Crossbar Switches Crossbar switches are an important general architecture for fast switches. 2 x 2 Crossbar Switches A general N x N crossbar switch.

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

Analysis of the Packet Waiting Time

• The steady-state waiting time probabilities:

1 20

0 1

0 0

Pr[ ] Pr[ ]

1

1[1 ]

k

n

k

n in i k n

k k n

n in i

W k W n and W k n

q ap

q ap