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Functional Large Deviation Principles for Waiting and Departure Processes

Published online by Cambridge University Press:  27 July 2009

Anatolii A. Puhalskii  and
Ward Whitt
Anatolii A. Puhalskii
Affiliation:
Department of Mathematics, University of Colorado at Denver, Denver, Colorado 80217-3364, puhalski@math.cudenver.edu, Institute for Problems in Information Transmission, 19 Bolshoi Karetnii, Moscow 101447, Russia
Ward Whitt
Affiliation:
AT&T Labs, Florham Park, New Jersey 07932-0971, wow@research.att.com
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Abstract

We establish functional large deviation principles (FLDPs) for waiting and departure processes in single-server queues with unlimited waiting space and the first-in first-out service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the nonuniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition, and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit expressions for the rate function of the departure process, but not more generally. In general, the rate function for the departure process evidently must be calculated numerically. We also obtain an FLDP for the departure process of completed work, which has important application to the concept of effective bandwidths for admission control and capacity planning in packet communication networks.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 12 , Issue 4 , October 1998 , pp. 479 - 507
Copyright
Copyright © Cambridge University Press 1998

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