Knockout Concentration Principle SINGLE-STAGE KNOCKOUT SWITCH

KNOCKOUT-BASED SWITCHES 144 Fig. 6.3 Operation of a barrel shifter.

6.1.2 Knockout Concentration Principle

All cells passing through the cell filters enter the concentrator, with an N-to-L concentration. If there are k F L cells arriving in a time slot for a given output, these k cells will emerge from the concentrator on outputs 1 to k after getting out of the concentrator. If k L, then all L outputs of the Ž . concentrator will have cells, and k y L cells will be dropped i.e., lost within the concentrator. The cell loss probability is evaluated as follows. It is assumed that, in every time slot, there is a fixed and independent probability ␳ that a cell arrives at an input. Every cell is equally likely destined for each output. Denote by P k the probability of k cells arriving in a time slot all destined for the same output, which is binomially distributed as follows: k Nyk ␳ ␳ N P s 1 y , k s 0, 1, . . . , N. 6.1 Ž . k ž ž ž k N N SINGLE-STAGE KNOCKOUT SWITCH 145 Fig. 6.4 Concentrator cell loss performance. KNOCKOUT-BASED SWITCHES 146 It then follows that the probability of a cell being dropped in a concentrator with N inputs and L outputs is given by N k Nyk 1 ␳ ␳ N w x Pr cell loss s k y L ⭈ 1 y . 6.2 Ž . Ž . Ý ž ž ž k ␳ N N ksLq1 Taking the limit as N ™ ⬁, and with some manipulations, L k y␳ L y␳ L ␳ e ␳ e w x Pr cell loss s 1 y 1 y q 6.3 Ž . Ý ž ž ␳ k L ks0 Ž . Ž . Ž . Using 6.2 and 6.3 , Figure 6.4 a shows a plot of the cell loss probability versus L, the number of outputs on the concentrator, for ␳ s 0.9 and N s 16, 32, 64, ⬁. Note that a concentrator with only eight outputs achieves a cell loss probability less than 10 y6 for arbitrately large N. This is comparable to the probability of losing a 500-bit cell from transmission errors with a bit y9 Ž . error rate of 10 . Also note from Figure 6.4 a that each additional output added to the concentrator beyond eight results in an order of magnitude decrease in the cell loss probability. Hence, independent of the number of Ž . inputs N , a concentrator with 12 outputs will have a cell loss probability y10 Ž . less than 10 . Figure 6.4 b illustrates, for N ™ ⬁, that the required number of concentrator outputs is not particularly sensitive to the load on the switch, up to and including a load of 100. It is also important to note that, assuming independent cell arrivals on each input, the simple, homoge- neous model used in the analysis corresponds to the worst case, making the cell loss probability performance results shown in Figure 6.4 upper bounds on w x any set of heterogeneous arrival statistics 10 .

6.1.3 Construction of the Concentrator