Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T21:35:21.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Limits and Approximations for the Busy-Period Distribution in Single-Server Queues

Published online by Cambridge University Press:  27 July 2009

Joseph Abate  and
Ward Whitt
Joseph Abate
Affiliation:
AT&T Bell Laboratories, Room 2C-178, Murray Hill, New Jersey 07974-0636
Ward Whitt
Affiliation:
AT&T Bell Laboratories, Room 2C-178, Murray Hill, New Jersey 07974-0636
Get access Rights & Permissions [Opens in a new window]

Abstract

Limit theorems are established and relatively simple closed-form approximations are developed for the busy-period distribution in single-server queues. For the M/G/l queue, the complementary busy-period c.d.f. is shown to be asymptotically equivalent as t → ∞ to a scaled version of the heavy-traffic limit (obtained as p → 1), where the scaling parameters are based on the asymptotics as t → ∞. We call this the asymptotic normal approximation, because it involves the standard normal c.d.f. and density. The asymptotic normal approximation is asymptotically correct as t → ∞ for each fixed p and as p → 1 for each fixed t and yields remarkably good approximations for times not too small, whereas the direct heavy-traffic (p → 1) and asymptotic (t → ∞) limits do not yield such good approximations. Indeed, even the approximation based on three terms of the standard asymptotic expansion does not perform well unless t is very large. As a basis for generating corresponding approximations for the busy-period distribution in more general models, we also establish a more general heavy-traffic limit theorem.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 4 , October 1995 , pp. 581 - 602
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abate, J., Choudhury, G.L., & Whitt, W. (1995). Calculating the M/G/l busy-period density and the LIFO waiting-time distribution by direct numerical transform inversion. Operations Research Letters, in press.CrossRefGoogle Scholar
2.Abate, J. & Whitt, W. (1987). Transient behavior of regulated Brownian motion, I and II. Advances in Applied Probability 19: 560–631.CrossRefGoogle Scholar
3.Abate, J. & Whitt, W. (1988). Approximations for the M/M/l busy-period distribution. In Boxma, O.J. & Syski, R. (eds.), Queueing theory and its applications, Liber Amicorum Professor J. W. Cohen. Amsterdam: North-Holland, pp. 149–191.Google Scholar
4.Abate, J. & Whitt, W. (1988). Simple spectral representations for the M/M/l queue. Queueing Systems 3: 321–346.Google Scholar
5.Abate, J. & Whitt, W. (1988). Transient behavior of the M/M/l queue via Laplace transforms. Advances in Applied Probability 20: 145–178.CrossRefGoogle Scholar
6.Abate, J. & Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10: 5–88.Google Scholar
7.Abate, J. & Whitt, W. (1992). Solving probability transform functional equations for numerical inversion. Operations Research Letters 12: 245–251.Google Scholar
8.Abate, J. & Whitt, W. (1994). Transient behavior of the M/G/l workload process. Operations Research 42: 750–764.CrossRefGoogle Scholar
9.Abate, J. & Whitt, W. (1996). An operational calculus for probability distributions via Laplace transforms. Advances in Applied Probability, in press.CrossRefGoogle Scholar
10.Abramowitz, M. & Stegun, I.A. (eds.). (1972). Handbook of mathematical functions. New York: Dover.Google Scholar
11.Choudhury, G.L. & Lucantoni, D.M. (1996). Numerical computation of the moments of a probability distribution from its transform. Operations Research (to appear).CrossRefGoogle Scholar
12.Cohen, J.W. (1982). The single server queue, 2nd ed.Amsterdam: North-Holland.Google Scholar
13.Cox, D.R. & Smith, W.L. (1961). Queues. London: Methuen.Google Scholar
14.De Meyer, A. & Teugels, J.L. (1980). On the asymptotic behavior of the distributions of the busy period and service time in M/G/l. Advances in Applied Probability 17: 802–813.CrossRefGoogle Scholar
15.Ethier, S.N. & Kurtz, T.G. (1986). Markov processes, characterization and convergence. New York: Wiley.CrossRefGoogle Scholar
16.Feller, W. (1971). An introduction to probability theory and its applications. Vol. II, 2nd ed.New York: Wiley.Google Scholar
17.Halfin, S. (1985). Delays in queues, properties and approximations. In Akiyama, M. (ed.), Teletraffic issues in an advanced information society, Proceedings of ITC-II. Amsterdam: Elsevier, pp. 47–52.Google Scholar
18.Heyman, D.P. (1974). An approximation for the busy period of the M/G/l queue using a diffusion model. Journal of Applied Probability 11: 159–169.CrossRefGoogle Scholar
19.Iglehart, D.L. & Whitt, W. (1970). Multiple channel queues in heavy traffic, I and II. Advances in Applied Probability 2: 150–177, 355–369.Google Scholar
20.Keilson, J. (1978). Exponential spectra as a tool for the study of server-systems with several classes of customers. Journal of Applied Probability 15: 162–170.Google Scholar
21.Kella, O. & Taksar, M.I. (1994). A heavy traffic limit for the cycle counting process in G/G/l, optional interruptions and elastic screen Brownian motion. Mathematics of Operations Research 19: 132–151.CrossRefGoogle Scholar
22.Kingman, J.F.C. (1961). The single server queue in heavy traffic. Proceedings of the Cambridge Philosophical Society 57: 902–904.CrossRefGoogle Scholar
23.Kingman, J.F.C. (1962). On queues in heavy traffic. Journal of the Royal Statistical Society Series B 24: 383–392.Google Scholar
24.Kingman, J.F.C. (1962). The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/1. Journal of the Australian Mathematical Society 2: 345–356.CrossRefGoogle Scholar
25.Kraemer, W. & Langenbach-Belz, M. (1976). Approximate formulae for the delay in the queueing system GI/G/1. Proceedings of the Eighth International Teletraffic Congress, Melbourne, Australia, 235-1/8.Google Scholar
26.Ott, T.J. (1977). The stable M/G/l queue in heavy traffic and its covariance function. Advances in Applied Probability 9: 169–186.Google Scholar
27.Rice, S.O. (1962). Single server systems-II. Busy periods. Bell System Technical Journal 41: 279–310.CrossRefGoogle Scholar
28.Riordan, J. (1962). Stochastic service systems. New York: Wiley.Google Scholar
29.Szczotka, W. (1990). Exponential approximation of waiting time and queue size for queues in heavy traffic. Advances in Applied Probability 22: 230–240.Google Scholar
30.Takács, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes. New York: Wiley.Google Scholar
31.Whitt, W. (1971). Weak convergence theorems for priority queues: Preemptive-resume discipline. Journal of Applied Probability 8: 74–94.Google Scholar
32.Whitt, W. (1983). The queueing network analyzer. Bell System Technical Journal 62: 2779–2815.CrossRefGoogle Scholar
33.Whitt, W. (1984). Minimizing delays in the GI/G/1 queue. Operations Research 32: 41–51.CrossRefGoogle Scholar