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Maximum Size Matchings & Input Queued Switches
Sundar Iyer, Nick McKeownHigh Performance Networking Group,Stanford University, http://yuba.stanford.edu
Allerton 2002Wednesday, Oct 2nd 2002
2
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.
3
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
4
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.
5
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
7
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?
9
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
10
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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Non Pre-emptive Scheduling … 2Batch Scheduling
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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Non Pre-emptive Scheduling … 2Batch Scheduling
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
3
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
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
1
2
3
01
021
0
001
1
2
3
Batch Request Graph with Dk =3
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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
18
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
19
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
24
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..
26
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..
27
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
28
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.
29
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
30
Example of a Uniform Graph
1
2
3
11
111
1
111
1
2
3
Batch Request Graph with Dk =3
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
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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
32
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
33
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.