Queueing Systems

, Volume 2, Issue 1, pp 41–65

Transient behavior of the M/M/l queue: Starting at the origin

  • Joseph Abate
  • Ward Whitt
Contributed Paper

Abstract

This paper presents some new perspectives on the time-dependent behavior of the M/M/1 queue. The factorial moments of the queue length as functions of time when the queue starts empty have interesting structure, which facilitates developing approximations. Simple exponential and hyperexponential approximations for the first two moment functions help show how the queue approaches steady state as time evolves. These formulas also help determine if steady-state descriptions are reasonable when the arrival and service rates are nearly constant over some interval but the process does not start in steady state.

Keywords

Transient behavior approach to steady state relaxation times birth- and-death process queues Brownian motion first passage times busy period complete monotonicity 

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References

  1. [1]
    J. Abate and W. Whitt, Transient behavior of regulated Brownian motion, I: starting at the origin, Adv. Appl. Prob., 19 (1987), to appear.Google Scholar
  2. [2]
    J. Abate and W. Whitt, Transient behavior of regulated Brownian motion, II: non-zero initial conditions, Adv. Appl. Prob., 19 (1987), to appear.Google Scholar
  3. [3]
    J. Abate and W. Whitt, Transient behavior of the M/M/1 queue via Laplace transforms, Adv. Appl. Prob., 20 (1988), to appear.Google Scholar
  4. [4]
    J.P.C. Blanc, The relaxation time of two queueing systems in series, Commun. Statist.-Stochastic Models 1 (1985) 1–16.Google Scholar
  5. [5]
    J.W. Cohen,The Single Server Queue, 2nd ed. (North-Holland, Amsterdam, 1982).Google Scholar
  6. [6]
    D.R. Cox,Renewal Theory (Methuen, London, 1962).Google Scholar
  7. [7]
    D.R. Cox and W.L. Smith,Queues (Methuen, London, 1961).Google Scholar
  8. [8]
    R.A. Doney, Letter to the Editor, J. Appl. Prob. 21 (1984) 673–674.Google Scholar
  9. [9]
    W. Feller,An Introduction to Probability Theory and its Applications, I, 3rd ed. (Wiley, New York, 1968).Google Scholar
  10. [10]
    D.P. Gaver, Jr., Diffusion approximations and models for certain congestion problems, J. Appl. Prob. 5 (1968) 607–623.Google Scholar
  11. [11]
    D.P. Gaver, Jr. and P.A. Jacobs, On inference and transient response for M/G/1 models, Naval Postgraduate School, Monterey, CA, 1986.Google Scholar
  12. [12]
    D.P. Heyman, An approximation for the busy period of the M/G/1 queue using a diffusion model, J. Appl. Prob. 11 (1974) 159–169.Google Scholar
  13. [13]
    D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic II: sequences, networks and batches, Adv. Appl. Prob. 2 (1970) 355–369.Google Scholar
  14. [14]
    N.L. Johnson and S. Kotz,Distributions In Statistics, Discrete Distributions (Wiley, New York, 1969).Google Scholar
  15. [15]
    J. Keilson,Markov Chain Models — Rarity and Exponentiality (Springer-Verlag, New York, 1979).Google Scholar
  16. [16]
    W.D. Kelton and A.M. Law, The transient behavior of the M/M/S queue, with implications for steady-state simulation, Opns. Res. 33 (1985) 378–396.Google Scholar
  17. [17]
    I. Lee, Stationary Markovian queueing systems: an approximation for the transient expected queue length, M.S. dissertation, Department of Electrical Engineering and Computer Science, MIT, Cambridge, 1985.Google Scholar
  18. [18]
    I. Lee and E. Roth, Stationary Markovian queueing systems: an approximation for the transient expected queue length, unpublished paper, 1986.Google Scholar
  19. [19]
    M. Mori, Transient behavior of the mean waiting time and its exact forms in M/M/1 and M/D/1. J. Opns. Res. Soc. Japan 19 (1976) 14–31.Google Scholar
  20. [20]
    P.M. Morse, Stochastic properties of waiting lines, Opns. Res. 3 (1955) 255–261.Google Scholar
  21. [21]
    G.F. Newell,Application of Queueing Theory, 2nd ed. (Chapman and Hall, London, 1982).Google Scholar
  22. [22]
    A.R. Odoni and E. Roth, An empirical investigation of the transient behavior of stationary queueing systems, Opns. Res. 31 (1983) 432–455.Google Scholar
  23. [23]
    N.U. Prabhu,Queues and Inventories (Wiley, New York, 1965).Google Scholar
  24. [24]
    E. Roth, An investigation of the transient behavior of stationary queueing systems, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, 1981.Google Scholar
  25. [25]
    C. Stone, Limit theorems for random walks, birth and death processes, and diffusion processes. Ill. J. Math. 7 (1963) 638–660.Google Scholar
  26. [26]
    D. Stoyan,Comparison Methods for Queues and Other Stochastic Models, ed. D.J. Daley (Wiley, Chichester, 1983).Google Scholar
  27. [27]
    L. Takacs,Combinatorial Methods in the Theory of Stochastic Processes (Wiley, New York, 1967).Google Scholar
  28. [28]
    E. Van Doom,Stochastic Monotonicity and Queueing Applications of Birth-Death Proceses, Lecture Notes in Statistics 4 (Springer-Verlag, New York, 1980).Google Scholar

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1987

Authors and Affiliations

  • Joseph Abate
    • 1
  • Ward Whitt
    • 2
  1. 1.AT&T Bell LaboratoriesWarrenUSA
  2. 2.AT&T Bell LaboratoriesMurray HillUSA

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