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Large deviations, moderate deviations, and queues with long-range dependent input

Part of: Special processes Limit theorems Operations research and management science Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Cheng-Shang Chang ,
David D. Yao  and
Tim Zajic
Cheng-Shang Chang*
Affiliation:
National Tsing Hua University
David D. Yao*
Affiliation:
Columbia University
Tim Zajic*
Affiliation:
University of Minnesota
*
∗ Postal address: Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C.
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, USA.
∗∗∗ Postal address: School of Mathematics, University of Minnesota, 514 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, USA. Email address: zajic@math.umn.edu
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Abstract

Long-range dependence has been recently asserted to be an important characteristic in modeling telecommunications traffic. Inspired by the integral relationship between the fractional Brownian motion and the standard Brownian motion, we model a process with long-range dependence, Y, as a fractional integral of Riemann-Liouville type applied to a more standard process X—one that does not have long-range dependence. When X takes the form of a sample path process with bounded stationary increments, we provide a criterion for X to satisfy a moderate deviations principle (MDP). Based on the MDP of X, we then establish the MDP for Y. Furthermore, we characterize, in terms of the MDP, the transient behavior of queues when fed with the long-range dependent input process Y. In particular, we identify the most likely path that leads to a large queue, and demonstrate that unlike the case where the input has short-range dependence, the path here is nonlinear.

Keywords

Sample path large deviations and moderate deviationslong-range dependenceweak convergencequeues

MSC classification

Primary: 60F10: Large deviations 60K25: Queueing theory 90B22: Queues and service
Secondary: 60G18: Self-similar processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 254 - 278
Copyright
Copyright © Applied Probability Trust 1999 

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