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A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

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Abstract

We consider an M X/G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system, including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Apaolaza, M.N., Artalejo, J.R. On the time to reach the certain orbit level in multi-server retrial queues. Applied Mathematics and Computation, 168: 686–703 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Artalejo, J.R. Accessible bibliography on retrial queue. Mathematical and Computer Modelling, 30: 1–6 (1999)

    Article  Google Scholar 

  4. Artalejo, J.R. A classical bibliography of research on retrial queues: Progress in 1990–1999. TOP, 7: 187–211 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Artalejo, J.R., Atencia, I. On the single server retrial queue with batch arrivals. Sankhyâ, 66: 140–158 (2004)

    MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Artalejo, J.R., Falin, G.I. Standard and retrial queueing systems: A comparative analysis. Revista Mathematica Computense, 15: 101–129 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Artalejo, J.R., Gomez-Corral, A. Steady state solution of a single server queue with linear repeated request. Journal of Applied Probability, 34: 223–233 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Choi, B.D., Shin, Y.W., Ahn, W.C. Retrial queues with collision arising from unslotted CDMA/CD protocol. Queueing Systems, 11: 335–356 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Choudhury, G., Madan, K.C. A two phase batch arrival queueing system with a vacation time under Bernoulli schedule. Applied Mathematics and Computation, 149: 337–349 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Choudhury, G., Madan, K.C. A two stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy. Mathematical and Computer Modelling, 42: 71–85 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Choudhury, G., Paul, M. A two phases queueing system with Bernoulli vacation schedule under multiple vacation policy. Statistical Methodology, 3: 174–185 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Choudhury, G., Tadj, L., Paul, M. Steady state analysis of an M X /G/1 queue with two phases of service and Bernoulli vacation schedule under multiple vacation policy. Applied Mathematical Modelling, 31: 1079–1091 (2007)

    Article  MATH  Google Scholar 

  14. Choudhury, G. Steady state analysis of an M/G/1 queue with linear retrial policy and two phase service under Bernoulli vacation schedule. Applied Mathematical Modelling, 32: 2480–2489 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cinlar, E. Introduction to stochastic processes. Prentice-Hall, Englewood Cliffs, New Jersy, 1975

    MATH  Google Scholar 

  16. Cooper, R.B. Introduction to queueing theory. Elsevier, Amsterdam, 1981

    Google Scholar 

  17. Doshi, B.T. Analysis of a two phase queueing system with general service times. Operation Research Letters, 10: 265–272 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dimitriou, I., Langaris, C. Analysis of a retrial queue with two phase service and server vacations. Queueing Systems 60: 111–129 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Falin, G.I. Aggregate arrival of customers in one line systems with repeated calls. Ukrain Mathematical Journal, 28: 337–340 (1976)

    MathSciNet  Google Scholar 

  20. Falin, G.I. A survey of retrial queues. Queueing Systems, 7: 127–168 (1986)

    Article  MathSciNet  Google Scholar 

  21. Falin, G.I., Templeton, J.G.C. Retrial queues. Chapman and Hall, London, 1997

    Book  MATH  Google Scholar 

  22. Farahmand, K. Single line queue with repeated demands. Queueing Systems, 6: 223–228 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fayolle, G. A simple telephone exchange with delayed feedbacks. Teletraffic Analysis and Computer Performance Evaluation. Ed. O.J. Boxma., J.W. Cohen. and H.C. Times. Elsevier, Amsterdam., 1986

    Google Scholar 

  24. Fuhrmann, S.W., Cooper, R.B. Stochastic decomposition in M/G/1 queue with generalized vacations. Operations Research, 33: 1117–1129 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ghafir, H.M., Silio, C.B. Performance analysis of a multiple access ring network. IEEE Transactions on Communications, 41: 1496–1506 (1993)

    Article  Google Scholar 

  26. Jaing, Y., Tham, C.K., Ko, C.C. Delay analysis of a probabilistic priority discipline. European Transaction on Telecommunications, 13(6): 563–577 (2002)

    Article  Google Scholar 

  27. Katayama, T., Takahashi, Y. Analysis of a two-class priority queue with Bernoulli schedules. Journal of Operations Research Society of Japan, 35: 236–249 (1992)

    MathSciNet  MATH  Google Scholar 

  28. Keilson, J., Servi, L.D. Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedule. Journal of Applied Probability, 23: 790–802 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Krishna, C.M., Lee, Y.H. A study of two phase service. Operations Research Letters, 9: 91–97 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Krishnakumar, B., Arivudainambi, D., Vijayakumar, A. On the M X /G/1 retrial queue with Bernoulli schedules and general retrial time. Asia-Pacific Journal of Operations Research, 19: 177–194 (2002)

    Google Scholar 

  31. Kulkarni, V.G. Expected waiting times in a multiclass batch arrival retrial queues. Journal of Applied Probability, 23: 144–159 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Levi, H. Binomial gated service: A method for effective operation and optimization of polling systems. IEEE Transactions on Communications, 39: 1341–1350 (1991)

    Article  Google Scholar 

  34. Madan, K.C., Choudhury, G. Steady state analysis of an M X/(G 1,G 2)/1 queue with restricted admissibility of arriving batches and modified Bernoulli schedule server vacations. Journal of Probability and Statistical Sciences, 2: 167–185 (2005)

    MathSciNet  Google Scholar 

  35. Mukherjee, B., Meditch, J.S. The pi-persistent protocol for unidirectional broadcast bus networks. IEEE Transactions on Communications, 36: 1277–1286 (1988)

    Article  Google Scholar 

  36. Selvam, D.D., Sivasankaran, V. A two phase queueing system with server vacations. Operations Research Letters, 15: 163–168 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sennott, L.I., Humblet, P.A., Tweedie, R.L. Mean drifts and the non-ergodicity of Markov chain. Operations Research, 31: 783–789 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  39. Wenhui, Z. Analysis of a single server retrial queue with FCFS orbit and Bernoulli vacations. Applied Mathematics and Computation, 161: 353–364 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wolff, R.W. Poisson arrivals see time averages. Operations Research, 30: 223–231 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yang, T., Templeton, J.G.C. A survey on retrial queues. Queueing Systems, 2: 201–233 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Guwahati, 781035, Assam, India

    Gautam Choudhury & Kandarpa Deka

Authors
  1. Gautam Choudhury
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  2. Kandarpa Deka
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Correspondence to Gautam Choudhury.

Additional information

Supported by Department of Science and Technology, Govt. of India through Project No.:SR/S4/MS:229/04.

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Choudhury, G., Deka, K. A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule. Acta Math. Appl. Sin. Engl. Ser. 29, 15–34 (2013). https://doi.org/10.1007/s10255-007-7083-9

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  • DOI: https://doi.org/10.1007/s10255-007-7083-9

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