Control algorithm (N = 1 case) (1) De ne a virtual queueing process q1 (n) and set q1 (0) = 0. (2) q1 (n) = [q1 (n 1) + V1 (n) p ]+ . For n 0, we let V (n) = 0. (3) a1 (n) = [ p V1 (n + n 1 ) q1 (n + n 1 1)]+ . For n 0, we let a1 (n) = 0. For N NRT traf c sources the algorithm is modi ed as follows. Control algorithm (N > 1 case) (1) Set qi (0) = 0, 1 i N . (2) qi (n) = [qi (n 1) + Vi (n) p ]+ . For n 0, we let V (n) = 0. (3) ai (n) = [ p Vi (n + n i ) qi (n + n i 1)]+ /N . (4) For n 0, we let ai (n) = 0. 21.5.1.2 Performance example Three different alternatives for predictor are used [81]. In A1, the xed low-pass lter is chosen as HLPF (z) = (1/4)(1 + z 1 + z 2 + z 3 ). The MMSE predictor is designed as follows. First, a low-pass lter A1 is applied to the high-priority RT traf c. Next, a standard Mi (i) minimizing mean square error linear predictor HMMSE (z) with the form m=0 Bm z ni m is calculated based on the low-frequency part of the RT traf c. The nal MMSE predictor is z ni Hi (z) = HLPF (z)HMMSE (z). Note that HMMSE (z) will require explicit knowledge of the round-trip delays per- ow information n i . In A3 that information is approximated by an average value n 0 for all ows. The prediction error is de ned as ( j) = V ( j) V1 ( j). The queue length at the node of interest is bounded by [79]

( j)

Under the de nitions and predictive ow control algorithm de ned above, if Vmax < and [( p V )/( V )] H1 (1) 1, we have [81] q(n) q0 (n) 2C1 , where C1 is a constant that does not depend on n. For l = n n 0 we de ne the accumulated error as

( j) =

( j)

( j)

C ( j1 j2) =

C ( j1 j2)

(21.36)

For the results shown below, V (n) is a generated Gaussian process which is multitimescale-correlated with Cv (k) = 479.599 0.999|k| + 161.787 0.99|k| +

10 4 10 8 Var {Xn,1} 6 4 2 0

40 I (ms)

Figure 21.13 Var{X n,l } with different predictors. (Reproduced by permission of IEEE [81].)

100 10 1 10 2 P (Q>x) 10 3 10 4 10 5 10 6

600 x (b)

Figure 21.14 Tail (cumulative queue length) probabilities with different control algorithms. (Reproduced by permission of IEEE [81].) 498.033 0.9|k| and V = 100 kb/s. This type of source has often been used to represent the multiple time-scale correlation in network traf c [71, 80]. The link capacity is 200 kb/s, and the utilization is set to 98%. The time unit is 1 ms and the unit of the queue length is 1 bit [81]. Var{X n,l }is shown in Figure 21.13. We can see that, for A1 and A3 predictors, Var{X n,l } converges to some constant when l is large enough. The asymptotic variance of A3 is also smaller than that of Al. In Figure 21.14, we compare the above control algorithm with the control algorithm that uses Equation (21.34). As we mentioned before, when V1 (n) p for all n, the control algorithms reduce to the same linear equation. However,