Skip to main content
SpringerLink
Log in

Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ C is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by λ∈[0,λ C ]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ∈[0,λ C ]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always λ C -transient. Some deep properties regarding λ C -transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

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. Anderson, W.: Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York (1991)

    Google Scholar 

  2. Armero, C., Conesa, D.: Prediction in Markovian bulk arrival queues. Queueing Syst. Theory Appl 34, 327–350 (2000)

    Article  Google Scholar 

  3. Arumuganathan, R., Ramaswami, K.S.: Analysis of a bulk queue with state dependent arrivals and multiple vacations. Indian J. Pure Appl. Math. 36, 301–317 (2005)

    Google Scholar 

  4. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)

    Google Scholar 

  5. Bayer, N., Boxma, O.J.: Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks. Queueing Syst. 23, 301–316 (1996)

    Article  Google Scholar 

  6. Chang, S.H., Choi, D.W., Kim, T.S.: Performance analysis of a finite-buffer bulk-arrival and bulk-service queue with variable server capacity. Stoch. Anal. Appl. 22, 1151–1173 (2004)

    Article  Google Scholar 

  7. Chaudhry, M.L., Templeton, J.G.C.: A First Course in Bulk Queues. Wiley, New York (1983)

    Google Scholar 

  8. Chen, A.Y., Renshaw, E.: The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Probab. 34, 192–207 (1997)

    Article  Google Scholar 

  9. Chen, A.Y., Renshaw, E.: Markovian bulk-arriving queues with state-dependent control at idle time. Adv. Appl. Probab. 36, 499–524 (2004)

    Article  Google Scholar 

  10. Chen, A.Y., Pollett, P.K., Li, J.P., Zhang, H.J.: Markovian bulk-arrival and bulk-service queues with state-dependent control. Queueing Syst. 64, 267–304 (2010)

    Article  Google Scholar 

  11. Chen, A.Y., Pollett, P., Li, J.P., Zhang, H.J.: Uniqueness, extinction and explosivity of generalised Markov branching processes with pairwise interaction. Methodol. Comput. Appl. Probab. 12, 511–531 (2010)

    Article  Google Scholar 

  12. Chung, K.L.: Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York (1967)

    Google Scholar 

  13. Darroch, J.N., Seneta, E.: On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab. 4, 192–196 (1967)

    Article  Google Scholar 

  14. Dudin, A., Nishimura, S.: A BMAP/SM/1 queueing system with Markovian arrival input of disasters. J. Appl. Probab. 36, 868–881 (1999)

    Article  Google Scholar 

  15. Fakinos, D.: The relation between limiting queue size distributions at arrival and departure epochs in a bulk queue. Stoch. Process. Appl. 37, 327–329 (1991)

    Article  Google Scholar 

  16. Flaspohler, D.C.: Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Stat. Math. 26, 351–356 (1974)

    Article  Google Scholar 

  17. Gelenbe, E.: Product from networks with negative and positive customers. J. Appl. Probab. 28, 656–663 (1991)

    Article  Google Scholar 

  18. Gelenbe, E., Glynn, P., Sigman, K.: Queues with negative arrivals. J. Appl. Probab. 28, 245–250 (1991)

    Article  Google Scholar 

  19. Gross, D., Harris, C.M.: Fundamentals of Queueing Theory. Wiley, New York (1985)

    Google Scholar 

  20. Harrison, P.G., Pitel, E.: Sojourn times in single-server queues with negative customers. J. Appl. Probab. 30, 943–963 (1993)

    Article  Google Scholar 

  21. Henderson, W.: Queueing networks with negative customers and negative queue lengths. J. Appl. Probab. 30, 931–942 (1993)

    Article  Google Scholar 

  22. Jain, G., Sigman, K.: A Pollaczek–Khintchine formula for M/G/1 queues with disasters. J. Appl. Probab. 33, 1191–1200 (1996)

    Article  Google Scholar 

  23. Kelly, F.P.: Invariant measures and the generator. In: Kingman, J.F.C., Reuter, G.E. (eds.) Probability, Statistics, and Analysis. London Math. Soc. Lecture Notes Series, vol. 79, pp. 143–160. Cambridge University Press, Cambridge (1983)

