Skip to main content
SpringerLink
Log in

Discrete time analysis of MAP/PH/1 vacation queue with gated time‐limited service

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We analyse a single‐server queue in which the server goes through alternating periods of vacation and work. In each work period, the server attends to the queue for no more than a fixed length of time, T. The system is a gated one in which the server, during any visit, does not attend to customers which were not in the system before its visit. As soon as all the customers within the gate have been served or the time limit has been reached (whichever occurs first) the server goes on a vacation. The server does not wait in the queue if the system is empty at its arrival for a visit. For this system the resulting Markov chain, of the queue length and some auxiliary variables, is level‐dependent. We use special techniques to carry out the steady state analysis of the system and show that when the information regarding the number of customers in the gate is not critical we are able to reduce this problem to a level‐independent Markov chain problem with large number of boundary states. For this modified system we use a hybrid method which combines matrix‐geometric method for the level‐independent part of the system with special solution method for the large complex boundary which is level‐dependent.

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. A.S. Alfa, A discrete MAP/PH/1 queue with vacations and exhaustive time-limited service, Oper. Res. Lett. 18 (1995) 31-40.

    Article  Google Scholar 

  2. A.S. Alfa and M.F. Neuts, Modelling vehicular traffic using the discrete time Markovian arrival process, Transportation Sci. 29(2) (1995) 109-117.

    Article  Google Scholar 

  3. C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belg. J. Oper. Res. Statist. Comput. Sci. 32(3,4) (1992) 3-23.

    Google Scholar 

  4. L. Bright and P.G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic Models 11(3) (1995) 497-525.

    Google Scholar 

  5. B.T. Doshi, Queueing systems with vacations — A survey, Queueing Systems 1 (1986) 29-66.

    Article  Google Scholar 

  6. B.T. Doshi, Single server queues with vacations, in: Stochastic Analysis of Computer and Communications Systems, ed. H. Takagi (1990).

  7. D.P. Gaver, P.A. Jacobs and G. Latouche, Finite birth-and-death models in randomly changing environments, Adv. in Appl. Probab. 16 (1984) 715-731.

    Article  Google Scholar 

  8. G. Latouche and V. Ramaswami, A logarithmic reduction algorithm for quasi-birth-death processes, J. Appl. Probab. 30 (1993) 650-674.

    Article  Google Scholar 

  9. K.K. Leung and M. Eisenberg, A single-server queue with vacations and gated time-limited service, IEEE Trans. Commun. 38(9) (1990) 1454-1462.

    Article  Google Scholar 

  10. D. Liu and M.F. Neuts, A queueing model for an ATM rate control scheme, Telecommunication Systems 2 (1994) 321-348.

    Article  Google Scholar 

  11. D.M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Stochastic Models 7 (1991) 1-46.

    Google Scholar 

  12. D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A single-server queue with server vacations and a class of nonrenewal arrival processes. Adv. in Appl. Probab. 22 (1990) 676-705.

    Article  Google Scholar 

  13. S.K. Matendo, A single-server queue with server vacations and a batch Markovian arrival process, Cahiers du C.E.R.O 35(1–2) (1993) 87-114.

    Google Scholar 

  14. M.F. Neuts, A versatile Markovian point process, J. Appl. Probab. 16 (1979) 764-779.

    Article  Google Scholar 

  15. M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models — An Algorithmic Approach (Johns Hopkins Univ. Press, Baltimore, MD, 1981).

    Google Scholar 

  16. M.F. Neuts and B.M. Rao, Numerical investigation of a multiserver retrial model, Queueing Systems 7 (1990) 169-190.

    Article  Google Scholar 

  17. V. Ramaswami and P.G. Taylor, Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases, Stochastic Models 12(1) (1996) 143-164.

    Google Scholar 

  18. K. Sriram, Dynamic bandwidth allocation and congestion control schemes for voice and data multiplexing in wideband packet technology, in: Proc. of IEEE ICC '90, Atlanta, GA (1990) pp. 1003-1009.

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 5V6

    Attahiru Sule Alfa

Authors
  1. Attahiru Sule Alfa
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alfa, A.S. Discrete time analysis of MAP/PH/1 vacation queue with gated time‐limited service. Queueing Systems 29, 35–54 (1998). https://doi.org/10.1023/A:1019123828374

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019123828374

Search

Navigation