Throughput of Internally Buffered Crossbar Switch

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Throughput of Internally Buffered Crossbar Switch
Thursday, October 22, 2020
Mingjie Lin mingjie_at_stanford.edu www.stanford.edu/
mingjie
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Contents

• Motivation
• High throughput performance crossbar switch
• What is the impact of crosspoint buffer on
  throughput of crossbar switch?
• Problem Statement and Notations
• The structure of a internally buffered crossbar
  switch (IBCS)
• Two cases 1. With blocking, 2. Without blocking
• Markov Chain model
• Analysis approach
• Results summary

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Background
IQ
OQ
Switching Fabric
CQ
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Motivations
Classic results 58.3(blocking) and
63(non-blocking) throughput for IQ crossbar
switch
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Motivations
What will happen to the throughput if we add
Crosspoint buffer?
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Contents

• Motivation
• High throughput performance crossbar switch
• What is the impact of crosspoint buffer on
  throughput of crossbar switch?
• Problem Statement and Notations
• The structure of a internally buffered crossbar
  switch (IBCS)
• Two cases 1. With blocking, 2. Without blocking
• Markov Chain model
• Analysis approach
• Results summary

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The structure of an internally buffered crossbar
switch (IBCS)

• Input Traffic i.i.d uniform Bernoulli type,
  independent at each input.
• Scheduling Algorithm (2 phases in 1 time slot)
• Buffer In Phase For each input queue i, each
  HOL packet goes to its destined crosspoint buffer
  cell if it is vacant.
• Buffer Out Phase For each output port j,
  randomly pick one cell from all occupied
  crosspoint buffer cells, and output its packet.

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Throughput Analysis
2 Cases 1. Non-Blocking Mode 2.
Blocking Mode
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Throughput Analysis

• 2 Cases
• 1. Non-Blocking Mode
• 2. Blocking Mode
• Idea
• Using Markov Chain to model the crossbar switch
  behavior.

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Notation
Internal buffer cell
Input queue at input port I
For any column of buffer cells, the probability
of having k packets in total at
time n state transition
probability of Markov chain
model.
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Contents

• Motivation
• High throughput performance crossbar switch
• What is the impact of crosspoint buffer on
  throughput of crossbar switch?
• Problem Statement and Notations
• The structure of a internally buffered crossbar
  switch (IBCS)
• Two cases 1. With blocking, 2. Without blocking
• Markov Chain model
• Analysis approach
• Results summary

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Observation

• Symmetry
• a) traffic b) switching fabric
  structure

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Observation

• Symmetry
• a) traffic b) switching fabric
  structure
• 2. During each time slot, if there is at least 1
  packet in B,j, then there will be a packet to
  output

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Observation

• Symmetry
• a) traffic b) switching fabric
  structure
• 2. During each time slot, if there is at least 1
  packet in B,j, then there will be a packet to
  output
• 3. Saturation Throughput

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Markov Chain
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Derivation
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Key Equation
Total probability
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Key Equation
Total probability
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Key Equation (cont.)
N linear equations
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IBCS without blocking
Solution of transition probability
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IBCS without blocking (cont.)
Solve those N linear equations, we can compute
through for any N.
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IBCS without blocking (cont.)
Solve those N linear equations, we can compute
through for any N. Question what happens to
throughput if N goes to infinity?
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IBCS without blocking (cont.)
We know
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IBCS without blocking (cont.)
We know
when
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IBCS without blocking (cont.)
Therefore
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IBCS without blocking (cont.)
Add them up
finally
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IBCS without blocking (cont.)
Which leads to
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IBCS with blocking
Markov chain model, however, state space too
large to manage
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IBCS with blocking
Markov chain model, however, state space too
large to manage What is key difference between
with blocking and without blocking?
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IBCS with blocking
Markov chain model, however, state space too
large to manage What is key difference between
with blocking and without blocking? What is
thoughput if N goes to infinity?
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IBCS with blocking (cont.)
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IBCS with blocking (cont.)
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IBCS with blocking (cont.)
when
Therefore
finally
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Contents

• Motivation
• High throughput performance crossbar switch
• What is the impact of crosspoint buffer on
  throughput of crossbar switch?
• Problem Statement and Notations
• The structure of a internally buffered crossbar
  switch (IBCS)
• Two cases 1. With blocking, 2. Without blocking
• Markov Chain model
• Analysis approach
• Results summary

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Results Summary
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Results Summary (cont.)
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Results Summary (cont.)

1. Crosspoint buffer cells have a significant impact
   on throughput of crossbar switch Symmetry.

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Results Summary (cont.)

1. Crosspoint buffer cells have a significant impact
   on throughput of crossbar switch Symmetry.
2. Without crosspoint buffer, throughput will
   decrease while N increases, the opposite is true
   for ICBS.

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Results Summary (cont.)

1. Crosspoint buffer cells have a significant impact
   on throughput of crossbar switch Symmetry.
2. Without crosspoint buffer, throughput will
   decrease while N increases, the opposite is true
   for ICBS.
3. When N goes infinity, throughput of crossbar
   switch without crosspoint buffer will converge to
   63 without HOL blocking, but ICBSs will
   converge to 100.

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Results Summary (cont.)

1. Crosspoint buffer cells have a significant impact
   on throughput of crossbar switch Symmetry.
2. Without crosspoint buffer, throughput will
   decrease while N increases, the opposite is true
   for ICBS.
3. When N goes infinity, throughput of crossbar
   switch without crosspoint buffer will converge to
   63 without HOL blocking, but ICBSs will
   converge to 100.
4. When N goes infinity, throughput of crossbar
   switch without crosspoint buffer will converge to
   58 without HOL blocking, but ICBSs will
   converge to 100.

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Thank you!