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An LIL Version of L = λW

Peter W. Glynn and Ward Whitt
Mathematics of Operations Research
Vol. 13, No. 4 (Nov., 1988), pp. 693-710
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/3689952
Page Count: 18
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An LIL Version of L = λW
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Abstract

This paper establishes a law-of-the-iterated-logarithm (LIL) version of the fundamental queueing formula L = λW: Under regularity conditions, the continuous-time arrival counting process and queue-length process jointly obey an LIL when the discrete-time sequence of interarrival times and waiting times jointly obey an LIL, and the limit sets are related. The standard relation L = λW appears as a corollary. LILs for inverse processes and random sums are also established, which are of general probabilistic interest because the usual independence, identical-distribution and moment assumptions are not made. Moreover, an LIL for regenerative processes is established, which can be used to obtain the other LILs.

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