A TWO-STAGE MULTICAST OUTPUT-BUFFERED ATM SWITCH
163
Note that the in MTTs that are connected to the same MGN2 contain identical information, because the copy of a multicast cell can appear
randomly at any one of L M output links to MGN2. As compared to those in
1
w x
18, 26 , the translation table size in the MTT is much smaller. This is because the copy of a multicast cell can only appear at one of L M output
1
w x
links in the MOBAS, vs. N links in 18, 26 , resulting in fewer table entries in the MTT. In addition, since the VCI values of replicated copies are not
stored in the MTT’s, the content of each table entry in the MTT is also less.
6.3.4 Multicast Knockout Principle
A new multicast knockout principle, an extension of generalized knockout principle, has been applied to the two-stage MOBAS to provide multicasting
Ž .
capability. Since the switch module SM in the MGN performs a concentra- Ž
. tion function e.g., N to L M , it is also called a concentrator.
1
6.3.4.1 Cell Loss Rate in MGN1
In the analysis, it is assumed that the traffic at each input port of the MOBAS is independent of that at the other
inputs, and replicated cells are uniformly delivered to all output groups. The average cell arrival rate, , is the probability that a cell arrives at an input
port in a given cell time slot. It is assumed that the average cell replication in
w x
w x
MGN1 is E F , the average cell replication in MGN2 is E F , the average
1 2
w x cell duplication in the OPC is E D , and the random variables F , F , and D
1 2
are independent of each other. Ž
. Every incoming cell is broadcast to all concentrators SMs , and is prop-
erly filtered at each concentrator according to the multicast pattern in the cell header. The average cell arrival rate, p, at each input of a concentrator is
w x
E F
1
p s ,
K Ž
. where K s NrM is the number of concentrators in MGN1. The probabil-
Ž .
ity A that k cells are destined for a specific concentrator of MGN1 in a
k
given time slot is N
Nyk k
A s p
1 y p
Ž .
k
ž
k
k Nyk
w x w x
E F
⭈ M
E F
⭈ M
1 1
N s
1 y ,
0 F k F N, 6.8
Ž .
ž
ž ž
k N
N w
x where E F ⭈ MrN is the probability of a cell arriving at the input of a
1
specific concentrator in MGN1.
KNOCKOUT-BASED SWITCHES
164
Ž .
As N ™ ⬁, 6.8 becomes
k
w x
E F
⭈ M
Ž .
1 y Ew F x⭈M
1
A s e 6.9
Ž .
k
k where should satisfy the following condition for a stable system:
w x w x w x
E F E F
E D - 1.
1 2
Since there are only L M routing links available for each output group, if
1
more than L M cells are destined for this output group in a cell time slot,
1
excess cells will be discarded and lost. The cell loss rate in MGN1, P , is
1
N
w x w x
k y L M A E F E D
Ž .
Ý
1 k
2
ksL Mq1
1
P s 6.10
Ž .
1
w x w x
NpE F E D
2
N
1 s
k y L M
Ž .
Ý
1
w x
E F
⭈ M
1
ksL Mq1
1
=
k Nyk
w x w x
E F
⭈ M
E F
⭈ M
1 1
N 1 y
. 6.11
Ž .
ž
ž ž
k N
N Ž
. Both the denominator and the numerator in
6.10 include the factor
w x w x
E F E D to allow for the cell replication in MGN2 and the OPC. In other
2
words, a cell lost in MGN1 could be a cell that would have been replicated Ž
. w
x w x in MGN2 and the OPC. The denominator in 6.10 , NpE F E D , is the
2
average number of cells effectively arriving at a specific concentrator Ž
.
N
Ž .
during one cell time, and the numerator in 6.10 , Ý k y L M
ksL Mq1 1
1
w x w x
A E F E D , is the average number of cells effectively lost in the specific
k 2
concentrator. Ž
. As N ™ ⬁, 6.11 becomes
k L M
L M
yN p yN p
1
1
L M Np
e Np
e
Ž .
Ž .
1
P s 1 y 1 y
q
Ý
1
ž
ž
Np k
L M
Ž .
1
ks0
k
L M
y Ew F x⭈M
1
1
w x
L M
E F ⭈
M e
Ž .
1 1
s 1 y 1 y
Ý
ž
ž
w x
E F M
k
1
ks0
L M
1
y Ew F x⭈M
1
w x
E F
⭈ M
e
Ž .
1
q 6.12
Ž .
L M
Ž .
1
A TWO-STAGE MULTICAST OUTPUT-BUFFERED ATM SWITCH
165
Ž .
Note that 6.12 is similar to the equation for the generalized knockout principle, except that the parameters in the two equations are slightly
different due to the cell replication in MGN1 and MGN2 and to the cell duplication in the OPC.
Figure 6.20 shows the plots of the cell loss probability at MGN1 vs. L for
1
Ž w
x w x w x.
various fanout values and an offered load s E F E F E D of 0.9 at
1 2
Ž each output port. Again it shows that as M increases i.e., more outputs are
. sharing their routing links , the required L value decreases for a given cell
1
loss rate. The requirements on the switch design parameters M and L are
1
more stringent in the unicast case than in the multicast. Since the load on MGN1 decreases as the product of the average fanouts in MGN2 and the
Fig. 6.20 Cell loss probability vs. the group expansion ratio L in MGN1.
1
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
