Title: Throughput of Internally Buffered Crossbar Switch
1Throughput of Internally Buffered Crossbar Switch
Thursday, October 22, 2020
Mingjie Lin mingjie_at_stanford.edu www.stanford.edu/
mingjie
2Contents
- 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
3Background
IQ
OQ
Switching Fabric
CQ
4Motivations
Classic results 58.3(blocking) and
63(non-blocking) throughput for IQ crossbar
switch
5Motivations
What will happen to the throughput if we add
Crosspoint buffer?
6Contents
- 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
7The 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.
8Throughput Analysis
2 Cases 1. Non-Blocking Mode 2.
Blocking Mode
9Throughput Analysis
- 2 Cases
- 1. Non-Blocking Mode
- 2. Blocking Mode
- Idea
- Using Markov Chain to model the crossbar switch
behavior.
10Notation
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.
11Contents
- 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
12Observation
- Symmetry
- a) traffic b) switching fabric
structure
13Observation
- 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
14Observation
- 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
15Markov Chain
16Derivation
17Key Equation
Total probability
18Key Equation
Total probability
19Key Equation (cont.)
N linear equations
20IBCS without blocking
Solution of transition probability
21IBCS without blocking (cont.)
Solve those N linear equations, we can compute
through for any N.
22IBCS 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?
23IBCS without blocking (cont.)
We know
24IBCS without blocking (cont.)
We know
when
25IBCS without blocking (cont.)
Therefore
26IBCS without blocking (cont.)
Add them up
finally
27IBCS without blocking (cont.)
Which leads to
28IBCS with blocking
Markov chain model, however, state space too
large to manage
29IBCS with blocking
Markov chain model, however, state space too
large to manage What is key difference between
with blocking and without blocking?
30IBCS 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?
31IBCS with blocking (cont.)
32IBCS with blocking (cont.)
33IBCS with blocking (cont.)
when
Therefore
finally
34Contents
- 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
35Results Summary
36Results Summary (cont.)
37Results Summary (cont.)
- Crosspoint buffer cells have a significant impact
on throughput of crossbar switch Symmetry.
38Results Summary (cont.)
- Crosspoint buffer cells have a significant impact
on throughput of crossbar switch Symmetry. - Without crosspoint buffer, throughput will
decrease while N increases, the opposite is true
for ICBS.
39Results Summary (cont.)
- Crosspoint buffer cells have a significant impact
on throughput of crossbar switch Symmetry. - Without crosspoint buffer, throughput will
decrease while N increases, the opposite is true
for ICBS. - When N goes infinity, throughput of crossbar
switch without crosspoint buffer will converge to
63 without HOL blocking, but ICBSs will
converge to 100.
40Results Summary (cont.)
- Crosspoint buffer cells have a significant impact
on throughput of crossbar switch Symmetry. - Without crosspoint buffer, throughput will
decrease while N increases, the opposite is true
for ICBS. - When N goes infinity, throughput of crossbar
switch without crosspoint buffer will converge to
63 without HOL blocking, but ICBSs will
converge to 100. - When N goes infinity, throughput of crossbar
switch without crosspoint buffer will converge to
58 without HOL blocking, but ICBSs will
converge to 100.
41Thank you!