Based on the above terminology we state below a set of
useful identities, for any switch Sk,j , in an N x N CNET:
( N - 2) > k > 0 (1)
{ H L }= {LH}
{H}"{L}" = {L}"{H}" m, n < rnin (k, N - 2
-
k) (2)
= {H}m{L}m
m < rnin (k, N - 2 - k ) ( 3 )
{HL}" = { H m m o d N } { p m o d N }
{HL}"
m < rnin (k, N
{HL}"
Lm mod N = { H L }m mod N
{H,r}mmodN
-
2 - k ) (4)
for all m (5)
= (HLJmmodN
reduction rule at level 0 (6)
- L mmodN
inverse reduction rule at level 0 (7)
ON THE CNET: A REARRANGEABLY
NO N B LOCK1NG 0PTICAL
INTERCON N ECTlON NETWORK
Indexing terms: Switching and switching circuits, Optical
switching, Optical communications, Networks
A planar, circular, rearrangeably nonblocking optical switching architecture using N ( N - 1) Ti : LiNbO, directional
coupler switches is suggested. Various figures of merit like
the signal-to-crosstalk ratio (SXR), insertion loss and fault
tolerance are presented. The outline for a distributed routing
algorithm in O ( N )time is given.
We propose an interconnection network the CNET, with a
hierarchical planar ring structure and N ( N - 1) optical
switches, which is rearrangeably nonblocking (Fig. 1). The
CNET and its properties are described first and its possible
optical implementation is outlined later on. In an N x N
CNET, there are N - 1 levels numbered from 0 to N - 2.
Each switch Si, (jth switch in the ith level) is connected to
two other switches, as shown in Fig. 1. Let the connection to
level k + 1 from level k be a H-connection, and the connection
to level k - 1 from level k be an L-connection. Any path @ is
a sequence of H- and L-connections, denoted as {H}"{L}",
called an 'operator sequence', for that source and destination
pair. The sequence { X X X .. . (n terms)} is referred to as { X } " .
The number of X operator sequences in a path @ is denoted
by n(X, @), or simply n(X). Weighted X indicates the
weighted sum w where w = weight associated with the position of X counted from the right with the LSB as 0.
1
A
03
Fig. 1 Digraph representation of 6 x 6 CNET (Ii = ith input; O i= ith
output)
For an N x N CNET to be nonblocking, there must be at
least ( N - 1) levels of N switches each. Consider a digraph of
degree 4 representing an N x N CNET with i levels (e.g.
i = N - 1 in Fig. 1) with a cut QQ' across all i levels. We
notice that cyclic shifts are permutations that require the
longest paths, and therefore, the largest number of switches.
We choose one such bijection from the family of
permutations' d;, k : (a2, al) + (a2, nt-j)(al)), j < N (cyclic
shifts within segments of size 2'"-j)) where n f ) : x + x + (k
mod 2") (cyclic shift of amplitude k). For 0 < j < N , d;, can
be realised with fewer than ( N - 1) levels of N switches each.
However, for j = 0, k = N - 1, across QQ, there must be at
least ( N - 1) edges, as there are (IV - 1) distinct inputs to be
connected to (N - 1) distinct outputs on either side of QQ'.
Thus there must be at least ( N - 1) levels in an N x N CNET.
If a path @ is to be found from Sk, to Sk+,. j + m , then n(H,
@) - n(L, @) = n, and there are at least (m mod N ) { H L }
sequences in @ if n < 0, and at least ((m- n) mod N ) { H L }
pairs if n > 0. If the difference in level is n, then there must be
at least n level transitions, i.e. I n(H) - n(L)I = I n I. If n > 0,
then n(H) > n(L) and if n < 0 then n(H) < n(L). Thus, n(H,
@) - n(L, @) = n. If n, L level transitions to S,,,,, j + m are
necessary, then they must occur from Sk,j + m . To reach Sk,j + m
from Sk,j , at least (m mod N ) { H L } pairs are required. If nHtraversals are necessary to reach S,,,, j + m then they must
occur from Sk,j + m - n . To reach S,, j + m - n from Sk, we require
at least (m - n) mod N { H L } pairs.
@', ...,
for all n > 0 and n E I, be a set
Let @ =
of paths between any input-output pair. Let Si.xbe a switch
in
i = 1, ..., n. We choose the path that includes min
(Si,x) for all
E @, where the min (Sx,J function is defined
as min (Sx,y , Sm,") = S x , y ,if x < m and min (S,...,,, Sx, = Sx,y
if y < z. If more than one
has the same minimum element
Si, x , i.e.
