Exercise Set 10:
Switching Fabric Topologies
Assigned: Mon. 16 May 2005 (week 11)  Due: Wed. 25 May 2005 (week 12)
[Lecture:
5.2 Switching Fabrics] 
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10.1
Rearrangably NonBlocking Benes Network
Consider an 8x8 Benes network made of 2x2 switches
(i.e. apply both steps of the recursive construction
not just the first step),
in an oldstyle telephony (circuit switching) context.
Make the following input>output connections
(a permutation of the numbers 0 through 7):

0>3, 1>5, 2>6, 3>0,
4>1, 5>2, 6>7, 7>4.
Next, verify that if inputs 0 and 3
wish to mutually exchange destinations
(i.e. their previous connections are torn down
and we wish to make the new connections 0>0 and 3>3),
then we also need to modify the routing of several other connections
as well.
10.2
Clos/Benes made of 4x4 Switching Elements
Draw a 64x64 rearrangeably nonblocking fabric
made of 4x4 switching elements.
Repeatedly use the Clos network construction,
with parameters
(4, x, 4, x, 4).
You will endup with a Benesstyle network
(I am not sure if the Benes network is only defined using 2x2 switches,
or if it is also generalized for larger switching elements;
if the latter is true, then I guess you will end up with a Benes network
not just a Benesstyle network).
Reminder:
a Clos network with parameters
(IN, N1, N2, N3, OUT)
has
N1 swithing elements in its first stage,
N2 elements in its second stage,
N3 elements in its third stage,
and each firststage swithing element has
IN input ports,
while each thirdstage swithing element has
OUT output ports.
10.3
NonBlocking Clos Network using Inverse Multiplexing
Show that a packetswitching
(k, m, k, m, k) Clos Network,
made out of
kxk and
mxm nonblocking switches,
is itself nonblocking,
assuming that the first and thirdstage switches
implement
kway inverse multiplexing
through the
k middlestage switches.
Assume that inverse multiplexing
is at the granularity of individual flows,
where flows are defined by the pair of
input
and output port numbers.
Your proof should be a simple generalization
of the proof given in class for the Benes network
which happens to be the case
k=2.
10.4
Maximum Degree of Internal Blocking in Banyan Networks
Draw a 16x16 banyan fabric made of 2x2 switching elements.
[
Reminder:
an
NxN banyan network made of 2x2 switches
has log
_{2}N stages made of
N/2 switches each,
and looks like
N balanced binary trees
with
N/2 leafswitches (and
N output ports) each,
where the
N trees share their leaf nodes
as well as a number of other internal nodes,
with the amount of sharing progressively decreasing
from the leaves to the root].
Following that, find and draw on the fabric
a set of telephonystyle connections
from the N=16 inputs to the N=16 outputs
(i.e. a permutation of the numbers 0 through 15)
that produces the worst possible amount of internal blocking
that you can come up with,
i.e. where there is a link through which
as many different connections as possible need to pass.
You should be able to find such a permutation
that requires 4 different connections to go through a single link
in the 16x16 banyan.
In general, the theory says that this worst possible congestion
is equal to the square root of N
for an NxN banyan;
try to proove this, or give an argument for it.
10.5
16Port Full Fat Tree
Draw a 16port nonblocking fat tree made of 4x4 switching elements.
In order for the fat tree to be nonblocking,
each switching element must dedicate
as many links to the "upwards" direction
(the direction to "remote" ports)
as the number of links that it dedicates to the "downwards" direction
(the direction to "local" ports)
hence, in our case, two (2) links upwards and two (2) links downwards.
Show a set of telephonystyle connections (a permutation)
that uses up all of the links up to the third level of switches;
then, show another permutation that uses up
all of the links across the entire tree.
10.6
Cost Comparison of Full Fat Tree versus Benes Network
A "full" fat tree, i.e. a nonblocking fat tree,
resembles very much a folded Benes network
and has approximately half the number of levels (or stages)
when compared to its corresponding Benes network;
remote traffic traverses these levels twice
once going upwards and once going downwards
so remote traffic traverses as many stages as in the Benes network,
but local traffic traverses much fewer levels (or switches).
Because of this relationship in the number of levels (stages),
fat trees give the impression that they offer nonblocking operation
at a lower cost, when compared to Benes networks;
this impression is
not true, however, as this exercise shows.
The reason is that fat trees use switching elements with
twice the number of links,
as compared to their corresponding Benes network.
Consider the ratio:
(Number of switching elements in a network) /
(Number of ports of the network)
as a function of the number of ports of the network,
for networks built out of switching elements
of a given size (2k)x(2k).
Plot this ratio for NxN full fat trees and Benes networks,
as a function of N, on a logarithmic horizontal scale;
assume that both networks are made of 4x4 switching elements
(notice that a fat tree that uses 4x4 elements is a binary tree,
whereas the Benes network made of 4x4 elements
has a quad fanout).
The comparison is not straightforward
because not all sizes N produced by one network
are possible sizes of the other network too;
however, a general trend should be observable
once you compute and plot a few points on each curve.
If you have enough time,
repeat for 6x6 or 8x8 switching elements,
and/or compute a general formula that shows the relationship.
Give an intuitive explanation
why an input port can "see" more output ports,
when it goes through a given number of switches,
in a Benes (or banyan) network,
as compared to the number of output ports that it can "see"
when it goes up and down a variable number of similar switches
up to the same maximum number of switches in a tree.
10.7
3/5Ratio Fat Tree
Use 8x8 switching elements to make a fat tree network
where each switching node dedicates 5 of its 8 ports
to the "downwards" direction,
and the remaining 3 of its ports to the "upwards" direction
hence, this network will have
some internal blocking.
Actually, notice that this network will exhibit no internal blocking
as long as the nonlocal traffic out of or into each subtree
does not exceed
3^{h} times the port throughput,
where
h is the heigth of the subtree.