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A queue with semiperiodic traffic

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Juan Alvarez  and
Bruce Hajek
Juan Alvarez*
Affiliation:
University of Illinois at Urbana-Champaign
Bruce Hajek*
Affiliation:
University of Illinois at Urbana-Champaign
*
∗ Postal address: Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 1308 West Main Street, Urbana, IL 61801, USA.
∗ Postal address: Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 1308 West Main Street, Urbana, IL 61801, USA.
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Abstract

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In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

Keywords

Semiperiodic trafficdiffusion limitoverflowGaussian processqueueing

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60F99: None of the above, but in this section 60G35: Signal detection and filtering
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 160 - 184
Copyright
Copyright © Applied Probability Trust 2005 

Footnotes

Supported by the National Science Foundation under grants NSF ANR 99-80544.

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