    Google Scholar 

  24. Kijima, M.: Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous-time. J. Appl. Probab. 30, 509–517 (1963)

    Article  Google Scholar 

  25. Kingman, J.F.C.: The exponential decay of Markov transition probability. Proc. Lond. Math. Soc. 13, 337–358 (1963)

    Article  Google Scholar 

  26. Kleinrock, I.: Queueing Systems, vol. 1. Wiley, New York (1975)

    Google Scholar 

  27. Li, J.P., Chen, A.Y.: Decay property of stopped Markovian bulk-arriving queues. Adv. Appl. Probab. 40, 95–121 (2008)

    Article  Google Scholar 

  28. Li, J.P., Chen, A.Y.: Decay parameter and invariant measures for Markovian bulk-arrival queues with control at idle time. Methodol. Comput. Appl. Probab. (2010, to appear)

  29. Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Stoch. Models 7, 1–46 (1991)

    Article  Google Scholar 

  30. Lucantoni, D.M., Neuts, M.F.: Some steady-state distributions for the MAP/SM/1 queue. Stoch. Models 10, 575–589 (1994)

    Article  Google Scholar 

  31. Medhi, J.: Stochastic Models in Queuing Theory. Academic Press, San Diego (1991)

    Google Scholar 

  32. Nair, M.G., Pollett, P.K.: On the relationship between μ-invariant measures and quasi-stationary distributions for continuous-time Markov chains. Adv. Appl. Probab. 25, 82–102 (1993)

    Article  Google Scholar 

  33. Neuts, M.F.: A versatile Markovian point process. J. Appl. Probab. 16, 764–779 (1979)

    Article  Google Scholar 

  34. Neuts, M.F.: Matrix-Geometric Solution in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore (1981)

    Google Scholar 

  35. Nishimura, S., Sato, H.: Eigenvalue expression for a batch Markovian arrival process. J. Oper. Res. Soc. Jpn. 40, 122–132 (1997)

    Google Scholar 

  36. Pollett, P.K.: Reversibility, invariance and mu-invariance. Adv. Appl. Probab. 20, 600–621 (1988)

    Article  Google Scholar 

  37. Srinivasan, L., Renganathan, N., Kalyanaraman, R.: Single server, bulk arrival, Bernoulli feedback queue with vacations—some performance measures. Int. J. Inf. Manag. Sci. 13, 45–54 (2002)

    Google Scholar 

  38. Stadje, W.: Some exact expressions for the bulk-arrival queue MX/M/1. Queuing Syst. 4, 85–92 (1989)

    Article  Google Scholar 

  39. Sumita, U., Masuda, Y.: Tandem queues with bulk arrivals, infinitely many servers and correlated service times. J. Appl. Probab. 34, 248–257 (1997)

    Article  Google Scholar 

  40. Ushakumari, P.V., Krishnamoorthy, A.: On a bulk arrival bulk service infinite server queue. Stoch. Anal. Appl. 16, 585–595 (1998)

    Article  Google Scholar 

  41. Tweedie, R.L.: Some ergodic properties of the Feller minimal process. Q. J. Math. Oxf. (2) 25, 485–493 (1974)

    Article  Google Scholar 

  42. Van Doorn, E.A.: Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Probab. 17, 514–530 (1985)

    Article  Google Scholar 

  43. Van Doorn, E.A.: Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Probab. 23, 683–700 (1991)

    Article  Google Scholar 

  44. Vere-Jones, D.: Geometric ergodicity in denumerable Markov chains. Q. J. Math. Oxf. (2) 13, 7–28 (1962)

    Article  Google Scholar 

  45. Yaglom, A.M.: Certain limit theorems of the theory of branching processes. Dokl. Acad. Nauk SSSR 56, 795–798 (1947)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Statistics and Probability, Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

    Anyue Chen

  2. Department of Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China

    Anyue Chen

  3. School of Mathematical Science and Computing Technology, Central South University, Changsha, 410075, China

    Junping Li & Zhenting Hou

  4. Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam, Hong Kong

    Kai Wang Ng

Authors
  1. Anyue Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junping Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhenting Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kai Wang Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anyue Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, A., Li, J., Hou, Z. et al. Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues. Queueing Syst 66, 275–311 (2010). https://doi.org/10.1007/s11134-010-9194-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-010-9194-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation