Maximum Size Matchings & Input Queued Switches Sundar Iyer, Nick McKeown High Performance Networking...

Maximum Size Matchings & Input Queued Switches

Sundar Iyer, Nick McKeownHigh Performance Networking Group,Stanford University, http://yuba.stanford.edu

Allerton 2002Wednesday, Oct 2nd 2002

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Definition - 100% Throughput

A switch gives 100% throughput if the expected size of the queues is finite for any admissible (no input or output is oversubscribed) load.

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A Characteristic Switch

N=4 N=4

1 1R

R

An input queued switch with a crossbar switching fabric

Crossbar

R

R

1

N=4

1

N=4

VOQs

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Maximum Size Matching

Maximum Size Matching (MSM)

Choose a matching which maximizes the size Contrary to intuition, MSM does not give 100%

throughput

Ref: [McKeown, Anantharam, Walrand - 1996], “Achieving 100% Throughput in an Input-Queued Switch“, IEEE Infocom '96.

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Contents

1. Background & Motivation

2. Non Pre-emptive Scheduling

3. Achieving 100% throughput with CMSM

• Bernoulli i.i.d. uniform traffic• Bernoulli i.i.d. non-uniform traffic

4. Achieving 100% throughput with MSM

• Bernoulli i.i.d. uniform traffic

5. Conclusion

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An ExampleMSM does not give 100% throughput

N=2 N=2

1 1R

R

Crossbar

R

R

11=0.49

12=0.50

21=0.50

22=0.00

Ref: [Keslassy, Zhang, McKeown - 2002], “MSM is unstable for any input queued switch”, In Preparation.

VOQs

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Motivation

“To understand the conditions under which the class of MSMs give 100% throughput”

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Questions

Do all MSMs not achieve 100% throughput?

Is there a sub class of MSMs which achieve 100% throughput?

Do all MSMs achieve 100% throughput under uniform load?

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Contents







5. Conclusion

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Non Pre-emptive Scheduling … 1Batch Scheduling

Main Idea

Scheduling cells in batches increases the choice for the matching and hence increases throughput

Allow the batch size to grow

Ref: [Dolev, Kesselman - 2000], “Bounded latency scheduling scheme for ATM cells", Computer Networks, vol. 32(3) pp.325-331, 2000.

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N N

1 1R

R

Priority-2

Crossbar

R

R

1

N

1

N

Priority-1

Batch-(k+1)

Batch-(k)

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N N

1 1R

R

Priority-2

R

R

1

N

1

N

Priority-1

CrossbarBatch-(k+1)

Batch-(k)

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Degree of a Batch

1

2

3

01

021

0

001

1

2

3

Batch Request Graph Degree (dv,k):

The number of cells departing from (destined to) a vertex in batch k.

Maximum Degree (Dk) The maximum degree

amongst all inputs/outputs in batch k.

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2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

1

2

3

01

021

0

001

1

2

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Batch Request Graph with Dk =3

2

3

1

2

3

1

Maximum Size MatchingWhy may MSM not give 100% throughput?

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Critical Maximum Size MatchingA sub-class of MSM

2

3

1

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1

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1

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01

021

0

001

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CMSM achieves 100% throughput under non pre- emptive scheduling, if the traffic is constrained to less than cells for any input/output in B timeslots.

This introduces deterministic constraints on the arrival traffic We are interested in the traditional stochastic traffic

Previous Results

Ref: [Weller, Hajek - 1997], “Scheduling non-uniform traffic in a packet-switching system with small propagation delay,” IEEE/ACM Transactions on Networking 5(6): 813-823, 1997.

, 1B

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Arrival Traffic

, :

1, 1

1. Traffi c matrix:

where expected number of

arrivals in one timeslot

2. I f ; we say the traffi c is "admissible".

3. For a Bernoulli i.

ij ij

ij iji j

A

, ( , );

i.d arrival process:

I f we say the traffi c is unif orm.ij i jN

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Contents







5. Conclusion

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CMSM with Uniform Traffic

Theorem 1:

CMSM gives 100% throughput under non pre-emptive scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform

Informal Arguments:

Let Tk be the time to schedule batch k

Then for batch k+1 we buffer new arrivals for time Tk

We expect about Tk packets at every input/output

Hence, the maximum degree of batch k +1, i.e. Dk+1 Tk

Hence for a CMSM, Tk+1 = Dk+1 Tk < Tk

Hence Tk is bounded in mean.

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1( ) , 0, k k k k k cE T T T T T T

1( ) (1- ) , 0, k k k k cE T T T T T

We are going to show that

Alternatively we will first show that

1 k kT NT

Observe that

Formal ArgumentsOutline

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We shall use the Chernoff bound to get

If we want to bound Dk+1, we require that all the 2N vertices are bounded

(1 ), 1{ (1 ) }

(1 )

kT

v k k k veP d T T p

1{ (1 ) } 1 2k k k vP D T T Np Q

Formal Arguments … 1Bounding the degree of a batch

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Choose > 0, such that .

Choose such that

We get

1(1 )

{ }2

k k kP T T T Q

(1 )

2

(1 )

4

1(1 ) 1

2

1

Formal Arguments … 2 Bounding the deviation of the service time of a batch

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Hence

1( ) (1- )

(1- ) , if

(1 )

2k k

k

k kE T T Q T Q NT

T

2 2 2 1

2 1

NQ

N

kNT(1 )

2kT

0 (1- ) kT kT

1( ) <k kE T T1( ) <k kE T T

Formal Arguments … 3 Bounding the service time of a batch

{.}P Q

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Choose < (1- )/2,

This gives

Observe that Q is now a function of Tk only for a constant We can make Q as close to 1, by choosing a large Tk

4 3 1

4 2 2

ifN

QN

1

(3 )( ) ,

4k k kE T T T

(1 )

4

Formal Arguments …4 Tightening the bound

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Hence, there is a constant Tc such that

Formally, using a linear Lyapunov function V(Tk) = Tk, we can say that Tk (averaged over the batch index) is bounded in mean.

1

1

(3 )( ) ,

4

(1 )( ) ,

4

k k k k c

k k k k k c

E T T T T T

E T T T T T T

Formal Arguments …5Finishing Off..

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In the paper we use a quadratic Lyapunov function V(Tk) = (Tk)2 , and show that Tk

2 (averaged

over the batch index) is bounded in mean.

There are a few technical steps after this to show that the queue size (averaged over time) is bounded in mean.

Then, it follows that CMSM gives 100% throughput for Bernoulli i.i.d. uniform traffic.

Formal Arguments …6Some Final Points..

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Contents







5. Conclusion

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CMSM with Non-Uniform Traffic

Theorem 2:

CMSM achieves 100% throughput under non pre-emptive scheduling, if the input traffic is admissible and Bernoulli i.i.d.

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Contents







5. Conclusion

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Example of a Uniform Graph

1

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11

111

1

111

1

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2

3

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MSM with Non-Uniform Traffic

Theorem 3:

MSM achieves 100% throughput under non pre-emptive scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform

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Contents







5. Conclusion

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Conclusions

We have used the more traditional stochastic arrivals and shown using batch scheduling that

CMSM gives 100% throughput for Bernoulli i.i.d. traffic MSM gives 100% throughput for Bernoulli i.i.d. uniform traffic

It would be nice to understand the stability of MSM with uniform load with continuous scheduling.

Maximum Size Matchings & Input Queued Switches Sundar Iyer, Nick McKeown High Performance Networking...

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