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The serial correlation coefficients of waiting times in a stationary single server queue

Published online by Cambridge University Press:  09 April 2009

D. J. Daley
D. J. Daley
Affiliation:
Statistical Laboratory CambridgeEngland
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In a stationary GI/G/1 queueing system in which the waiting time variance is finite, it can be shown that the serial correlation coefficients {ρn} of a (stationary) sequence of waiting times are non-negative and decrease monotonically to zero. By means of renewal theory we find a representation for Σ∞0 ρn from which necessary and sufficient condition for its finiteness can be found. In M/G/1 rather more can be said: {ρn} is convex sequence, the asymptotic form of ρ n can be given in a nearly saturated queue, and a simple explicit expression for Σ∞0 ρn exists. For the stationary M/M/1 queue we find the ρn's explicitly, illustrate them numerically, and derive a representation which shows that {ρn} is completely monotonic.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 4 , November 1968 , pp. 683 - 699
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Benes, V. E., ‘The convariance function of a simple trunk group, with applications to traffic measurements’, Bell System Tech. J. 40 (1961), 117–148.CrossRefGoogle Scholar
[2]Breny, H., ‘Quelques propriétés des files d'attentes où les clients arrivent en grappes’, Mem. Soc. Roy. Sci. Liège 6 (1961), No. 4.Google Scholar
[4]Cheong, C. K. and Heathcote, C. R., ‘On the rate of convergence of waiting times’ J. Aust. Math. Soc. 5 (1965), 365–373.Google Scholar
[5]Copson, E. T., An Introduction to the Theory of Functions of a Complex Variable (Oxford U.P., 1935).Google Scholar
[6]Craven, B. D., ‘Serial dependence of a Markov process’, J. Aust. Math. Soc. 5 (1965), 299–314.Google Scholar
[7]Daley, D. J., ‘Stochastically monotone Markov chains’ (in preparation).Google Scholar
[8]Feller, W., An Introduction to Probability Theory and its Applications, Vol. II (Wiley, 1966).Google Scholar
[9]Fishman, G. S. and Kiviat, P. J., ‘Spectral analysis of time series generated by simulation models’, RAND Memorandum RM-4393-PR (1963).Google Scholar
[10]Jackson, J. R., ‘Distributions d'échantillonnage du temps moyen d'attente dans une file’, Bull. Soc. Roy. Sci. Liège 30 (1961), 243–246.Google Scholar
[11]Kiefer, J. and Wolfowitz, J., ‘On the characteristics of the general queueing process, with applications to random walks’, Ann. Math. Statist. 27 (1956), 147–161.Google Scholar
[12]Klingman, J. F. C., ‘On queues in heavy traffic’, J. Roy. Statist. Soc. B 24 (1962), 383–392.Google Scholar
[13]Kingman, J. F. C., ‘The heavy traffic approximation in the theory of queues’. pp. 137–169 ofGoogle Scholar
Proceedings of the Chapel Hill Symposium on Congestion Theory (University of North Carolina Press, 1965).Google Scholar
[14]Le Gall, P., Les Systèmes avec ou sans Attente et les Processus Stochastiques, Tome I (Dunod, 1962).Google Scholar
[15]Morse, P. M., ‘Stochastic properties of waiting lines’, Operat. Res. 3 (1955), 255–261.Google Scholar
[16]Prabhu, N. U., Queues and Inventories: A Study of their Basic Stochastic Processes (Wiley, 1965).Google Scholar
[17]Riordan, J., Stochastic Service Systems (Wiley, 1962).Google Scholar
[18]Saaty, T. L., Elements of Queueing Theory with Applications (McGraw-Hill, 1961).Google Scholar
[19]Widder, D. V., The Laplace Transform (Princeton U.P., 1941).Google Scholar
[20]Zygmund, A., Trigonometrical Series (Cambridge U.P., 1959).Google Scholar
[21]Daley, D. J., ‘The correlation structure of the output process of some single server queuing systems’ (submitted to Ann. Math. Statist., 1967)Google Scholar