Achieving 100% throughput Where we are in the course…. Switch model Uniform traffic Technique: Uniform schedule (easy) Non-uniform traffic, but known traffic matrix Technique: Non-uniform schedule (Birkhoff-von Neumann) Unknown traffic matrix Technique: Lyapunov functions (MWM) - PowerPoint PPT Presentation
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Achieving 100% throughputWhere we are in the course…
1. Switch model2. Uniform traffic
Technique: Uniform schedule (easy)
3. Non-uniform traffic, but known traffic matrix Technique: Non-uniform schedule (Birkhoff-von Neumann)
Delay Guarantees Traditional Crossbar – Emulating an OQ Switch Buffered Crossbar – Emulating an OQ Switch
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Work-conserving Property If there is a cell for a given output in the system, that
output is busy.
Work Conservation
Output Queued (OQ) Switch
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?
Emulating an OQ switch
Under identical inputs, the departure time of every cell from both switches is identical
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Input Priority List
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Label each cell with their corresponding departure times Arrange input cells into an input priority list Output selects crosspoint with earliest departure time
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Input Priority List
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Good guy
Bad guysBad guy
Label each cell with their corresponding departure times Arrange input cells into an input priority list Output selects crosspoint with earliest departure time
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Definitions
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Output Margin – cells at its output with earlier departure time Input Margin – cells ahead in input priority list destined to
different outputs Total Margin – Output Margin minus Input Margin
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2 good guys2 bad guys
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Emulation of FIFO OQ Switch
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Scheduling Phase Crosspoint is full – Output Margin will increase by one Crosspoint is empty – Input Margin will decrease by one
Total Margin increases by two
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Emulation of FIFO OQ Switch
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Arrival Phase Input Margin might increase by one
Departure Phase Output Margin will decrease by one
Total Margin decreases by at most two
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Emulation of FIFO OQ Switch
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Lemma 1 For every time slot, total margin does not decrease
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FIFO Insertion Policy
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857 6 323
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Arrival Phase Cell for non-empty VOQ, insert behind cells for same
output Cell for empty VOQ, insert at head of input priority list
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FIFO Insertion Policy
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Lemma 2 An arriving cell will have a non-negative total margin
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Theorem 2 A buffered crossbar with speedup of 2 can exactly emulate a
FIFO OQ switch.
Result was shown independently B. Magill, C. Rohrs, R. Stevenson, “Output-Queued Switch
Emulation by Fabrics With Limited Memory”, in IEEE Journal on Selected Areas in Communications, pp.606-615, May. 2003.
Theorem 3 A buffered crossbar with speedup of 2 can be work-conserving
with a distributed algorithm.
Emulation of FIFO OQ Switch
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Summary
Buffered crossbars Uses crosspoints to relieve contention Inputs and outputs schedule independently and in
parallel
Performance guarantees Throughput – any work-conserving input/output
schedule Work Conservation – simple insertion policy
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Relevant Papers
Crossbars Shang-Tse Chuang, Ashish Goel, Nick McKeown,
Balaji Prabhakar, “Matching Output Queuing with a Combined Input Output Queued Switch,” IEEE Journal on Selected Areas in Communications, vol.17, n.6, pp.1030-1039, Dec.1999.
Buffered Crossbars Shang-Tse Chuang, Sundar Iyer, Nick McKeown,
“Practical Algorithms for Performance Guarantees in Buffered Crossbars,” in preparation for IEEE/ACM Transactions on Networking.