Skip to main content
SpringerLink
Log in

On M/M/1 queues with a smart machine

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper discusses a class of M/M/1 queueing models in which the service time of a customer depends on the number of customers served in the current busy period. It is particularly suited for applications in which the server has kind of learning ability and warms up gradually. We present a simple and computationally tractable scheme which recursively determines the stationary probabilities of the queue length. Other performance measures such as the Laplace transform of the busy period are also obtained. For the firstN exceptional services model which can be considered as a special case of our model, we derive a closed-formula for the generating function of the stationary queue length distribution. Numerical examples are also provided.

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. J.A. Buzacott and J.R. Callahan, The capacity of the soaking pit-rolling mill complex in steel production,INFOR 9 (1971) 87–95.

    Google Scholar 

  2. A. Federgruen and H.C. Tijms, Computation of the stationary distribution of the queue size in an M/G/1 queueing system with variable service rate,J. Appl. Prob. 17 (1980) 515–522.

    Google Scholar 

  3. D. Gross and C.M. Harris,Fundamentals of Queueing Theory, 2nd ed. (Wiley, New York, 1985).

    Google Scholar 

  4. S.K. Gupta, Queues with hyper-Poisson input and exponential arrival and service rates,Oper. Res. 15 (1967) 847–856.

    Google Scholar 

  5. S. Halfin, Steady-state distribution of the buffer content of an M/G/1 queue with varying service rate,SIAMJ. Appl. Math. 23 (1972) 356–363.

    Google Scholar 

  6. C.M. Harris, Queues with state-dependent stochastic service rate,Oper. Res. 16 (1968) 117–130.

    Google Scholar 

  7. M.F. Neuts, The M/M/1 queue with randomly varying arrival and service rates,Opsearch 15 (1978) 139–157.

    Google Scholar 

  8. M. Posner, Single-server queue with service time depended on waiting time,Oper. Res. 24 (1973) 610–616.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wireless Networks, NORTEL, MS-D207 2201 Lakeside Blvd., 75083-3871, Richardson, TX, USA

    Huan Li, Yixin Zhu, Ping Yang & Seshu Madhavapeddy

Authors
  1. Huan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yixin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ping Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Seshu Madhavapeddy
    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

Li, H., Zhu, Y., Yang, P. et al. On M/M/1 queues with a smart machine. Queueing Syst 24, 23–36 (1996). https://doi.org/10.1007/BF01149078

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01149078

Keywords

Search

Navigation