1

Input Queued Switches:Cell Switching vs. Packet Switching

Abtin Keshavarzian

Joint work with

Yashar Ganjali, Devavrat Shah

Stanford University

2

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

3

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

Spl

itte

r

Com

bine

r

Switch

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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– PB Algorithms Classification

• Work Conserving• Waiting

– Waiting PB Algorithms

• Conclusion

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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

ijij

n n

nA

)(lim

)(nZ ij

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,

][ ij

N

iij Nj

1

,...,11

N

jij Ni

1

,...,11

ijij

n n

nD

)(lim

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MWM algorithm• A matching

• MWM: At each time slot, select the matching with maximum weight

)(maxarg)( nWnm

mm

ji ijij nZmnnW

,)()(,)( Zmm

NNijm ][m

otherwise 0

output toconnected is input if1 jimij

N

iij jm

1

1

N

jij im

1

1

)()(max)( nWnWnW

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 the whole packet is received at output port

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Packet-Based Switching

• Once the scheduler starts transmitting the first cell of a packet, it continues until the whole packet is received at output port

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Packet-Based Switching

• Once the scheduler starts transmitting the first cell of a packet, it continues until the whole packet is received at output port.

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Cell-based to Packet-based

• Consider cell-based algorithm X

• At each time slot:– Busy ports : middle of sending a packet– Free ports : i/o ports can be assigned freely

• PB-X – Keep the assignments used by busy ports– Find a sub-matching for free ports using

algorithm X.

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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,” INFOCOM 2001, pp. 1085-1094

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Proof Outline

• Matching m(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

)(2)()(|)( * TNnWnZnW EE

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Question

• CB-MWM is stable under any admissible traffic

• PB-MWM is stable under any admissible regenerative 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

Time

A22

A11

A12

A21

18

Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

Time

A22

A11

A12

A21

19

Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

Time

A22

A11

A12

A21

20

Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

Time

A22

A11

A12

A21

21

Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

Time

A22

A11

A12

A21

22

Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

Time

A22

A11

A12

A21

23

Counter-example

1 2

2

1 3 3

1

1 2 2

3

3 4

4

Time

A22

A11

A12

A21

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Classification of PB algorithms

• Work Conserving (non-waiting):– No input is left unmatched when it has a packet

for an unmatched output.

• Waiting : – Input ports may wait (do not start sending a

packet) for infinite number of time slots.

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

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PB-wMWM

1

/L

Segment #1 Segment #2

/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[ LL

kL

kn

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Modified PB-wMWM

Segment #1 Segment #2

/)2()2( eLM

)2(eL

/)1()1( eLM

)1(eL

• The packet length is not known but has bounded expectation

• : 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 admissible

regenerative traffic.• Work-conserving packet based algorithms can not

be rate stable for all admissible traffics Waiting is essential• PB-wMWM and its modified version are stable

under any admissible traffic (with bounded mean packet length)

• Future work:– Find simpler algorithms that are stable for any

admissible traffic.

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Fluid model• : number of time slots matching m being used

up to time n

)(nTm

nnT

nTnTmnDnD

nDnAnZ

ijZijijij

ijijij

mm

mmm

)(

)1()(1)1()(

)()()(

}0{

ttT

t

tTm

t

tD

tDttZ

ijZijij

ijijij

mm

m

m

)(~

)(~

1)(

~)(

~)(

~

}0~

{

tnA

tDnD

ijij

ijij

)(

)(~

)(

r

rtZtZ ij

rij

)(lim)(

~