crossbar switches crossbar switches are an important general architecture for fast switches. 2 x 2...
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Crossbar SwitchesCrossbar Switches• Crossbar switches are an important general
architecture for fast switches. • 2 x 2 Crossbar Switches
• A general N x N crossbar switch
Input Queueing versus Output Input Queueing versus Output QueueingQueueing
• Input Queueing -- "If we come in together then we wait together"
• Output Queueing -- "We wait at the destination (output) together"
The queueing will be at the The queueing will be at the input or at the outputinput or at the output??
• The switch fabric speed is equal to the input line speed – To avoid collision on the single speed switch fabric, only one input
line can can place a packet on the switch fabric at a time. This requires the other inputs to stop the packet from entering the switch
fabric. This is implemented using an queue at the input.
• The switch fabric speed is N times faster than the input line speed – The internal switch has slot times which are N times as fast as
those of the input lines. The packets enter the crossbar switch together and are shifted to the outputs together. This requires queueing at the outputs to avoid collisions.
General Assumptions for General Assumptions for AnalysisAnalysis
• In any given time slot, the probability that a packet will arrive on a particular input is p. Thus p represents the average utilization of each input.
• Each packet has equal probability 1/N of being addressed to any given output, and successive packets are independent.
Analysis of Output Queueing
mA
1111
p= load
Pr[ ] ( )( ) (1 )ip pN i N i
m i N Na A i
!!( )!N
i N ipe as N
!i pp ei
Poisson Distribution.
• Switch with Speedup factor of N.• Arriving packets reach the targeted output ”immediately”.• = # arriving packets at the tagged queue during a given time slot m
Analysis of the Output Queue Size : the number of packets in the tagged queue at the end of
the time slot m
Using a standard approach in queueing analysis
mQ
1max(0, 1)m mmQ Q A
(1 )
(1 )(1 )(1 )
(1 )(1 )( )( ) (1 )(1 )
N
p z
p z if Np pz zN Np zQ z
A z z p z if Ne z
2/ /1
1
( 1) ( 1)[ ( )]( ) 2(1 ) M D
z
N p NdQ Q z QN Nd z p
The mean stead-state queue size
The mean queue size for an M/D/1
queue
As / /1, M DN Q Q
The State transition diagram for the output queue size
1a
2
1a
0 1
2a
0a0 1a a
2a
0a
…2a
0a
3a
4a
3a
The Steady-State Queue Size The Steady-State Queue Size ProbabilitiesProbabilities
0Pr( 0) (1 ) pq Q p e
0 2 3Pr( 1) ( ...) Pr( 0)a Q a a Q 0 11 00
(1 )Pr( 1)
a aq Q qa
20 0 2 3Pr( 2) Pr( 0) ( ...) Pr( 1)a Q a Q a a a Q
1 22 1 00 0
(1 )Pr( 2)
a aq Q q qa a
…
20 0 2 3Pr( ) Pr( ) ( ...) Pr( 1)
n
ii
a Q n a Q n i a a a Q n
110 02
(1 )Pr( )n n
nin i
i
aq Q n qa
aqa
Analysis of the Packet Waiting Time
1 2W W W
The time slots that packet must wait while packets that arrived in earlier time slots are transmitted
The time slots that packet must wait additionally until it is randomly selected out of the packet arrivals in the time slot m
Analysis of the Packet Waiting Time
• b: the size of the batch the packet arrives in
1(1 )(1 )( ) ( )
( )p zW z Q zA z z
1 1mW Q
2 20
0 1
1 1
Pr[ ] Pr[ | ] Pr[ ]
10 Pr[ ] Pr[ ]
1 1Pr[ ]
i
k
i i k
ii k i k
W k W k b i b i
b i b ii
b i ai p
Pr[ ] i iia iab i
pA
21 ( )( )
(1 )A zW z
p z
Analysis of the Packet Waiting Time
• the mean steady-state waiting time
1 21( ) ( ) ( ) ( )
1 )A zW z W z W z Q zz
( )(
2/ /1
( 1) ( 1)1 [ ]2 2(1 )
M DN p NW Q A A WN Np p
The mean waiting time for an M/D/1
queue
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