Cell Loss Rate in MGN2

KNOCKOUT-BASED SWITCHES 166 OPC increases, the cell loss rate of MGN1 in the multicast case is lower than that in the unicast case. The replicated cells from a multicast call will never Ž . contend with each other for the same output group concentrator , since the MOBAS replicates at most one cell for each output group. In other words, an MGN1 that is designed to meet the performance requirement for unicast calls will also meet the one for calls.

6.3.4.2 Cell Loss Rate in MGN2

Special attention is required when analyzing the cell loss rate in MGN2, because the cell arrival pattern at the inputs of MGN2 is determined by the number of cells passing through the Ž . corresponding concentrator in MGN1. If there are l l F L M cells arriving 1 at MGN2, these cells will appear at the upper l consecutive inputs of MGN2. If more than L M cells are destined for MGN2, only L M cells will arrive at 1 1 Ž . MGN2’s inputs one cellper input port , while excessive cells are discarded in MGN1. The cell loss rate of a concentrator in MGN2 is assumed to be P , and the 2 probability that l cells arrive at a specific concentrator in MGN2 to be B . l Both P and B depend on the average number of cells arriving at the inputs 2 l Ž of MGN2 i.e., the number of cells passing through the corresponding . Ž . concentrator in MGN1 . This implies that P is a function of A in 6.8 . In 2 k order to calculate P , the probability of j cells arriving at MGN2, denoted as 2 I , is found to be j A for j - L M , ° j 1 N ~ I s j A for j s L M. Ý k 1 ¢ ksL M 1 Ž . If j j F L M cells arrive at MGN2, they will appear at the upper j 1 consecutive inputs of MGN2. Since how and where the cells appear at the MGN2’s inputs does not affect the cell loss performance, the analysis can be simplified by assuming that a cell can appear at any input of the MGN2. Let us denote by B the probability that l cells arrive at the inputs of a l j specific concentrator in MGN2 for given j cells arrived at MGN2. Then, j jyl l B s q 1 y q , 0 F j F L M, 0 F l F j, Ž . l j 1 ž l w x where q is equal to E F rM under the assumption that replicated cells are 2 uniformly delivered to the M concentrators in MGN2. Ž . If no more than L cells arrive at MGN2 0 F j F L , no cell will be 2 2 discarded in MGN2, because each concentrator can accept up to L cells 2 Ž during one cell time slot. If more than L cells arrive at MGN2 L F j F 2 2 . L M , cell loss will occur in each concentrator with a certain probability. 1 A TWO-STAGE MULTICAST OUTPUT-BUFFERED ATM SWITCH 167 Since replicated cells are assumed to be uniformly distributed to all M outputs in MGN2, the probability B that l cells are destined for a specific l output port of MGN2 in a cell time slot is L M 1 B s B I . Ý l l j j jsl The cell loss rate in MGN2, P , is 2 L M 1 w x l y L B E D Ž . Ý 2 l lsL q1 2 P s 6.13 Ž . 2 w x ␳ E F 1 w x N ⭈ 1 y P E F Ž . 1 2 K w x ⭈ E D M L M 1 l y L B Ž . Ý 2 l lsL q1 2 s . w x w x ␳ E F ⭈ 1 y P E F Ž . 1 1 2 w x Here N ⭈ ␳ E F rK is the average number of cells destined for a concentra- 1 Ž w x .Ž . tor in MGN1 from the inputs of the MOBAS, and N ⭈ ␳ E F rK 1 y P is 1 1 the average number of cells that have survived in this concentrator, which in turn becomes the average number of cells arriving at the correspond- Ž . Ž w x .Ž . ing MGN2. Thus, the denominator in 6.13 , N ⭈ ␳ E F rK 1 y P 1 1 w x w x E F E D rM, is the average number of cells effectively arriving at a specific 2 Ž . L 1 M Ž . w x output port. The numerator in 6.13 , Ý l y L B E D , is the average lsL q1 2 l 2 number of cells effectively lost in a specific concentrator in MGN2, because the lost cell can be a cell that will be duplicated in the OPC. Figure 6.21 shows the plots of the cell loss probability at MGN2 vs. L 2 for various average duplication values and an effective offered load Ž w x w x w x. s ␳ E F E F E D of 0.9. The average fanout on MGN1 is assumed to be 1 2 Ž w x . Ž . 1.0 E F s 1.0 , the group size to be 32 M s 32 , and the expansion ratio 1 Ž . to be 2.0 L s 2.0 . Since the traffic load on MGN2 decreases as the 1 Ž w x. average cell duplication E D increases, the cell loss rate in MGN2 de- w x creases as E D increases. Therefore, the switch design parameter L is 2 more stringently restricted in the unicast case than in the multicast. Conse- quently, if MGN2 is designed to meet the performance requirement for unicast calls, it will also satisfy multicast calls’ performance requirement. KNOCKOUT-BASED SWITCHES 168 Fig. 6.21 Cell loss probability vs. group expansion ratio L in MGN2. 2

6.3.4.3 Total Cell Loss Rate in the MOBAS