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Fig. 12.16
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Mean buffer size versus traf c intensity for lm.
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12.4 WHY LONG-RANGE DEPENDENCE DOES NOT MATTER FOR VBR VIDEO
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In Eq. (12.8) assume that Xn is the superposition of N statistically identical and independent sources all of which have Gaussian N l; s2 marginal distributions and autocorrelation function r k . The normal distribution is needed for the large deviations estimate of the steady-state solution of Eq. (12.8). We make it to gain insight into the effect of r k on the solution and do not claim it is valid as a source model. We have been using d and c as the service rate and buffer capacity to which the source Xn is offered. In this section we emphasize that N sources are superposed. We take d and c as the service rate and buffer capacity per source, and we scale the service rate and the buffer size via D Nd and C Nc;
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which are the parameters in physical dimensions. In the stationary regime, let V be the buffer content, and F c F d; c; N PfV cg. The ``large N asymptotic'' is [5, 28] lim 1 log F d; c; N I d; c ; N 12:14
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Let mc be the in mizing m in Eq. (12.15); it is the value of m that maximizes the probability that the buffer over ows in m time periods, so Ryu and Elwalid call mc the the critical time scale. Ryu and Elwalid [28] make the following observations about mc . Only those correlations smaller than the critical time scale enter Eq. (12.14) explicitly (through Eq. (12.16), so only the rst mc correlations are meaningful in evaluating F d; c; N . Thus, if long-range dependence is to be an important effect, mc must be large. They show that when r is monotonically decreasing (which is a property observed in our video traces), then mc is nite. It is easily established that m0 1, so a continuity argument suggests that mc is small when c is small. When Xn is an exact long-range dependent (LRD) process, they prove that m c H ; % 1 H d l c 12:17
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LONG-RANGE DEPENDENCE AND QUEUEING EFFECTS FOR VBR VIDEO
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where H is the Hurst parameter. The traf c intensity is a l=d and d c=d is the time to empty the buffer. Rewriting Eq. (12.17) as m c % H d 1 H 1 a 12:18
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shows that long-range dependence can matter only when the buffer drain time is large and the traf c intensity is close to one. These conditions are not likely to occur in practice because d is limited by maximum delay considerations and the traf c intensity is limited by packet-loss rate requirements. 12.4.5 Simulation Experiments
An empirical comparison of short- and long-range dependent processes is to use them to drive simulations of Eq. (12.8) and compare output statistics. Ryu and Elwalid [28] use the superposition of a fractal-binomial-noise-driven Poisson process and a DAR(1) process to emulate a process with large short-range correlations that decay geometrically and also possess long-range dependence. They nd that the short-term correlations have the dominant impact on the value of the critical time scale and on the over ow probability. Rao et al. [27] take a scaling of a fractional ARIMA process to be the ``true'' X-process and approximate it by autoregressive processes that match autocorrelation functions at certain points. They conclude that when these processes are such that the initial portions of their autocorrelation functions match, then the buffer over ow probabilities will be comparable. A negative feature of these experiments is that the short- and long-range correlations cannot be varied independently, so the effect on the performance measures from changing one while holding the other xed cannot be obtained. Heyman et al. [14] used the discrete-time M =G=I queue to eliminate this feature. The arrival rate is l and the mean service time is 1=m. Makowski and Parulekar in 9 show that when r is a given autocorrelation function that is monotonic, then the service time distribution gk r k 1 r k 1 2r k ; m k ! 1; 12:19
will achieve r . In particular, we can choose r k to be ( rk ak ; const ; k 2 1 H k kx ; 12:20
k ! kx ;
which has geometrically decaying short-term correlations and hyperbolically decaying long-term correlations. We can use Eq. (12.19) as the service time distribution in