Skip to main content
SpringerLink
Log in

The MAP/PH/N retrial queue in a random environment

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

We consider the MAP/PH/N retrial queue with a finite number of sources operating in a finite state Markovian random environment. Two different types of multi-dimensional Markov chains are investigated describing the behavior of the system based on state space arrangements. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated, based on which the main performance measures are obtained, and numerical examples are presented as well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alfa, A., Isotupa, K. An M/PH/k retrial queue with finite number of sources. Computers & Operations Research, 31: 1455–1464 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Artalejo, J. R. Accessible bibliography on retrial queues. Mathematical and Computer Modelling, 30: 223–233 (1999)

    MathSciNet  Google Scholar 

  3. Artalejo, J.R. A classified bibliography of research on retrial queues: Progress in 1990–1999, 7(2): 187–211 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Artalejo, J. R., Lopez-Herrero, M.J. A simulation study of a discrete-time multiserver retrial queue with finite population. Journal of Statistical Planning and Inference, 137: 2536–2542 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bocharov, P. P., D’Apice, C., Pechinkin, A.V., Salerno, S. Queueing Theory. Utrecht, Boston, 2004

    MATH  Google Scholar 

  6. Breuer, L., Dudin, A., Klimenok, V. Aretrial BMAP/PH/N system. Queueing Systems, 40: 433–457 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chakravarthy, S., Dudin, A. A multi-server retrial queue with BMAP arrivals and group service. Queueing Systems, 42: 5–31 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chakravarthy, S., Dudin, A. Analysis of a retrial queueing model with MAP arrivals and two types of customers. Mathematical and Computer Modelling, 37: 343–364 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Choi, B., Chang, Y. MAP1, MAP2/M/C retrial queue with the retrial group of finite capacity and geometric loss. Mathematical and Computer Modelling, 30: 99–114 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Choi, B., Chung, Y., Dudin, A. The BMAP/SM/1 retrial queue with controllable operation modes. European Journal of Operational Research, 131: 16–30 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dudin, A., Kazimirsky, A., Klimenok, A., Breuer, L., Krieger, U. Queueing model MAP/PH/1/N with feedback operating in a Markovian random environment. Austrian Journal of Statistics, 34(2): 101–110 (2005)

    Google Scholar 

  12. Dudin, A., Klimenok, V. Retrial BMAP/SM/1 system operating in a synchronous random environment, Probabilistic Analysis of Rare Events: Theory and Problems of Safety. Insurance and Ruin, Riga Aviation University, Riga, 143–148, 1999

    Google Scholar 

  13. Dudin, A., Krishnamoorthy, A., Joshua, V., Tsarenkov, G. Analysis of the BMAP/G/1 retrial system with search of customers from the orbit. European Journal of Operational Research, 157: 169–179 (2004)

    Article  MATH  Google Scholar 

  14. Falin, G. A multiserver retrial queue with finite number of sources of primary calls. Mathematical and Computer Modelling, 30: 33–49 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Falin, G. The M/M/∞ queue in a random environment. Queueing Systems, 58: 65–76 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Falin, G., Artalejo, J. A finite source retrial queue. European Journal of Operational Research, 108: 409–424 (1998)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  18. Falin, G. I. A survey of retrial queues. Queueing Systems, 7: 127–167 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gaver, D., Jacobs, P., Latouche, G. Finite birth-and-death models in randomly changing environments. Advances in Applied Probability, 16: 715–731 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Graham, A. Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Cichester, 1981

