In this chapter, we evaluate the impact of long-range dependence on a single network element (multiplexer or router) of a data network, this element being modeled as a ¯uid queueing system. Such a system has constant output rate C and is fed by a number N of . on=off traf®c sources. The case where the number of sources N is ®xed is treated, for instance, in Boxma and Dumas [4]. | Self-Similar Network Traffic and Performance Evaluation Edited by Kihong Park and Walter Willinger Copyright 2000 by John Wiley Sons Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X 5 HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON OFF SOURCES F. Brichet and A. Simonian France Télécom CNET 92794 Issy-Moulineaux Cédex 9 France L. MASSOULUB Microsoft Research Ltd. Cambridge Cfi2 3NH United Kingdom D. Veitch Software Engineering Research Centre Carlton Victoria 30S3 Australia INTRODUCTION In this chapter we evaluate the impact of long-range dependence on a single network element multiplexer or router of a data network this element being modeled as a fluid queueing system. Such a system has constant output rate C and is fed by a number N of . on off traffic sources. The case where the number of sources N is fixed is treated for instance in Boxma and Dumas 4 . For N sufficiently large C exceeds the peak rate of an individual source. If the ratio source peak rate total output rate C decreases with N . is proportional to 1 N then we are in the realm of small sources. In the meantime we consider the heavy load case when the output rate C is slightly larger than the total mean input rate so that the queue is almost always nonempty. This heavy load situation associated with small sources in a fluid queueing system motivates the derivation of limit theorems for both input traffic and queue occupancy processes when the number N increases to infinity as detailed below. More precisely represent an on off source by mutually independent alternating silence periods A and activity periods B. When active the source emits data at 115 116 HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON OFF SOURCES constant rate its peak rate taken as unity. Given E d 1 a and E t 1 the activity probability of a source is then v a a and we require that C Nv so that the queue has a stationary regime. Provided that the probability density of duration A B satisfies a simple regularity condition we
