Switching - Fabric

1
Switching - Fabric

• Based on
• An Engineering Approach to Computer Networking/
  Keshav

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

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

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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Example - TSI
sessions (1,2) (2,4) (3,1) (4,3)
2 4 1 3
TSI
4 3 2 1
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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 gt impossible
  with current technology
• Need to look to other techniques

8
Space division switching

• Each sample takes a different path through the
  switch, depending on its destination

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Crossbar

• Simplest possible space-division switch
• Crosspoints can be turned on or off

10
Crossbar - example
sessions (1,2) (2,4) (3,1) (4,3)
1
2
input ports
3
4
4
1
2
3
output ports
11
Crossbar

• Advantages
• simple to implement
• simple control
• strict sense non-blocking
• Drawbacks
• number of crosspoints, N2
• large VLSI space
• vulnerable to single faults

12
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 switchs schedule

MUX
1
2 1
2
3
MUX
4 3
4
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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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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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Time-space-time (TST) switching

• Allowed to flip samples both on input and output
  trunk
• Gives more flexibility gt lowers call blocking
  probability

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Circuit switching - Space division

• graph representation
• transmitter nodes
• receiver nodes
• internal nodes
• Feasible schedule
• edge disjoint paths.
• cost function
• number of crosspoints
• internal nodes

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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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Clos Network
Clos(N, n , k) N - inputs/outputs
kxn
nxk
(N/n)x(N/n)
2x2
3x3
N6 n2 k2
2x2
3x3
2x2
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Clos Network - strict sense non-blocking

• Holds for k gt 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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Proof
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Example
Clos(8,2,3)
2x3
4x4
3x2
2x3
4x4
3x2
N8 n2 k3
3x2
2x3
4x4
3x2
2x3
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Clos Network - rearrangable

• Holds for k gt 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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Clos Network
An example for blocking when kn
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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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Recursive constructions
1
1
N/2 x N/2
N/2 x N/2
n
n
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Algorithm Complexity

• N routing steps for each level
• Log n levels to do
• gt N log n
• Given parallel hardware O(N) time

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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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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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Algorithm Principle

• For e0,1,2,
• 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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The Matrix Meaning

0

1
2x3
4x4
3x2
2x3
4x4
3x2
3x2
2x3
4x4
3x2
2x3
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The Matrix Meaning

0

1
2x3
4x4
3x2
1 3 4 0 7 5 6 2
2x3
4x4
3x2
3x2
2x3
4x4
3x2
2x3
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The Matrix Meaning

0

1
2x3
4x4
3x2
0 1 2 0 3 2 3 1
2x3
4x4
3x2
3x2
2x3
4x4
3x2
2x3
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Illegal example
0 1
0 2
3 2
3 1
2x2
2x2
0 1 2 0 3 2 3 1
2x2
4x4
2x2
2x2
2x2
4x4
2x2
2x2
34
Legal example
1 0
0 2
2 3
3 1
2x2
2x2
0 1 2 0 3 2 3 1
2x2
4x4
2x2
2x2
2x2
4x4
2x2
2x2
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Notation
Nnxr
j

e


nxm
rxr
mxn
nxm
rxr
mxn
i
r rows
mxn
nxm
rxr
mxn
nxm
m columns
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Data Structures

• (j,e), j0,1,n-1, e0,1,,r-1 - the set of rows
  i such that sije
• All the input switches routed to e thru j
• 0(e), e0,1,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), e0,1,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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Algorithm

• (Next Simple Swap)
• If egt sik
• i ? 2nd element of (j,e)
• repeat step 3 on i
• if egt sik 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 eltv
• 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

• Init e0
• If 2(e) empty
• e?e1
• if er 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 elt 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)

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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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Clos network size

• Number of switching elements is given by
• for kn (rearrangeable non-blocking)
• Optimal value for n is nsqrtN/2, which yields

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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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Benes Networks

• Size
• 2 log N 1 stages
• N/2 switches in each stage
• N log2 N N/2
• Rearrangeable
• Clos network with k2 n2
• Symmetry
• Example.
• proof

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Recursive constructions - Benes Network
1
1
N/2 x N/2
N/2 x N/2
n
n
43
Example 16x16
44
Strict Sense non-Blocking
N/2 x N/2
. . .
. . .
N/2 x N/2
N/2 x N/2
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Cantor Networks

• m copies of Benes network.
• For m gt log N it is strict sense non-blocking
• Network size N log2 N
• Example

m4
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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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Banyan
111
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Other Banyans
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How do deal with internal blocking in Banyan

• speed up
• internal buffers
• Batcher bitonic sorter

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Batcher sorter size

• nlog N sorters with 1,2,3,.. Columns
• Each column has N/2 switches