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Discrete-time GI/G/1 retrial queues with time-controlled vacation policies

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Abstract

A discrete-time GI/G/1 retrial queue with Bernoulli retrials and time-controlled vacation policies is investigated in this paper. By representing the inter-arrival, service and vacation times using a Markov-based approach, we are able to analyze this model as a level-dependent quasi-birth-and-death (LDQBD) process which makes the model algorithmically tractable. Several performance measures such as the stationary probability distribution and the expected number of customers in the orbit have been discussed with two different policies: deterministic time-controlled system and random time-controlled system. To give a comparison with the known vacation policy in the literature, we present the exhaustive vacation policy as a contrast between these policies under the early arrival system (EAS) and the late arrival system with delayed access (LAS-DA). Significant difference between EAS and LAS-DA is illustrated by some numerical examples.

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References

  1. Aissani, A.A. An M X /G/1 retrial queue with exhaustive vacations. Journal of Statistics Management Systems, 3(3): 269–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alfa, A.S., Li, W. Matrix-geometric analysis of the discrete time GI/G/1 system. Stochastic Models, 17: 541–554 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alfa, A.S. Vacation models in discrete time. Queueing Systems, 44: 5–30 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alfa, A.S. Markov chain representations of discrete distributions applied to queueing methods. Computers & Operations Research, 31: 2365–2385 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alfa, A.S. Discrete-time analysis of the GI/G/1 system with Bernoulli retrials: an algorithmic approach. Annals of Operations Research, 141: 51–66 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Artalejo, J.R. Analysis of an M/G/1 queue with constant repeated attempts and server vacations. Computers & Operations Research, 24(6): 493–504 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Artalejo, J.R., Rodrigo, A. On the Single Server Queue with Linear Retrial Policy and Exaustive Vacations. In: Stochastic Processes and their Applications, Vijayakumar A, Sreenivasan M (eds). Narosa Publishing House, 1999, 196–205

    Google Scholar 

  8. Artalejo, J.R., Atencia, I., Moreno, P. A discrete-time Geo [X] /G/1 retrial queue with control of admission. Applied Mathematical Modelling, 29: 1100–1120 (2005)

    Article  MATH  Google Scholar 

  9. Artalejo, J.R., Gómez-Corral, A. Retrial Queueing Systems: a Computational Approach. Springer-Verlag, Berlin, 2008

    Book  Google Scholar 

  10. Atencia, I., Moreno, P. A discrete-time Geo/G/1 retrial queue with general retrial times. Queueing Systems 48: 5–21 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Atencia, I., Moreno, P. A discrete-time Geo/G/1 retrial queue with server breakdowns. Asia-Pacific Journal of Operational Research 23(2): 247–271 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Atencia, I., Moreno, P. A Discrete-Time Geo/G/1 retrial queue with the server subject to starting failures. Annals of Operations Research, 141: 85–107 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Choi, B.D, Kim, J.W. Discrete-time Geo 1,Geo 2 /G/1 retrial queueing system with two types of calls. Computers and Mathematics with Applications, 33(10): 79–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Diamond, J.E., Alfa, A.S. The MAP/PH/1 retrial queue. Stochastic Models, 14: 1151–1177 (1995)

    MathSciNet  Google Scholar 

  15. Falin, G.I. Calculation of probability characteristics of a multiline system with repeated calls. Moscow University Computational Mathematics and Cybernetics, 3(1): 43–49 (1983)

    Google Scholar 

  16. Falin, G.I., Templeton, J.G.C. Retrial Queues. Chapman & Hall, London, 1997)

    Book  MATH  Google Scholar 

  17. Gómez-Corral A. A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Annals of Operations Research, 141: 163–191 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gravey A, Hébuterne G. Simultaneity in discrete time single server queues with Bernoulli inputs. Performance Evaluation, 14: 123–131 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hunter, J.J. Mathematical Techniques of Applied Probability, Vol.2, Discrete-time Models: Techniques and Applications. Academic Press, New York, 1983

