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Asymptotic results for multiplexing subexponential on-off processes

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Predrag R. Jelenković  and
Aurel A. Lazar
Predrag R. Jelenković*
Affiliation:
Columbia University
Aurel A. Lazar*
Affiliation:
Columbia University
*
∗ Postal address: Department of Electrical Engineering and Center for Telecommunications Research, Columbia University, New York, NY 10027, USA.
∗ Postal address: Department of Electrical Engineering and Center for Telecommunications Research, Columbia University, New York, NY 10027, USA.
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Abstract

Consider an aggregate arrival process AN obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period.

Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process At∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable QtP observed at the beginning of the arrival process activity periods

where ρ = 𝔼At∞ < c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation.

In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate 𝔼et. This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value 𝔼et.

Keywords

Non-Cramér type conditionssubexponential distributionslong-tailed distributionslong-range dependencynetwork multiplexerfluid flow queueM/G/∞ queue

MSC classification

Secondary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 394 - 421
Copyright
Copyright © Applied Probability Trust 1999 

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