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A second-order heavy traffic approximation for the queue GI/G/1

Published online by Cambridge University Press:  01 July 2016

Julian Köllerström
Julian Köllerström*
Affiliation:
University of Kent
*
∗ Postal address: Mathematical Institute, University of Kent, Canterbury, Kent, CT2 7NF, U.K.
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Abstract

A second-order heavy traffic approximation for the stationary waiting-time d.f. G for GI/G/1 queues is derived, the first-order term of which is Kingman's (1961), (1962a), (1965) exponential approximation. On the way to this result there are others of independent interest, such as a convolution equation relating this waiting time d.f. G with the d.f. of a related ladder height, an integral equation for G and some stochastic bounds for G. The main result requires a particular type of functional convergence that may also be of interest.

Keywords

QUEUE GI/G/1WAITING TIMEHEAVY TRAFFIC APPROXIMATIONSSECOND-ORDER TERMSTOCHASTIC BOUNDSINTEGRAL EQUATION

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 1 , March 1981 , pp. 167 - 185
Copyright
Copyright © Applied Probability Trust 1981 

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References

Blomqvist, N. (1969) Estimation of waiting-time parameters in the GI/G/1 queueing system. Part II. Skand. Akt. Tidskr., 125–136.Google Scholar
Daley, D. J. (1980) Tight bounds for the renewal function of a random walk. Ann. Prob. 8,CrossRefGoogle Scholar
Delbrouck, L. E. N. (1975) Convexity properties, moment inequalities and asymptotic exponential approximations for delay distributions in GI/G/1 systems. Stoch. Proc. Appl. 3, 193–207.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
Kiefer, J. and Wolfowitz, J. (1956) On the characteristics of the general queueing process with applications to random walk. Ann. Math. Statist. 27, 147–161.CrossRefGoogle Scholar
Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.CrossRefGoogle Scholar
Kingman, J. F. C. (1962a) On queues in heavy traffic. J. R. Statist. Soc. B 24, 383–392.Google Scholar
Kingman, J. F. C. (1962b) Some inequalities for the queue GI/G/1. Biometrika 49, 315–324.CrossRefGoogle Scholar
Kingman, J. F. C. (1965) The heavy traffic approximation in the theory of queues. In Proc. Symp. Congestion Theory, ed. Smith, W. L. and Wilkinson, W. E., University of North Carolina Press, Chapel Hill, 137–169.Google Scholar
Köllerström, J. (1976) Stochastic bounds for the single-server queue. Math. Proc. Camb. Phil. Soc. 80, 521–525.CrossRefGoogle Scholar
Köllerström, J. (1978) Stochastic bounds for the queue GI/G/1 in heavy traffic. Math. Proc. Camb. Phil. Soc. 84, 361–375.Google Scholar
Köllerström, J. (1979) Heavy traffic theory for queues with several servers. II. J. Appl. Prob. 16, 393–401.Google Scholar
Lemoine, A. J. (1976) On random walks and stable GI/G/1 queues. Maths. Operat. Res. 1, 159–164.CrossRefGoogle Scholar
Lindley, D. V. (1952) Theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
Rudin, W. (1964) Principles of Mathematical Analysis, 2nd edn. McGraw-Hill, New York.Google Scholar
Whitt, W. (1974) Heavy traffic limit theorems for queues: a survey. In Mathematical Methods in Queueing Theory. Proceedings 1973, ed. Clarke, A. B.. Lecture Notes in Economics and Mathematical Systems 98, Springer-Verlag, Berlin, 307–350.CrossRefGoogle Scholar