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Admission Control for Multidimensional Workload input with Heavy Tails and Fractional Ornstein-Uhlenbeck Process

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  22 February 2016

Amarjit Budhiraja ,
Vladas Pipiras  and
Xiaoming Song
Amarjit Budhiraja*
Affiliation:
University of North Carolina
Vladas Pipiras*
Affiliation:
University of North Carolina
Xiaoming Song*
Affiliation:
University of North Carolina
*
∗ Postal address: Department of Statistics and OR, University of North Carolina, Hanes Hall, Chapel Hill, NC 27599, USA.
∗ Postal address: Department of Statistics and OR, University of North Carolina, Hanes Hall, Chapel Hill, NC 27599, USA.
∗ Postal address: Department of Statistics and OR, University of North Carolina, Hanes Hall, Chapel Hill, NC 27599, USA.
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Abstract

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The infinite source Poisson arrival model with heavy-tailed workload distributions has attracted much attention, especially in the modeling of data packet traffic in communication networks. In particular, it is well known that under suitable assumptions on the source arrival rate, the centered and scaled cumulative workload input process for the underlying processing system can be approximated by fractional Brownian motion. In many applications one is interested in the stabilization of the work inflow to the system by modifying the net input rate, using an appropriate admission control policy. In this paper we study a natural family of admission control policies which keep the associated scaled cumulative workload input asymptotically close to a prespecified linear trajectory, uniformly over time. Under such admission control policies and with natural assumptions on arrival distributions, suitably scaled and centered cumulative workload input processes are shown to converge weakly in the path space to the solution of a d-dimensional stochastic differential equation driven by a Gaussian process. It is shown that the admission control policy achieves moment stabilization in that the second moment of the solution to the stochastic differential equation (averaged over the d-stations) is bounded uniformly for all times. In one special case of control policies, as time approaches ∞, we obtain a fractional version of a stationary Ornstein-Uhlenbeck process that is driven by fractional Brownian motion with Hurst parameter H > ½.

Keywords

Poisson random measureGaussian random measureself-similarityheavy-tailed distributionfractional Brownian motionfractional Ornstein-Uhlenbeck processadmission control

MSC classification

Primary: 60G18: Self-similar processes 60G57: Random measures
Secondary: 60G15: Gaussian processes 90B15: Network models, stochastic

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 476 - 505
Copyright
© Applied Probability Trust 

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