18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFIC

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Fig. 18.1 1.95.

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Hurst parameter estimates (R=S and variance time) for a varying from 1.05 to

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the protocol stack since the observed traf c pattern is a direct consequence of hosts exchanging les whose transport was mediated through protocols (e.g., TCP, owcontrolled UDP) in the protocol stack. This provides us with a natural environment where the impact of control actions by a congestion control protocol can be discerned and evaluated under self-similar traf c conditions.

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18.3 18.3.1

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PREDICTABILITY OF SELF-SIMILAR TRAFFIC Predictability Setup

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In this section, we show that the correlation structure present in long-range dependent (LRD) traf c can be detected and used to predict the future over time scales relevant to congestion control. Time series analysis and prediction theory have long histories with techniques spanning a number of domains from estimation theory to regression theory to neural network based techniques to mention a few [3, 17, 22, 40, 44, 45, 49]. In many senses, it is an ``art form'' with different methods giving variable performance depending on the context and modeling assumptions. Our goal is not to perform optimal time series prediction but rather to choose a simple, easyto-implement scheme, and use it as a reference for studying congestion control techniques and their ef cacy at exploiting the correlation structure present in LRD traf c for improving network performance. Our prediction method, which is described next, is a time domain technique and can be viewed as an instance of conditional expectation estimation.

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CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC

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Assume we are given a wide-sense stationary stochastic process xt tPZ and two numbers T1 ; T2 > 0. At time t, we have at our disposal a P

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iP t T1 ;t

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qi ;

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where qi is a sample path of xt over time interval t T1 ; t . For notational clarity, let V1 P

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iP t T1 ;t

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V2

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iP t;t T2

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xi :

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a may be thought of as the aggregate traf c observed over the ``recent past'' t T1 ; t and V1, V2 are composite random variables denoting the recent past and near future. We are interested in computing the conditional probability PrfV2 bjV1 ag 18:3

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for b in the range of V2 . For example, if a represented a ``high'' traf c volume, then we may be interested in knowing what the probability of encountering yet another high traf c volume in the near future would be. Let

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t Vmax max

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iP t T 1;t

qi ;

t Vmin min

iP t T1 ;t

t t where t t kT1 ; k 0; 1; . . . ; Vmax and Vmin denote the highest and lowest traf c volume seen so far at time t, respectively. t To make sense of ``high'' and ``low,'' we will partition the range between Vmax and t t t Vmin into h levels with quantization step m Vmax Vmin =h: t t t t t 0; Vmin m ; Vmin m; Vmin 2m ; Vmin 2m; Vmin 3m ; . . . t t t Vmin h 2 m; Vmin h 1 m ; Vmin h 1 m; I ;

We will de ne two new random variables L1 ; L2 where

t Lk 1 D Vk P 0; Vmin m ; t t Lk 2 D Vk P Vmin m; Vmin 2m ; . . . t t Lk h 1 D Vk P Vmin h 2 m; Vmin h 1 m ; t Lk h D Vk Vmin h 1 m; I :

In other words, Lk is a function of Vk ; Lk Lk Vk ; and it performs a certain quantization. Thus if Lk % 1 then the traf c level is ``low'' relative to the mean, and if Lk % h, then it is ``high.''

18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFIC

In our case, eight levels h 8 were found to be suf ciently granular for t t prediction purposes. In practice, Vmax and Vmin are determined by applying a 3% threshold to the previously observed traf c volumes, i.e., the outliers corresponding to extraordinarily large or small data points are dropped to make the classi cation reasonable. Returning to Eq. (18.3) and prediction, for certain values of T1 , T2 , we are interested in knowing the conditional probability densities PrfL2 jL1 lg for l P 1; 8 . If PrfL2 jL1 8g were concentrated toward L2 8, and PrfL2 j L1 1g were concentrated toward L2 1, then this information could be potentially exploited for congestion control purposes. 18.3.2 Estimation of Conditional Probability Density

To explore and quantify the potential predictability of self-similar network traf c, we use TCP traf c traces used in Park et al. [33] whose Hurst parameter estimates are shown in Fig. 18.1 as the main reference point. First, we use off-line estimation of aggregate throughput traf c, which is then re ned to on-line estimation of aggregate traf c using per-connection traf c when performing predictive congestion control. Other traces including those collected from ow-controlled UDP runs yield similar results. The traces used are each 10,000 seconds long at 10 ms granularity. They represent the aggregate traf c of 32 concurrent TCP Reno connections recorded at a bottleneck router. We observe that the aggregate throughput series exhibit correlation structure at several time scales from 250 ms to 20 s and higher. To estimate PrfL2 jL1 lg from the aggregate throughput series Xt , we segment Xt into N 10;000 seconds T1 T2 seconds

contiguous nonoverlapping blocks of length T1 T2 (except possibly for the last block), and for each block j P 1; N compute the aggregate traf c V1 , V2 over the subintervals of length T1 , T2 . For l; l H P 1; 8 , let hl P 0; N denote the total number of blocks such that L1 V1 l and let hlH P 0; hl denote the size of the subset of those blocks such that L2 V2 l H . Then PrfL2 l H jL1 lg hl H : hl

Figure 18.2 shows the estimated conditional probability densities for a 1:05, 1.95 traf c for time scales 500 ms, 1 s, and 5 s. In the following, T1 T2 .

Fig. 18.2 Top row: Probability densities with L2 conditioned on L1 for a 1:05. Bottom row: Probability densities with L2 conditioned on L1 for a 1:95.