1. Based on An Engineering Approach to Computer Networking/ Keshav
Switching - Fabric
Based on
An Engineering Approach to Computer Networking/ Keshav

2. Communication Networks
Switching
Number of connections: from few (4 or 8) to huge (100,000s)
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Communication Networks

3. Switching - Basic Assumptions
continuous streams vs. packet by packet
telephone connections
no bursts
no buffers
connections change
multicast
Blocking
external
internal
re-arrangeable
strict sense non-blocking
wide sense non-blocking
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Communication Networks

4. Multiplexors and demultiplexors
Multiplexor: aggregates sessions
N input lines
Output runs N times as fast as input
Demultiplexor: distributes sessions
one input line and N outputs that run N times slower
Can cascade multiplexors
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Communication Networks

5. Time division switching
Key idea: when demultiplexing, position in frame determines output link
Time division switching interchanges sample position within a frame: time slot interchange (TSI)
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Communication Networks

6. Communication Networks
Example - TSI
sessions: (1,2) (2,4) (3,1) (4,3)
TSI
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Communication Networks

7. Communication Networks
TSI
Simple to build.
Multicast
Limit is the time taken to read and write to memory
For 120,000 circuits
need to read and write memory once every 125 microseconds
each operation takes around 0.5 ns => impossible with current technology
Need to look to other techniques
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Communication Networks

8. Space division switching
Each sample takes a different path through the switch, depending on its destination
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Communication Networks

9. Communication Networks
Crossbar
Simplest possible space-division switch
Crosspoints can be turned on or off
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Communication Networks

10. Communication Networks
Crossbar - example
sessions: (1,2) (2,4) (3,1) (4,3) 1 2
input
ports 3 4 4 1 2 3
output ports
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Communication Networks

11. Communication Networks
Crossbar
Advantages:
simple to implement
simple control
strict sense non-blocking
Drawbacks
number of crosspoints, N2
large VLSI space
vulnerable to single faults
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Communication Networks

12. Communication Networks
Time-space switching
Precede each input trunk in a crossbar with a TSI
Delay samples so that they arrive at the right time for the space division switch’s schedule 1
MUX
2 1 2 3
MUX
4 3 4
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Communication Networks

13. Communication Networks
Time-Space: Example
time 1
time 2
2 1
2 1
TSI
3 4
4 3 3 1 2 4
Internal speed = double link speed
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Communication Networks

14. Communication Networks
Finding the schedule
Build a graph
nodes - input links
session connects an input and output nodes.
Feasible schedule
Computing a schedule
compute perfect matching.
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Communication Networks

15. Time-space-time (TST) switching
Allowed to flip samples both on input and output trunk
Gives more flexibility => lowers call blocking probability
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Communication Networks

16. Circuit switching - Space division
graph representation
transmitter nodes
receiver nodes
internal nodes
Feasible schedule
edge disjoint paths.
cost function
number of crosspoints
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Communication Networks

17. Communication Networks
Example
sessions: (1,3) (2,6) (3,1) (4,4) (5,2) (6,5) 1 2 3 4 5 6 1 2 3 4 5 6
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Communication Networks

18. Communication Networks
Clos Network
Clos(N, n , k)
N - inputs/outputs;
nxk
(N/n)x(N/n)
kxn
2x2
3x3
2x2
N=6
n=2
k=2
2x2
3x3
2x2
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Communication Networks

19. Clos Network - strict sense non-blocking
Holds for k >= 2n-1
Proof:
Consider and idle input and output
Input box connected to at most n-1 middle layer switches
output box connected to at most n-1 middle layer switches
There exists a "free" middle switch.
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Communication Networks

20. Communication Networks
Proof
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Communication Networks

21. Communication Networks
Example
Clos(8,2,3)
2x3
4x4
3x2
2x3
4x4
3x2
N=8
n=2
k=3
2x3
3x2
4x4
2x3
3x2
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Communication Networks

22. Clos Network - rearrangable
Holds for k >= n
Proof:
Consider all input and output
find a perfect matching.
route the perfect matching
remaining network is Clos(N-n,n-1,k-1)
summary:
smaller circuit
weaker guarantee
Mulicast ?
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Communication Networks

23. An example for blocking when k=n
Clos Network
An example for blocking when k=n
nxk
(N/n)x(N/n)
kxn
N=6
n=2
k=2
3x3
2x2

24. Rearrangable Clos Network – routing algorithm
Start at some arbitrary 2x2 input switch, and route it to its destination through the upper switch.
Route the other output port of the 2x2 switch you reached to its input port through the lower switch.
If the other port in the input switched you reached has not been routed yet, route it through the upper middle switch.
Otherwise, select an arbitrary input port switch that was not yet used, and repeat the procedure.
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Communication Networks

25. Recursive constructions1 1 . .
N/2 x N/2
N/2 x N/2 n n
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Communication Networks

26. Communication Networks
Algorithm Complexity
N routing steps for each level
Log n levels to do
===> N log n
Given parallel hardware: O(N) time
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Communication Networks

27. Alternative View of the problem
A graph coloring problem:
Input/output switches = nodes
A match = link
Middle stage switches = colors
This is the well known “Coloring of bi-partite graph” problem.
Heuristics fail miserably!!!
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Communication Networks

28. Another algorithm – matrix based
Specification matrix:
Rows = input switches (nxm, 1..r)
Columns = middle switches (rxr, N/n = r, 1..m)
Content = output switches (1..r)
We need an algorithms that will fill up the matrix with a feasible routing:
Same number cannot appear in the same column
Works also for CLOS with redundancy
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Communication Networks

