doi:10.1016/j.peva.2004.05.002
Copyright © 2004 Published by Elsevier B.V.
Filtering and forecasting problems for aggregate traffic in Internet links*1
a Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA
b Department of Computer and Systems Science, University of Rome “La Sapienza”, Via Eudossiana 18, 00184, Rome, Italy
Received 30 July 2003;
Revised 22 March 2004.
Available online 14 July 2004.
Abstract
An important problem in bandwidth allocation and reservation over a communication link is to estimate the traffic bit rate in that link. This can be done by using specific tools for measurements of the traffic bit rate. However, the obtained measures are affected by some noise. Moreover, one might be interested in future traffic forecasting, when a prediction is needed. In this paper, an iterative filtering procedure is proposed for updating the traffic estimate upon the arrival of a new measurement. A birth and death stochastic model is assumed for the traffic bit rate to provide dynamical equations for the average behavior in the absence of information carried by measurements. Approximate solutions of the same updating problem are also given under the assumption that the posterior distribution of the traffic bit rate belongs to a specific class (beta or Gaussian distribution). This leads to approximate filtering procedures, which are expected to provide significant computational advantages. Finally, results obtained by processing simulated and real data are presented; stressing that the practical behavior of the approximate filters is quite satisfactory.
Author Keywords: Internet traffic measurement; Measurement noise; Filtering; Forecasting
Fig. 1. Simulated data. N=50;C=1;λ=0.05;μ=0.005;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 2. Simulated data. N=50;C=1;λ=0.05;μ=0.005;n(iT)=−2 (with prob. 0.2), −1 (with prob. 0.2), 0 (with prob. 0.2), 1 (with prob. 0.2), 2 (with prob. 0.2).
Fig. 3. Simulated data. N=50;C=1;λ=0.067;μ=0.003;n(iT)=−2 (with prob. 0.2), −1 (with prob. 0.2), 0 (with prob. 0.2), 1 (with prob. 0.2), 2 (with prob. 0.2).
Fig. 4. Simulated data. N=50;C=1;λ(t)=0.005,t
[0,400]
(800,1200](1600,2000],0.05,t(400,800](1200,1600];μ=0.005;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 5. Simulated data. N=50;C=1;λ(t)=0.05,t[0,800](1600,2000],0.08,t(800,1600];μ(t)=0.005,t[0,400](1200,2000],0.01,t(400,1200];n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 6. Input traffic on nordunet interface of Abilene router in NYC on 20 May 2003 with N=35;C=1;λ=0.05;μ=0.04;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 7. Output traffic on nordunet interface of Abilene router in NYC on 20 May 2003 with N=30;C=1;λ=0.01;μ=0.04;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 8. Input traffic on SOX interface of Abilene router in Atlanta on 20 May 2003 with N=30;C=1;λ=0.05;μ=0.05;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 9. Output traffic on SOX interface of Abilene router in Atlanta on 20 May 2003 with N=65;C=1;λ=0.05;μ=0.05;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 10. Robustness analysis by simulated data; exact parameter values: N=50,C=1,λ=0.05,μ=0.005; modified parameter values: N=60,C=1,λ=0.06,μ=0.006;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 11. Robustness analysis by simulated data; exact parameter values: N=50,C=1,λ=0.05,μ=0.005; modified parameter values: λ=0.075,μ=0.0075;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Fig. 12. Robustness analysis by simulated data; exact parameter values: N=50,C=1,λ=0.05,μ=0.005; modified parameter values: N=60;n(iT)=−1 (with prob. 0.15), 0 (with prob. 0.7), 1 (with prob. 0.15).
Corresponding author. Tel.: +1-404-894-6616; fax: +1-404-894-7883.
*1 The work of T. Anjali and C. Scoglio was supported by NSF under award number 0219829.
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