Switching - Fabric An Engineering Approach to Computer Networking

Switching - Fabric

An Engineering Approach to Computer Networking

April 3, 2000 Communication Networks 2

Switching

Number of connections: from few (4 or 8) to huge (100K)


Switching - Basic Assumptions

continuous streams

telephone connections no bursts no buffers

connections change

multicast

Blocking external internal

re-arrangeable strict sense non-blocking wide sense non-blocking


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


Time division switching Key idea: when demultiplexing, position in frame determines

output limk

Time division switching interchanges sample position within a frame: time slot interchange (TSI)


Example - TSI

sessions: (1,2) (2,4) (3,1) (4,3)

4 3 2 12 4 1 3TSI


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


Space division switching

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


Crossbar

Simplest possible space-division switch

Crosspoints can be turned on or off


Crossbar - example

1

2

3

4

1 2 3 4

sessions: (1,2) (2,4) (3,1) (4,3)


Crossbar

Advantages: simple to implement simple control strict sense non-blocking

Drawbacks number of crosspoints, N2

large VLSI space vulnerable to single faults


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

2 1

4 3

MUX

MUX

1

2

3

4


Time-Space: Example

2 1

3 4

2 1

4 3TSI

31

24

Internal speed = double link speed

time 1

time 2


Finding the schedule

Build a graph nodes - input links session connects an input and output nodes.

Feasible schedule

Computing a schedule compute perfect matching.


Time-space-time (TST) switching

Allowed to flip samples both on input and output trunk

Gives more flexibility => lowers call blocking probability


Circuit switching - Space division

graph representation transmitter nodes receiver nodes internal nodes

Feasible schedule edge disjoint paths.

cost function number of crosspoints internal nodes


Example

sessions: (1,3) (2,6) (3,1) (4,4) (5,2) (6,5)


Clos Network

Clos(N, n , k)N - inputs/outputs;

nxk (N/n)x(N/n) kxn

N=6n=2k=2

3x3

3x3

2x2

2x2

2x2

2x2

2x2

2x2


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.


Proof


Example

Clos(8,2,3)

N=8n=2k=3

4x4

4x4

3x2

3x2

3x2

2x3

2x3

2x3

2x3 4x4 3x2


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 ?


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.


Recursive constructions

N/2 x N/2

N/2 x N/2

.

.

.

.

.

.

1

n

1

n


Algorithm Complexity

N routing steps for each level

Log n levels to do

===> N log n

Given parallel hardware: O(N) time


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!!!


Another algorithm – matrix based

Specification matrix: Rows = input switches (n) Columns = middle switches (N/n = r) Content = output switches (1..n)

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


Algorithm Principle

For j=0,1,2,… balance the destination j among the columns

Challenges: efficiency termination

We describe an Algorithm by Lee, Hwang, and Carpinelli, T.COM. 44 (11), Nov 1996


The Matrix Meaning

0

1

4x4

4x4

3x2

3x2

3x2

2x3

2x3

2x3

2x3 4x4 3x2


Data Structures

(j,e), j=0,1,…n-1, e=0,1,…,r-1 :- the set of rows {i} such that sij=e

0(e), e=0,1,r-1 :- the set of columns {j} such that Si does not contain e

2(e), e=0,1,r-1 :- the set of columns {j} such that Si contains e at least twice


AlgorithmInit: e=0

1. If 2(e) empty

ee+1if e=r stop; else goto 1

2. if 2(e)

j 1st element of 2(e)k 1st element of 0(e)

3. (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

4. (Next Simple Swap)

If e> sik

i’ 2nd element of sik

repeat step 3 on i’

if e> si’k goto step 5

5. (Successive Swap)A. u e;

remove k from 0(u)if if |(j,u)|=2 remove j from 2(u)

B. v sik

swap sij with sik

remove i from (j,u) and (k,v)add i to (j,v) and (k,u)

C. if e<vif |(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

D. if e>vuvgoto step 5B


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)


Recursive constructions - Benes Network

N/2 x N/2

N/2 x N/2

.

.

.

.

.

.

1

n

1

n


Clos network size

Number of switching elements is given by

for k=n (rearrangeable non-blocking)

Optimal value for n is n=sqrt{2}N, which yields

22

22n

NNk

n

Nk

n

NknC

nNNn

n

NNnC

122 2

2

2/3)122( NC


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?

)log(33.1log2

144.1log

2!2

222

2

1

NNONNNNc

eNN NNC


Example 16x16


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


Strict Sense non-Blocking

N/2 x N/2

N/2 x N/2

.

.

.

.

.

.N/2 x N/2


Cantor Networks

m copies of Benes network.

For m >= log N its strict sense non-blocking

Network size N log2 N

Example

Proof.


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!


Banyan

111

110


Other Banyans

Omega or

shuffle exchange


How do deal with internal blocking in Banyan

speed – up

internal buffers

Batcher bitonic sorter


Documents

Switching - Fabric An Engineering Approach to Computer Networking