    MATH  Google Scholar 

  21. He, Q. M., Li, H., Zhao, Y.Q. Ergodicity of the BMAP/PH/S/S+K retrial queue with PH-retrial times. Queueing Systems, 35: 323–347 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kim, C., Dudin, A., Klimenok, V., Khramova, V. Erlang loss queueing system with batch arrivals operating in a random environment. Computers & Operations Research, 36: 674–697 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kim, C., Klimenok, V., Lee, S., Dudin, A. The BMAP/PH/1 retrial queueing system operating in random environment. Journal of Statistical Planning and Inference, 137: 3904–3916 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kim, C., Klimenok, V., Orlovsky, D. The BMAP/PH/N retrial queue with Markovian flow of breakdowns. European Journal of Operational Research, 189: 1057–1072 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Klimenok, V. Queueing system BMAP/SM/1 with hybrid mechanism of operation. Automation and Remote Control, 66: 111–124 (2005)

    Article  MathSciNet  Google Scholar 

  26. Klimenok, V., Dudin, A. Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Systems, 54: 245–259 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Klimenok, V., Dudin, A., Kim, C. A loss-retrial BMAP/PH/N system. Queues: Flows, Systems, Networks, 17: 129–135 (2003)

    Google Scholar 

  28. Klimenok, V., Kim, C., Dudin, A., Orlovsky, D. Lack of Invariant property of the Erlang loss model in case of MAP input. Queueing System, 49: 187–213 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Klimenok, V., Orlovsky, D., Kim, C. The BMAP/PH/N/N+R retrial queueing system with different disciplines of retrials. In: Proceedings of 11th International Conference on Analytical and Stochastic Modelling Techniques and Applications ASMTA 2004), Magdeburg, Germany: 93–98 (2004)

    Google Scholar 

  30. Krieger, U., Klimenok, V., Kazimirsky, A., Breuer, L. A BMAP/PH/1 queue with feedback operating in a random environment. Mathematical and Computer Modelling, 41: 867–882 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kumar, B. K., Rukmani, R., Thangaraj, V. On multiserver feedback retrial queue with finite buffer. Applied Mathematical Modelling, 33: 2062–2083 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Latouche, G., Ramaswami, V. Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia, PA, 1999

    Book  MATH  Google Scholar 

  33. 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)

    Article  MATH  Google Scholar 

  34. Lucantoni, D. M. New results on the single server queue with a batch Markovian arrival process. Stochastic Models, 7: 1–46 (1997)

    MathSciNet  Google Scholar 

  35. Neuts, M. F. A versatile Markovian point process. Journal of Applied Probability, 16:764–779 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  36. Neuts, M. F. Matrix-Geometric Solutions in Stochastic Models-an Algorithmic Approach. Johns Hopkins Press, Baltimore, MD, 1981

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  38. Ramaswami, V. The N/G/1 queue and its detailed analysis. Advances in Applied Probability, 12: 222–261 (1981)

    Article  MathSciNet  Google Scholar 

  39. Tran-Gia, P., Mandjes, M. Modeling of customer retrial phenomenon in cellular mobile networks. IEEE Journal on Selected Areas in Communications, 15, 1406–1414 (1997)

    Article  Google Scholar 

  40. Wang, J., Zhao, L., Zhang, F. Analysis of the finite source retrial queues with server breakdowns and repairs. Journal of Industrial and Management Optimization, 7(3), 655–676 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, F., Wang, J. Performance analysis of the retrial queues with finite number of sources and server interruptions. Journal of the Korean Statistical Society, 42(1): 117–131 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information, Hunan University of Commerce, Changsha, 410205, China

    Gang Yang, Luo-gen Yao & Zi-sheng Ouyang

Authors
  1. Gang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luo-gen Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zi-sheng Ouyang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gang Yang.

Additional information

Supported by National Social Science Foundation of China (No. 11BTJ011) and Humanities and Social Sciences

Foundation of Ministry of Education of China, 2012 (No. 12YJAZH173)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, G., Yao, Lg. & Ouyang, Zs. The MAP/PH/N retrial queue in a random environment. Acta Math. Appl. Sin. Engl. Ser. 29, 725–738 (2013). https://doi.org/10.1007/s10255-013-0251-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-013-0251-1

Keywords

2000 MR Subject Classification

Search

Navigation