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An LIL Version of L = W

Department of Operations Research, Stanford University, Stanford, California 94305
Room 2C-178, AT & T Bell Laboratories, Murray Hill, New Jersey 07974

This paper establishes a law-of-the-iterated-logarithm (LIL) version of the fundamental queueing formula L = W: 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 = W 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.

Key Words: queueing theory; Little's law; conservation laws; law of the iterated logarithm; limit theorems; inverse stochastic processes; random sums