E a1and Si,xzE
and Si,x l = rnin (a1)
=
min (a2),
then we take the sum of all x over Si,
x over all
Si,x E
as the deciding criterion, i.e. we choose that path
which has the maximum weighted L. The routing strings are
arranged in order of decreasing weighted L. Using eqns. 1-7,
it is possible to arrive at the optimal weighted L routing string
if the source and destination are specified. The value of the
'most significant bit' (either 'H'or 'L')in this string is taken as
the routing bit and the switch is set accordingly.
As regards fault tolerance, we assume a fault model as
follows: any network component (switch or link) in levels 0 to
N - 4 can fail, faults occur independently, link failures are
equivalent to partial or total switch failures and retention of
full access. The CNET is N ( N - 3) fault tolerant, as it can
tolerate N faults in each of the N - 3 levels. To provide a
basis for the comparison of switching architectures with
various degrees of fault tolerance and numbers of switches, we
introduce a figure of merit, the normalised fault tolerance
(NFT), which is defined as the ratio of the maximum number
of faults that can be tolerated to the total number of switches.
On this basis, the NFT of the CNET is ( N - 3)/(N - 1). The
upper bound on the number of passes is N .
Optical switching architectures have been reviewed, in
detail, by Spanke.' When implemented using integrated optic
directional couplers3 as the basic switching elements, the
worst case insertion loss for an N x N CNET is given by
I, = (2N - 2) . L + 2 . W + (2N - 3) . G (in dB), where
(al,
ai,
an),
ai
ai
ai)
(c
L = directional coupler insertion loss, W = waveguide to fibre
coupling loss and G = inter-switch waveguide loss. The differential loss is given by 1; = (2N - 3)L + (2N - 5)G, in dB (for
N 2 2). The worst case insertion loss occurs for path AB for a
6 x 6 CNET, while minimum loss occurs for the path A’B’
(Fig. 1).
In a nonfaulty CNET, we find that the worst case SXR
will arise when the signal path passes through (2N - 2)
switches (path AB as shown in Fig. 1). Thus, SXR =
10 log,, ((1 - 10x/10)(2N-2)/(1
- (1 - 10”’10)(ZN-2))},
where x
is the crosstalk per switch. Typical differential loss and signal
to crosstalk ratio values for the CNET as a function of the
number of inputs are shown in Figs. 2 and 3, respectively. It
may be mentioned here that if the CNET is faulty such that
all switches in levels 0 to N - 4 are unusable, the signal path
would run along the two outer-most levels. Under such fault
conditions only one path would be set up at a time. The
remaining N - 1 paths would be set up in successive passes.
L:07
/
the width of each level is Wd, interswitch waveguide length is a
fraction ‘a’ of the device length ‘D’,
the number of devices that
can be put in level 0 is given by N = n/{2 sin-’ (D(l + a)/
2(R - (N - I)&)}, where R is the radius of the outermost
level. For N = 32, u = 0.2 and D = 6mm, R is equal to
73.34 mm and thus the above configuration can be realised
easily.
4Or
31
c\
”::
‘
‘06
/
N , n u m b e r of Inputs
1891131
Fig. 3 Differential loss against number of inputs
However, due to the narrow annular geometry, much of
substrate real estate at the centre of the wafer goes unutilised.
This area could be effectively utilised for putting local electronic switch controllers if a suitable optoelectronic substrate
like GaAs/AlGaAs or InP/GaInAsP is chosen.
0
12
24
36
N, number of inputs
40
60
189112/
Fig. 2 Signal to crosstalk ratio against number ofinputs
G = 0,4dB/cm
W = 2.0dB
With a system budget of say 30dB and a 5dB system
margin, we can have a worst case insertion loss of 25dB.
Hence, from Fig. 2, taking L = 0.3dB, the maximum subsystem size would be a 32 x 32 CNET. For an SXR of 12dB,
the permissible subsystem size can be 32 x 32, assuming
x = -30dB. This is significantly larger than other reported
optical switching architectures realised using directional couplers. Assuming that the substrate wafer diameter is 76 mm,
24th November 1988
S. KAR
A. SELVARAJAN
Department of Electrical Communication Engineering
Indian Institute of Science
blangalore-560 012, India
References
1
LENFANT, I.: ‘Parallel permutations of data: a Benes network
control algorithm for frequently used permutations’, IEEE Trans.
Comput., 1983, pp. 637-647
2 SPANKE, R. A , : ‘Architectures for guided-wave optical spaceswitching systems’, IEEE Commun. Mag., 1987,25, pp. 42-48
R. c.: ‘Ti : LiNbO, integrated optic
3 KOROTKY, s. K.,and ALFERNFS,
technology’, in HUTCIIESON, L. D. (Ed.): ‘Integrated optical circuits
and componentsdesign and applications’ (Marcel Dekker, New
York, 1987), pp. 169-227