29. Communication Networks
Algorithm Principle
For e=0,1,2,…
balance the destination e among the columns
Challenges:
efficiency
termination
We describe an Algorithm by Lee, Hwang, and Carpinelli, T. Comm. 44 (11), Nov. 1996
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Communication Networks

30. Communication Networks
The Matrix Meaning 1
2x3
4x4
3x2
2x3
4x4
3x2
2x3
3x2
4x4
2x3
3x2
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Communication Networks

31. Communication Networks
The Matrix Meaning 1
2x3
4x4
3x2 1 3 4 7 5 6 2
2x3
4x4
3x2
2x3
3x2
4x4
2x3
3x2
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Communication Networks

32. Communication Networks
The Matrix Meaning 1
2x3
4x4
3x2 1 2 3
2x3
4x4
3x2
2x3
3x2
4x4
2x3
3x2
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Communication Networks

33. Communication Networks
Illegal example 1 2 3
2x2
2x2 1 2 3
2x2
4x4
2x2
2x2
2x2
4x4
2x2
2x2
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Communication Networks

34. Communication Networks
Legal example 1 2 3
2x2
2x2 1 2 3
2x2
4x4
2x2
2x2
2x2
4x4
2x2
2x2
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Communication Networks

35. Communication Networks
Notation
N=nxr j e
nxm
rxr
mxn
nxm i
rxr
mxn
r rows
nxm
mxn
rxr
nxm
mxn
m columns
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Communication Networks

36. Communication Networks
Data Structures
(j,e), j=0,1,…n-1, e=0,1,…,r-1 :- the set of rows {i} such that sij=e
All the input switches routed to e thru j
0(e), e=0,1,r-1 :- the set of columns {j} such that Si does not contain e
All middle switches that are not routing to e
2(e), e=0,1,r-1 :- the set of columns {j} such that Si contains e at least twice
All middle switches that have contention routing to e
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Communication Networks

37. Communication Networks
Algorithm
(Next Simple Swap)
If e> sik
i’  2nd element of (j,e)
repeat step 3 on i’
if e> si’k goto step 5
(Successive Swap)
u  e;
remove k from 0(u)
if |(j,u)|=2 remove j from 2(u)
v  sik
swap sij with sik
remove i from (j,u) and (k,v)
add i to (j,v) and (k,u)
if e<v
if |(k,v)|=0 add k to 0(v)
if |(k,v)|=1 remove k from 2(v)
if |(j,v)|=1 remove j from 0(v)
if |(j,v)|=2 add j to 2(v)
goto step 1
if e>v
uv
i element in (j,u)
goto step 5B
Init: e=0
If 2(e) empty
ee+1
if e=r stop; else goto 1
if 2(e)
j  1st element of 2(e)
k 1st element of 0(e)
(simple swap)
i 1st element of (j,e)
if e< sik swap sij with sik
e’  sik
remove i from (j,e) and (k,e’)
add i to (j,e’) and (k,e)
if |(j,e)|=1 remove j from 2(e)
if |(j,e’)|=1 remove j from 0(e’)
if |(j,e’)|=2 add j to 2(e’)
if |(k,e’)|=0 add k to 0(e’)
if |(k,e’)|=1 remove k from 2(e’)
remove k from 0(e)
goto step 1
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Communication Networks

38. Algorithm Complexity
adding an element to a set, choosing/removing the 1st/2nd element from a set take O(1)
 steps 5A 5B and 5D each take O(1) time
removing a generally positioned element from an r-set takes O(r) time  step 5C time complexity is in O(r) [ 2(v)-{k}, 0(v)-{j} ]
the looping in step 5 does not contain step 5C, only 5B and 5D 
the time complexity for step 5 is O(r)
Algorithm time complexity O(nr2)
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Communication Networks

39. Communication Networks
Clos network size
Number of switching elements is given by
for k=n (rearrangeable non-blocking)
Optimal value for n is n=sqrt{N/2}, which yields
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Communication Networks

40. a lower bound for the number of switching elements?
Assume we have 2x2 switching units.
We have N! switching permutation
Can we achieve this bound?
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Communication Networks

41. Communication Networks
Benes Networks
Size:
2 log N –1 stages
N/2 switches in each stage
N log2 N –N/2
Rearrangeable
Clos network with k=2 n=2
Symmetry
Example.
proof
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Communication Networks

42. Recursive constructions - Benes Network1 1 . .
N/2 x N/2
N/2 x N/2 n n
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Communication Networks

43. Communication Networks
Example 16x16
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Communication Networks

44. Strict Sense non-Blocking
N/2 x N/2 . .
N/2 x N/2
N/2 x N/2
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Communication Networks

45. Communication Networks
Cantor Networks
m copies of Benes network.
For m >= log N it is strict sense non-blocking
Network size N log2 N
Example:
m=4
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Communication Networks

46. Communication Networks
Banyan
Self routing!
Size:
log2 N stages
N/2 switches in each stage
0.5N log2 N elements
This is less than the lower bound!
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Communication Networks

47. Communication Networks
Banyan
111
110
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Communication Networks

48. Communication Networks
Other Banyans
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Communication Networks

49. How do deal with internal blocking in Banyan
speed – up
internal buffers
Batcher bitonic sorter
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Communication Networks

50. Communication Networks
Batcher sorter size
n=log N sorters with 1,2,3,.. Columns
Each column has N/2 switches
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Communication Networks