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

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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.

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References

  1. J. Abate and W. Whitt, Transient behavior of regulated Brownian motion, I: starting at the origin, Adv. Appl. Prob., 19 (1987), to appear.

  2. J. Abate and W. Whitt, Transient behavior of regulated Brownian motion, II: non-zero initial conditions, Adv. Appl. Prob., 19 (1987), to appear.

  3. J. Abate and W. Whitt, Transient behavior of the M/M/1 queue via Laplace transforms, Adv. Appl. Prob., 20 (1988), to appear.

  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. J.W. Cohen,The Single Server Queue, 2nd ed. (North-Holland, Amsterdam, 1982).

    Google Scholar 

  6. D.R. Cox,Renewal Theory (Methuen, London, 1962).

    Google Scholar 

  7. D.R. Cox and W.L. Smith,Queues (Methuen, London, 1961).

    Google Scholar 

  8. R.A. Doney, Letter to the Editor, J. Appl. Prob. 21 (1984) 673–674.

    Google Scholar 

  9. W. Feller,An Introduction to Probability Theory and its Applications, I, 3rd ed. (Wiley, New York, 1968).

    Google Scholar 

  10. D.P. Gaver, Jr., Diffusion approximations and models for certain congestion problems, J. Appl. Prob. 5 (1968) 607–623.

    Google Scholar 

  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. 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. 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. N.L. Johnson and S. Kotz,Distributions In Statistics, Discrete Distributions (Wiley, New York, 1969).

    Google Scholar 

  15. J. Keilson,Markov Chain Models — Rarity and Exponentiality (Springer-Verlag, New York, 1979).

    Google Scholar 

  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. 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. I. Lee and E. Roth, Stationary Markovian queueing systems: an approximation for the transient expected queue length, unpublished paper, 1986.

  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. P.M. Morse, Stochastic properties of waiting lines, Opns. Res. 3 (1955) 255–261.

    Google Scholar 

  21. G.F. Newell,Application of Queueing Theory, 2nd ed. (Chapman and Hall, London, 1982).

    Google Scholar 

  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. N.U. Prabhu,Queues and Inventories (Wiley, New York, 1965).

    Google Scholar 

  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. C. Stone, Limit theorems for random walks, birth and death processes, and diffusion processes. Ill. J. Math. 7 (1963) 638–660.

    Google Scholar 

  26. D. Stoyan,Comparison Methods for Queues and Other Stochastic Models, ed. D.J. Daley (Wiley, Chichester, 1983).

    Google Scholar 

  27. L. Takacs,Combinatorial Methods in the Theory of Stochastic Processes (Wiley, New York, 1967).

    Google Scholar 

  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 

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Authors and Affiliations

  1. AT&T Bell Laboratories, LC 2W-E06, 184 Liberty Corner Road, 07060, Warren, NJ, USA

    Joseph Abate

  2. AT&T Bell Laboratories, MH 2C-178, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA

    Ward Whitt

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  1. Joseph Abate
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  2. Ward Whitt
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Abate, J., Whitt, W. Transient behavior of the M/M/l queue: Starting at the origin. Queueing Syst 2, 41–65 (1987). https://doi.org/10.1007/BF01182933

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  • DOI: https://doi.org/10.1007/BF01182933

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