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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)
4. Unknown traffic matrix Technique: Lyapunov functions (MWM)
5. Faster scheduling algorithms Technique: Speedup (maximal matchings) Technique: Memory and randomization (Tassiulas) Technique: Twist architecture (buffered crossbar)
6. Accelerate scheduling algorithm Technique: Pipelining Technique: Envelopes Technique: Slicing
7. No scheduling algorithm Technique: Load-balanced router
Buffered CrossbarsWith Performance Guarantees
Taken from the 2004 Ph.D. defense of:
Shang-Tse (Da) ChuangDepartment of Electrical Engineering,Stanford University, http://yuba.stanford.edu/~stchuang
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Motivation
Network operators want performance guarantees Throughput guarantee Delay guarantee
High performance routers use crossbars
Hard to build crossbar-based routers with guarantees
My talk: How a crossbar with a small amount of internal
buffering can give guarantees
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Contents
Throughput Guarantees Buffered Crossbar - 100% Throughput Buffered Crossbar - Work Conservation
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Generic Crossbar-Based Architecture
Speedup of S
Scheduler
VOQs
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Admissible Traffic
1 , , j
iji
ij
Traffic Matrix
Traffic is admissible if
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100% Throughput An algorithm delivers 100% throughput if for any
admissible traffic the average backlog is finite
Throughput Guarantee
Speedup of S
Scheduler
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Previous Work
1985 1990 1995 2000 2005
Wave Front Arbiter [Tamir]
Parallel Iterative Matching [Anderson et al.]
iSLIP [McKeown]
Longest Port First [Mekkittikul et al.]
Maximum Weight Matching [McKeown et al.]
Maximal Matching S=2[Dai,Prabhakar]
Heuristics
TheoreticallyProven
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Maximal Matching Has Become Hard
TTX Switch Fabric Uses maximal matching Speedup less than 2 Consumes up to 8kW Limited to ~2.5Tb/s No 100% throughput guarantee
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Traditional Crossbar
Crossbar Requirements An input can send at most one cell An output can receive at most one cell
Scheduling Problem Must overcome two constraints simultaneously
New Crossbar Relieve contention Remove dependency between inputs and outputs
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Contents
Throughput Guarantees Buffered Crossbar - 100% Throughput Buffered Crossbar - Work Conservation
Delay Guarantees Traditional Crossbar – Emulating an OQ Switch Buffered Crossbar – Emulating an OQ Switch
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Buffered Crossbar
Arrival Phase Scheduling Phases – Speedup of 2 Departure Phase
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Scheduling Phase
Input Schedule Each input selects in parallel a cell for an empty crosspoint
Output Schedule Each output selects in parallel a cell from a full crosspoint
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Example of Input/Output Scheduling
Round-robin Policy Each input schedules in a round-robin order Each output schedules in a round-robin order
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Previous Work
Buffered Crossbar Simulations [Rojas-Cessa et al. 2001] 32x32 switch, Uniform Bernoulli Traffic, Round-Robin, S=1
0.01
0.1
1
10
100
1000
0.025 0.125 0.225 0.325 0.425 0.525 0.625 0.725 0.825 0.925
Offered Load p
Ave
rag
e D
elay
(C
ell
Tim
e)
1-SLIP
4-SLIP
Buffered Crossbar
Ideal Router
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Theorem 1 A buffered crossbar with speedup of 2 delivers 100%
throughput for any admissible Bernoulli iid traffic using any work-conserving input/output schedules.
100% Throughput
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Intuition of Proof
ε
<1-ε
<1-ε
1 2
1-ε 1-ε+ + ε = 2- ε
When a flow is backed up, the services for this backlog exceeds the arrivals
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Contents
Throughput Guarantees Buffered Crossbar - 100% Throughput Buffered Crossbar - Work Conservation
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
57 6
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1
1
2
9
2
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8 3
1
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
57 6
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132
9
4
2
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1
8
2
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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4
132
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4
2
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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
1
8
2 good guys2 bad guys
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Emulation of FIFO OQ Switch
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2
4
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9
4
2
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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
1
8 3
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Emulation of FIFO OQ Switch
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2
4
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9
4
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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
1
8 3
3
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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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4
2
9
4
2
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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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4
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4
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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.
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Thank you!