Syst., 7, 127-168 (1990) · Zbl 0709.60097
[4] Falin, G. I.; Templeton, J. G.C., Retrial Queues (1997), Chapman & Hall: Chapman & Hall London · Zbl 0944.60005
[5] Yang, T.; Templeton, J. G.C., A survey on retrial queues, Queue. Syst., 2, 201-233 (1987) · Zbl 0658.60124
[6] Artalejo, J. R.; Falin, G. I., Standard and retrial queueing systems: A comparative analysis, Rev Mat. Complut., 15, 101-129 (2002) · Zbl 1009.60079
[7] Yang, T.; Li, H., On the steady-state queue size distribution of the discrete-time \(Geo /G/1\) queue with repeated customers, Queue. Syst., 21, 199-215 (1995) · Zbl 0840.60085
[8] Atencia, I.; Moreno, P., A discrete-time \(Geo /G/1\) retrial queue with general retrial times, Queue. Syst., 48, 5-21 (2004) · Zbl 1059.60092
[9] Atencia, I.; Moreno, P., Discrete-time \(Geo^{[X]}/G_{H\) · Zbl 1061.60092
[10] Choi, B. D.; Kim, J. W., Discrete-time \(Geo_1, Geo_2/G/1\) retrial queueing system with two types of calls, Comput. Math. Appl., 33, 79-88 (1997) · Zbl 0878.90041
[11] Li, H.; Yang, T., \( Geo /G/1\) discrete time retrial queue with Bernoulli schedule, Eur. J. Operat. Res., 111, 629-649 (1998) · Zbl 0948.90043
[12] Li, H.; Yang, T., Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues, Math. Comput. Model., 30, 51-63 (1999) · Zbl 1042.60543
[13] Takahashi, M.; Osawa, H.; Fujisawa, T., \( Geo^{[X]}/G/1\) retrial queue with non-preemptive priority, Asia-Pacific J. Operat. Res., 16, 215-234 (1999) · Zbl 1053.90505
[14] Meisling, T., Discrete time queueing theory, Operat. Res., 6, 96-105 (1958) · Zbl 1414.90112
[15] Bruneel, H.; Kim, B. G., Discrete-time Models for Communication Systems Including ATM (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Boston
[16] Hunter, J. J., Mathematical Techniques of Applied Probability, Discrete-time models: techniques and applications, vol. 2 (1983), Academic Press: Academic Press New York · Zbl 0539.60065
[17] Takagi, H., Queueing analysis: A Foundation of Performance Evaluation, Discrete-time Systems, vol. 3 (1993), North-Holland: North-Holland Amsterdam
[18] Woodward, M. E., Communication and Computer Networks: Modelling with Discrete-time Queues (1994), IEEE Computer Society Press: IEEE Computer Society Press Los Alamitos, California
[19] Choi, B. D.; Shin, Y. W.; Ahn, W. C., Retrial queues with collision arising from unslotted CSMA/CD protocol, Queue. Syst., 11, 335-356 (1992) · Zbl 0762.60088
[20] Kitaev, M. Y.; Rykov, V. V., Controlled Queueing Systems (1995), CRC Press: CRC Press Boca Raton · Zbl 0876.60077
[21] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in Fortran. The Art of Scientific Computing (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0778.65002
[22] Artalejo, J. R.; Falin, G. I.; Lopez-Herrero, M. J., A second order analysis of the waiting time in the \(M/G/1\) retrial queue, Asia-Pacific J. Operat. Res., 19, 131-148 (2002) · Zbl 1165.60343
[23] Falin, G. I.; Martin, M.; Artalejo, J. R., Information theoretic approximations for the \(M/G/1\) retrial queue, Acta Inform., 31, 559-571 (1994) · Zbl 0818.68038
[24] Kouvatsos, D. D., Entropy maximization and queueing networks models, Ann. Operat. Res., 48, 63-126 (1994) · Zbl 0789.90032
[25] Gravey, A.; Hébuterne, G., Simultaneity in discrete-time single server queues with Bernoulli inputs, Perform. Eval., 14, 123-131 (1992) · Zbl 0752.60079
[26] Fuhrmann, S. W.; Cooper, R. B., Stochastic decomposition in the \(M/G/1\) queue with generalized vacations, Operat. Res., 33, 1117-1129 (1985) · Zbl 0585.90033
[27] Artalejo, J. R.; Falin, G. I., Stochastic decomposition for retrial queues, Top, 2, 329-342 (1994) · Zbl 0837.60084
[28] Yang, T.; Posner, M. J.M.; Templeton, J. G.C.; Li, H., An approximation method for the \(M/G/1\) retrial queue with general retrial times, Eur. J. Operat. Res., 76, 552-562 (1994) · Zbl 0802.60089
[29] Artalejo, J. R.; Atencia, I., On the single server retrial queue with batch arrivals, Sankhyā: The Indian J. Stat., 66, 140-158 (2004) · Zbl 1192.90042
[30] Tijms, H. C., A First Course in Stochastic Models (2003), Wiley: Wiley Chichester · Zbl 1088.60002
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.