Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua...
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![Page 1: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/1.jpg)
Optimal Load-Balancing
Isaac Keslassy (Technion, Israel),
Cheng-Shang Chang (National Tsing Hua University, Taiwan),
Nick McKeown (Stanford University, U.S.A.), Duan-Shin Lee (National Tsing Hua University, Taiwan)
![Page 2: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/2.jpg)
Router Designer Wishlist
1. Mesh Switch: avoid switch reconfiguration and complex scheduling algorithms. Practical for optics (AWGR).
2. 100% Throughput: router guaranteed to be stable under any admissible traffic matrix
3. Minimum Linecard Complexity Minimize maximum rate at which packets arrive to/depart from any input/output.
Buffering Speed Processing Speed
![Page 3: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/3.jpg)
In
In
In
Out
Out
Out
R
R
R
R
R
R
?
?
?
?
?
?
?
?
?
R
R
R
R
R
R
R
R
R
RRRR
Naive Mesh with 100% Throughput
Output Write Speed = NR
![Page 4: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/4.jpg)
Output-Queued Mesh
1. Mesh
2. 100% throughput
3. … but output write speed = NR
![Page 5: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/5.jpg)
R
In
In
In
Out
Out
Out
R
R
R
R
R
R/N
R/N
R/N
R/NR/N
R/N
R/N
R/N
R/N
If Traffic Is Uniform
RNR /NR /NR /
R
NR / NR /
![Page 6: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/6.jpg)
100% Throughput: Non-Uniform Traffic Matrices
R
In
In
In
Out
Out
Out
R
R
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
RNR /NR /NR /
R
RNR /NR /NR /
R
RNR /NR /NR /
R
R
R
R
?
![Page 7: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/7.jpg)
Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
Load-Balanced Router
Load-balancing mesh Forwarding mesh
In
In
In
Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R
R
R
Theorem: 100% Throughput [Val. 82, CLJ 01, K. et al. 03]
![Page 8: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/8.jpg)
Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
In
In
In
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
112233
Load-balancing mesh
Forwarding mesh
Load-Balanced Router
![Page 9: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/9.jpg)
Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
In
In
In
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N33
22
11
Load-balancing mesh
Forwarding mesh
Load-Balanced Router
![Page 10: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/10.jpg)
Load-Balanced Router
1. Mesh → 2 meshes
2. 100% throughput
3. Node speed?
?
≈
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Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
In
In
In
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
One linecard
In
Out In
Out
Combining the Two Meshes
![Page 12: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/12.jpg)
In
Out
In
Out
In
Out
In
Out
RIn
Out
In
Out
In
Out
In
Out
R2R/N
A Single Combined Mesh
![Page 13: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/13.jpg)
Matrix for the Combined Mesh
Combined mesh matrix:
The combined mesh matrix gets 100% throughput
![Page 14: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/14.jpg)
Node Speed for Combined Mesh
Max input/output read/write speed = 2R
In Out
In Out
In Out
In Out
R In Out
In Out
In Out
In Out
R2R/N
![Page 15: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/15.jpg)
Combined Mesh
1. Single Mesh
2. 100% Throughput
3. Max Node Speed = 2R
Question: is 2R optimal?
Any better architecture?
?
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Other Mesh Architectures We Consider
Any number of stages (e.g., 3 stages, 4 stages….)
Any mesh architecture (e.g., ring) Any link capacities (e.g., non-uniform mesh) Any packet routing algorithm (e.g., adaptive
algorithm)
Any mesh and any routing.
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Example 1: Add A Third Mesh?
In
In
In
Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R
R
R
In
In
In
Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
In
In
In
Out
Out
Out
R
R
R
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
R/N
1st stage 2nd stage 3rd stage
Combine the 3 meshes
Max speed = 3R (instead of 2R)
![Page 18: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/18.jpg)
Example 2: Use a Non-Uniform Mesh
This is actually a ring!
![Page 19: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/19.jpg)
Example 2: Unidirectional Ring
Assume that each node sends all traffic to itself. Then each packet goes through N nodes. To get 100% throughput, each node needs to
run N times faster.
1 23
i
N
Max speed = NR (instead of 2R)
![Page 20: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/20.jpg)
At First Glance…
… it seems that the uniform mesh is optimal with 2R!
Why: All links have the same capacity, And it is perfectly symmetric.
However…. uniform mesh is NOT optimal!
![Page 21: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/21.jpg)
Why Uniform Mesh is Not Optimal
Links between two different nodes used for spreading and forwarding
Same-node links only used for forwarding, not spreading need less capacity.
Example: packet from node 1 to node 2. No point in sending it from node 1 to node 1 before forwarding to node 2!
1 2
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Main Result
Slightly Non-Uniform Mesh
Slightly better than 2R
![Page 23: Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,](https://reader035.fdocuments.in/reader035/viewer/2022062314/56649d775503460f94a586d7/html5/thumbnails/23.jpg)
However…
The result is actually good for the load-balanced router with uniform mesh.
The uniform mesh is optimal as N → 1
In other words, asymptotically with N, the load-balanced router is at least as good as any other mesh architecture with any other routing algorithm.
The load-balanced router satisfies the wishlist goals.
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Generalization: Load-Balanced Network
1
3
…
N
Hotnets III, Nov. 2004: Zhang-Shen and
McKeown Kodialam, Lakshman
and Sengupta Two steps:
1. Uniform spreading of incoming packets (independently of destination)
2. Forwarding to destination
22
10
A uniform load-balanced backbone guarantees 100% throughput for any traffic matrix is at least as good as any other backbone design
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Thank you.