Cross-Stuck and Toggle-Stuck Cases

KNOCKOUT-BASED SWITCHES 182 X Ž . P k is defined as the cell loss rate of the k th input with an expansion L ratio L under a faulty SWE. Note that the cell loss rate discussed here is not for the entire MOBAS, but only for the SM that has a faulty SWE.

6.4.4.1 Cross-Stuck and Toggle-Stuck Cases

As discussed before, a TS fault can be isolated by setting the TS SWE to a cross state, which Ž . resembles the occurrence of the CS fault at this SWE see Fig. 6.27 . Thus, the effect on cell loss performance of the CS and TS faults is the same. We will analyze only the cell loss performance of the SM with a CS SWE. Ž . Ž . If the CS fault of SWE i, j is not in the last column j L , there is no Ž . performance degradation. When it is in the last column j s L , the CS SWE does not affect the cell loss performance of the inputs except for the one in the same row of the faulty SWE. Ž . Ž . For a fault CS i, L i G L , only one input arrival pattern contributes to the cell loss performance degradation. That is when there are exactly L cells destined for this output port. Among these cells are L y 1 cells from inputs 1 to i y 1; one cell from input i, and no cells from the inputs i q 1 to N Ž . Ž . L F i F N . The average number of additionally lost cells, AL i , due to the Ž . faulty SWE i, j depends on the fault row position i: Ly1 iyL Nyi ␳ ␳ ␳ ␳ ␳ i y 1 N y i AL i s 1 y 1 y Ž . ž ž ž ž ž ž L y 1 N N N N N L NyL ␳ ␳ i y 1 s 1 y , L q 1 F i F N. ž ž ž L y 1 N N Ž . Ž . Note that a faulty SWE i, L i F L y 1 does not introduce any cell loss performance degradation. Under the cross-state reconfiguration for a faulty Ž . SWE i, L , the cell loss rate of each input is as follows: ° P k , 1 F k F i y 1, Ž . L X ~ P k , k s i, Ž . P k s Ž . Ly1 L ¢ P k y P A , i q 1 F k F N. Ž . Ž . L Ž . where P A is the probability of the event A, the k th input’s active cell L inputs, located above the A s ½ 5 is routed successfully k th input, have active cells Ly1 iyL kyi ␳ ␳ ␳ ␳ ␳ i y 1 k y i P A s 1 y 1 y Ž . ž ž ž ž ž ž L y 1 N N N N N L kyL ␳ ␳ i y 1 s 1 y , i - k F N . ž ž ž L y 1 N N A FAULT-TOLERANT MULTICAST OUTPUT-BUFFERED ATM SWITCH 183 Ž . Fig. 6.32 Cell loss rate of each input port after cross-reconfiguration. 䊚1994 IEEE. Ž . The probability P A is considered as the cell loss rate improvement of the k th input Figure 6.32 shows the cell loss rate of each input after the SWE array is reconfigured. Here, the MOBAS is assumed to be a single stage with a size of 64 = 64 and an expansion ratio L of 12. The faulty SWE is assumed to be Ž . Ž . SWE 35, 12 . For those inputs above the faulty input 35 , the cell loss rates are the same as in the fault-free case, while the faulty input’s cell loss rate is equal to that of the L s 11 fault-free case. The inputs below the faulty one Ž . from 36 to 64 have almost the same cell loss rates as in the fault-free case, Ž . since the improvement P A is very small.

6.4.4.2 Vertical-Stuck Case