Cell Infocom Apr 2003

8/8/2019 Cell Infocom Apr 2003

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Input Queued Switches:

Cell Switching vs. Packet Switching

Abtin Keshavarzian

Joint work with

Yashar Ganjali, Devavrat Shah

Stanford University


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Background

Time is slotted

Data units of fixed size cells

Buffers at input ports (Input-Queued Switch)

To avoid HoL blocking , virtual output queues are

used

VOQ11

VOQ1N

VOQN1

VOQNN

Output 1

Output N

Input 1

Input N

Switching Fabric


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VOQ11

VOQ1N

VOQN1

VOQNN

Motivation

Packets have different lengths

Splitter module Combiner module (memory)

Packet delays are more important than Cell delays

Packet Based Scheduling algorithms

Switch


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Notation Arrival rate

: Number of cells arrived to VOQij up to time n

: Number of cells departed from VOQij up to time n : Number of cells queued at VOQij at time n

(SLLN) almost surely

)(nAij

)(nDij

ij

ij

n n

nAP!

gp

)(

lim

)(nZij

VOQ11

VOQ1N

VOQN1

VOQNN

Output 1

Output N

Input 1

Input N

Switching Fabric


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Admissibility and Rate Stability

The arrival rate matrix is admissible

iff

A switch under a matching algorithm is stable

(rate stable) if, almost surely,

][ ijP!0

! !ePN

i

ij Nj1

,...,11 ! !eP

N

j

ij Ni1

,...,11

ij

ij

n n

nP!

gp

)(lim


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MWM algorithm

A matching

MWM: At each time slot, select the matching

with maximum weight

maxarg nWnm

mm

1

!

!! ji ijij nmnnW , )()(,)( Z

NNijm v! ][m

!otherwise0

outputtoconnectedisinputif1 jimij

! !N

i

ij jm1

1 ! !N

j

ij i1

1

)()(max)( nWnWnW !!1

mm

m


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MWM Stability

McKeown et al showed that

MWM is stable under i.i.d. Bernoulli traffic

Dai and Prabhakar using Fluid model technique showed

MWM is stable for any admissible traffic

J. G. Dai and B. Prabhakar, The throughput of data switches with or without

speedup,INFOCOM 2000,pp. 556-564.

N. McKeown,V. Ananthram, and J. Walrand, Achieving 100% throughput in an

input-queued switch, INFOCOM 1996, pp. 296-302.


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Outline

Cell based algorithms review:

Stability concept

Maximum Weight Matching algorithm

Packet based algorithms Packet-Based Algorithms

PB-MWM and its stability

Packet Based Algorithms Classification

Work Conserving Waiting

Waiting Packet Based Algorithms

Conclusion


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Packet-Based Switching

Once the scheduler starts transmitting the

first cell of a packet, it continues until thewhole packet is received at output port


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first cell of a packet, it continues until thewhole packet is received at output port


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first cell of a packet, it continues until thewhole packet is received at output port.


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Stability of PB-MWM

PB-MWM is stable under regenerative

admissible traffic

Traffic is called regenerative if on average it requires a

finite time to reach the state where all ports are free if it

keeps using any fixed matching.

Bernoulli i.i.d. is a regenerative traffic.

M.A. Marsan, A. Bianco, P. Giaccone, E. Leonardi, and F. Nari, Packet Scheduling in Input-

Queued Cell-based switches, INFOCOM2001, pp. 1085-10

94


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Proof Outline

Matchingm(n) is k-imperfect if

For PB

-MWM:

Lemma: A scheduling algorithm is rate stable if

the average value of its weight is larger than

maximum weight matching minus a bounded

constant.

)) knn ! mm

_ a )(2)()(|)( * TNnWnZnW EE u


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Question

CB-MWM is stable under any admissible

traffic

PB-MWM is stable under any admissibleregenerative traffic.

Is the regenerative condition necessary?


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Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

me

22

11

12

21


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Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

me

22

11

12

21


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Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

me

22

11

12

21


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Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

me

22

A11

A12

A21


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Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

me

A22

A11

A12

A21


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Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

me

A22

A11

A12

A21


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2

3

Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

me

A22

A11

A12

A21


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4

Classification of PB algorithms

WorkConserving(non-waiting):

No input is left unmatched when it has a packetfor an unmatched output.

Waiting:

Input ports may wait(do not start sending apacket) forinfinite numberof time slots.

No work-conserving algorithm can be ratestable for all admissible traffic.


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PB-wMWM

I1

I/L

Segment #1 Segment #2

I/L

L L

Switch runs at speedup

Maximum packet length: L

If use usual PB-MWM Else wait till all ports are free.

PB-wMWM is rate stable for any admissible

traffic with known max packet length

1(,1[L

Lk

Lkn

I

I


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Modified PB-wMWM

Segment #1 Segment #2

I! /)2()2( eLM

)2(eL

I! /)1()1( eLM

)1(eL

The packet length is not known but has boundedexpectation

: the maximum length of packets left when

waiting starts during lth segment

Modified PB-wMWM is rate stable for any

admissible traffic with bounded packet length

)(lLe


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Conclusion

PB-MWM is rate stable under any admissibleregenerative traffic.

Work-conserving packet based algorithms can notbe rate stable for all admissible traffics

Waitingis essential PB-wMWM and its modified version are stable

under any admissible traffic (with bounded meanpacket length)

Future work: Find simpler algorithms that are stable for any

admissible traffic.


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Fluid model

: number of time slots matchingmbeing usedup to time n

)(nTm

!

!

!

1

1"

nnT

nTnTnn

nnAnZ

ijZijijij

ijijij

m

m

m

mm

)(

)1()(1)1()(

)()()(

}0{

!

x

x!

x

x

P!

"

ttT

t

tTm

t

tD

tDttZ

ijZij

ij

ijijij

m

m

m

m

)(~

)(~1

)(~

)(~

)(~

}0~

{

/

tn

tDnD

ijij

ijij

Pp

p

)(

)(~

)(

-

r

rtZtZ

ij

rij

)(lim)(

~

gp

!

Documents

Cell Infocom Apr 2003