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)
Optimal Load-Balancing Isaac Keslassy (Technion, Israel), Cheng-Shang Chang (National Tsing Hua University, Taiwan), Nick McKeown (Stanford University,
Dec 22, 2015
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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)
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
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
Output-Queued Mesh
1. Mesh
2. 100% throughput
3. … but output write speed = NR
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 /
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
?
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]
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
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
Load-Balanced Router
1. Mesh → 2 meshes
2. 100% throughput
3. Node speed?
?
≈
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
In
Out
In
Out
In
Out
In
Out
RIn
Out
In
Out
In
Out
In
Out
R2R/N
A Single Combined Mesh
Matrix for the Combined Mesh
Combined mesh matrix:
The combined mesh matrix gets 100% throughput
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
Combined Mesh
1. Single Mesh
2. 100% Throughput
3. Max Node Speed = 2R
Question: is 2R optimal?
Any better architecture?
?
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.
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)
Example 2: Use a Non-Uniform Mesh
This is actually a ring!
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)
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!
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
Main Result
Slightly Non-Uniform Mesh
Slightly better than 2R
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.
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
Thank you.
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