Skip to main content
SpringerLink
Log in

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching.

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. T. Collings and C. Stoneman, The M=M=∞ queue with varying arrival and departure rates, Oper. Res. 24 (1976) 760–773.

    Google Scholar 

  2. E.G. Landauer and L.C. Becker, Reducing waiting time at security checkpoints, Interfaces 19 (1989) 57–65.

    Google Scholar 

  3. P. Kolesar, Stalking the endangered CAT: A queueing analysis of congestion at automatic teller machines, Interfaces 14 (1984) 16–26.

    Google Scholar 

  4. B.O. Koopman, Air-terminal queues under time-dependent conditions, Oper. Res. 20 (1972) 1089–1114.

    Article  Google Scholar 

  5. L. Green and P. Kolesar, Testing the validity of a queueing model of police patrol, Management Science 35 (1989) 127–148.

    Google Scholar 

  6. G.F. Newell, Queues with time-dependent arrival rates I: the transition through saturation, J. Appl. Probab. 5 (1968) 436–451.

    Article  Google Scholar 

  7. G.F. Newell, Queues with time-dependent arrival rates II: the maximum queue and the return to equilibrium, J. Appl. Probab. 5 (1968) 579–590.

    Article  Google Scholar 

  8. G.F. Newell, Queues with time-dependent arrival rates III: a mild rush hour, J. Appl. Probab. 5 (1968) 591–606.

    Article  Google Scholar 

  9. J.B. Keller, Time-dependent queues, SIAM Review (1982) 401–412.

  10. W.A. Massey, Asymptotic analysis of the time dependent M/M/∞ queue, Math. Oper. Res. 10 (1985) 305–327.

    Google Scholar 

  11. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II (Wiley, New York, 1962).

  12. J.W. Cohen, The Single Server Queue (North-Holland, Amsterdam, 1982).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of St. Thomas, St. Paul, MN, 55105‐1096, USA

    Yongzhi (Peter) Yang

  2. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL, 60607‐7045, USA

    Charles Knessl

Authors
  1. Yongzhi (Peter) Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charles Knessl
    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

Yang, Y.(., Knessl, C. Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate. Queueing Systems 26, 23–68 (1997). https://doi.org/10.1023/A:1019116804750

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019116804750

Search

Navigation