    Google Scholar 

  20. Krishna, Kumar B., Arivudainambi, D. The M/G/1 retrial queue with Bernoulli schedules and general retrial times. Computers and Mathematics with Applications, 43: 15–30 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Krishna, Kumar B., Arivudainambi, D., Vijayakumar, A. On the M X /G/1 retrial queue with Bernoulli schedules and general retrial times. Asia-Pacific Journal of Operational Research, 19: 177–194 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Langaris, C. Gated polling models with customers in orbit. Mathematical and Computer Modelling, 30: 171–187 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Latouche, G., Ramaswami, V. Introduction to Matrix Analytic Methods in Stochastic Modeling. ASASIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999

    Book  Google Scholar 

  24. Li, H., Yang, T. A single-server retrial queue with server vacation and a finite number of input sources. European Journal of Operational Research, 85(1): 149–160 (1995)

    Article  MATH  Google Scholar 

  25. Li, H., Yang, T. Geo/G/1 discrete-time retrial queue with Bernoulli schedule. European Journal of Operational Research, 111(3): 629–649 (1998)

    Article  MATH  Google Scholar 

  26. Li, H., Yang, T. Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues. Mathematical and Computer Modelling, 30: 51–63 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Moreno, P. A discrete-time retrial queue with unreliable server and general server lifetime. Journal of Mathematical Sciences, 23(5): 643–655 (2006)

    Article  Google Scholar 

  28. Moutzoukis, E., Langaris, C. Non-preemptive priorities and vacations in a multiclass retrial queueing system. Stochastic Models, 12(3): 455–472 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models: an Algorithm Approach. The Johns Hopkins University Press, Baltimore, MD, 1982

    Google Scholar 

  30. Neuts, M.F. Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker Inc., New York, 1989

    Google Scholar 

  31. Neuts, M.F., Rao, B.M. Numerical investigation of a multi-server retrial model. Queueing Systems, 7: 169–190 (1990)

    Article  MATH  Google Scholar 

  32. Ramaswami, V., Taylor, P.G. Some properties of the rate operators in level dependent Quasi-Birth-and-Death processes with a countable number of phases. Comm. Statist. Stochastic Models, 12: 143–164 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Takahashi, M., Osawa, H., Fujisawa, T. Geo [X]/G/1 retrial queue with non-preemptive priority. Asia-Pacific Journal of Operational Research, 16: 215–234 (1999)

    MathSciNet  MATH  Google Scholar 

  34. 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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, J., Zhao, Q. A discrete-time Geo/G/1 retrial queue with starting failures and second optional service. Computers and Mathematics with Applications, 53(1): 115–127 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, J., Li, J. A repairable M/G/1 retrial queue with Bernoulli vacation and two-phase service. Quality Technology & Quantitative Management, 5(2): 179–192 (2008)

    MathSciNet  Google Scholar 

  37. Wang, J., Zhang, P. A discrete-time retrial queue with negative customers and unreliable server. Computers & Industrial Engineering, 56(4): 1216–1222 (2009)

    Article  Google Scholar 

  38. Wang, J., Zhang, P. A single-server discrete-time retrial G-queue with server breakdowns and repairs. Acta Mathematica Applicatae Sinica, 25(4): 675–684 (2009)

    Article  MATH  Google Scholar 

  39. 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)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Jin-ting Wang & Nan Wang

  2. Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 5V6

    Attahiru S. Alfa

Authors
  1. Jin-ting Wang
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  2. Nan Wang
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  3. Attahiru S. Alfa
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Correspondence to Jin-ting Wang.

Additional information

Research supported by the National Natural Science Foundation of China (Grant Nos. 10871020 and 11171019), Program for New Century Excellent Talents in University (No. NCET-11-0568) and the Fundamental Research Funds for the Central Universities (Nos. 2011JBZ012 and 2013JBZ019).

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Wang, Jt., Wang, N. & Alfa, A.S. Discrete-time GI/G/1 retrial queues with time-controlled vacation policies. Acta Math. Appl. Sin. Engl. Ser. 29, 689–704 (2013). https://doi.org/10.1007/s10255-013-0244-0

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  • DOI: https://doi.org/10.1007/s10255-013-0244-0

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