The GBAR Model in .NET

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The GBAR Model
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The DAR model may not be suitable for a single source (by a single source we mean a source that does not interact with other sources) as described above. Lucantoni et al. [22] give three areas where single source models are useful: studying what types of traf c descriptors make sense for parameter negotiation with the network at call setup, testing rate control algorithms, and predicting the quality of service degradation caused by congestion on an access link. For this reason, Heyman [11] proposed the GBAR model. Lakshman et al. [21] use the GBAR model to predict frame sizes in a rate control algorithm. Lucantoni et al. [22] propose a Markov-renewal process model to describe a single source. This model has the advantage of being very general, and the disadvantage that it is not parameterized by some simple summary statistics of the data trace. The GBAR model exploits the properties enjoyed by teleconferencing traf c described in Section 12.2.2, the geometrically decaying autocorrelation
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TABLE 12.3 CAC Performance for Video Conference A and Video Conference C Video Conference A B C Ksim KCDE Video Conference B
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57 9 50 44 5 0.5 9 23 7 1 8.5 9.5 10 11 45 67 81 103 125 145 195 245 270 310 110 185 280 375 20 30 40 50 60 70 98 128 139 156 16 30 49 66 16 25 33 44 53 63 90 120 130 150 15 30 50 70
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function and the negative-binomial (or gamma) marginal distributions, to produce a simple model based on the three parameters that describe these features. The GBAR(1) process was introduced by McKenzie [23], along with some other interesting autoregressive processes. (As with the DAR model we will drop the argument (1).) Two inherent features of this process are the marginal distribution is gamma and the autocorrelation function is geometric. Toward de ning the GBAR model, let Ga b; l denote a random variable with a gamma distribution with shape parameter b and scale parameter l; that is, the density function is fG t l lt b lt e ; G b 1 t > 0: 12:3
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Similarly, let Be p; q denote a random variable with a beta distribution with parameters p and q; that is, with density function fB t G p q t p 1 t q ; G p 1 G q 1 0 < t < 1; 12:4
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where p and q are both larger than 1. The GBAR model is based on two wellknown results: the sum of independent Ga a; l and Ga b; l random variables is a Ga a b; l random variable, and the product of independent Be a; b a and Ga b; l random variables is a Ga a; l random variable. Thus, if Xn 1 is Ga b; l , An is Be a; b a , and Bn is Ga b a; l , and these three are mutually independent, then Xn An Xn 1 Bn 12:5
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de nes a stationary stochastic process Xn with a marginal Ga b; l distribution. Furthermore, the autocorrelation function of this process is given by r k  k a ; b k 0; 1; 2; . . . 12:6
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The process de ned by Eq. (12.5) is called the GBAR processes. The G and B denote gamma and beta, respectively, and the AR stands for autoregressive. Since the current value is determined by only one previous value, this is an autoregressive process of order one. A possible physical interpretation of Eq. (12.5) is the following. Interpret An as the fraction of frame n 1 that is used in the predictor of frame n, so the rst term on the left of Eq. (12.5) is the contribution of interframe prediction. In hybrid PCM=DPCM coding [24], for example, resistance to transmission error is accomplished by periodically setting some differential predictor coef cients to zero and sending a PCM value. We can think of Bn as the number of cells to do that. If the
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