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A diffusion approximation for correlation in queues

Published online by Cambridge University Press:  14 July 2016

C. M. Woodside ,
B. Pagurek  and
G. F. Newell
C. M. Woodside*
Affiliation:
Carleton University
B. Pagurek*
Affiliation:
Carleton University
G. F. Newell*
Affiliation:
University of California
*
∗Postal address: Department of Systems Engineering and Computing Science, Carleton University, Ottawa, Canada K1S 5B6.
∗Postal address: Department of Systems Engineering and Computing Science, Carleton University, Ottawa, Canada K1S 5B6.
∗∗Postal address: Department of Civil Engineering, University of California, Berkeley, CA 94720, U.S.A.
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Abstract

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in any GI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.

Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system in M/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

Keywords

AUTOCORRELATIONQUEUEDIFFUSION APPROXIMATIONSPECTRAL ANALYSIS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 1033 - 1047
Copyright
Copyright © Applied Probability Trust 

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References

[1] Abramowitz, M. and Stegun, I. (1964) Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Benež, V. E. (1957) On queues and Poisson arrivals. Ann. Math. Statist. 28, 670–677.Google Scholar
[3] Cox, D. R. and Miller, H. D. (1965) Theory of Stochastic Processes. Methuen, London.Google Scholar
[4] Daley, D. J. (1968) The serial correlation coefficients of waiting times in a stationary single server queue. J. Austral. Math. Soc. 8, 683–699.Google Scholar
[5] Gaver, D. P. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5, 607–623.Google Scholar
[6] Gaver, D. P. and Shedler, G. S. (1973) Processor utilization in multiprogramming systems via diffusion approximations. Operat. Res. 21, 569–576.Google Scholar
[7] Gelenbe, E. (1975) On approximate computer system models. J. ACM 22, 261–269.CrossRefGoogle Scholar
[8] Iglehart, D. and Whitt, W. (1970) Multiple channel queues in heavy traffic I, II. Adv. Appl. Prob. 2, 150–177, 355–369.CrossRefGoogle Scholar
[9] Morse, P. M. (1955) Stochastic properties of waiting lines. Operat. Res. 3, 255–261.Google Scholar
[10] Newell, G. F. (1971) Applications of Queueing Theory. Chapman and Hall, London.Google Scholar
[11] Ott, T. J. (1977) The stable M/G/1 queue in heavy traffic and its covariance function Adv. Appl. Prob. 9, 169–186.Google Scholar
[12] Ott, T. J. (1976) The covariance function of the Brownian motion with drift, made ergodic by a reflecting boundary. Adv. Appl. Prob. 8, 234–236.Google Scholar
[13] Pagurek, B. and Woodside, C. M. (1977) A recursive algorithm for computing serial correlations of times in an M/G/1 queue. In North-Holland/TIMS Studies in the Management Sciences 7, North-Holland, Amsterdam.Google Scholar
[14] Pagurek, B. and Woodside, C. M. (1979) An expression for the sum of serial correlations of times in GI/G/1 queues with rational arrival processes. Operat. Res. 27, 755–766.CrossRefGoogle Scholar
[15] Reiser, M. and Kobayashi, H. (1974) Accuracy of the diffusion approximation for some queueing systems. IBM J. Research and Development 18, 110–124.Google Scholar
[16] Prohorov, Yu (1963) Transient phenomena in processes of mass service (in Russian). Litovsk. Mat. Sb. 3, 199–205.Google Scholar
[17] Reynolds, J. F. (1975) The covariance structure of queues and related processes — a survey of recent work. Adv. Appl. Prob. 7, 383–415.Google Scholar
[18] Whitt, W. (1974) Heavy traffic limit theorems for queues: a survey. In Mathematical Methods in Queueing Theory, ed. Clarke, A. B. Springer-Verlag, Berlin.Google Scholar
[19] Woodside, C. M. and Pagurek, B. (1979) An algorithm for computing serial correlations of times in GI/G/1 queues with rational arrival processes. Management Sci. 25, 54–63.Google Scholar