Sojourn time asymptotics in the M/G/1 processor sharing queue |
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Authors: | Zwart AP Boxma OJ |
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Institution: | (1) Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands;(2) CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands |
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Abstract: | We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer,
iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes
transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution.
We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time
distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic
limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm
expansions are provided for the case that the service time has a Pareto distribution.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | M/G/1 queue processor sharing service discipline sojourn time distribution heavy tailed distributions regular variation heavy traffic theory |
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