1. Providing 100% throughput for non-uniform Bernoulli traffic
Shang-Tse (Da) Chuang
Siva Gaggara

2. Motivation
Maximum Weight Matching can provide 100% throughput with a speedup of 1. (McKeown) (Dai,Prabhakar)
Any maximal matching algorithm can provide 100% throughput with a speedup of 2. (Dai,Prabhakar)
What speedup is required to provide 100% throughput using a practical maximal matching algorithm?

3. Problem Statement Find λmax
non-uniform Bernoulli traffic with rate λij
Σi λij < λmax and Σj λij < λmax
FPWWFA is the maximal matching algorithm

4. Wrapped Wave Front Arbiter (Tamir) reminder1 2 3

5. Friends, Strangers, Enemies1 2 3
λ21
λ31
λ11
λ22
λ32
λ12
λ23
λ33
λ13
Flows λ11,λ21,λ32, and λ33 are enemies
Flows λ12,λ13, and λ22 are strangers
Flow λ23 is a friend

6. The “L” Traffic Pattern1 3 2
(1-λ11)
λ21
λ31
λ11
λ32
λ33
λ21
Q31 service rate = =
Thus, for this traffic pattern, λmax = 0.75

7. Throughput Formula λ21 λ31 λ11 λ32 λ23 λ33 Q31 service rate =
(1-λ11)(1-λ23)
λ21
λ31
λ11
λ32
λ23
λ33
λ33
(1-λ23)(1-λ32)
Q31 service rate =
Greater than L-traffic analysis for all λ23

8. Simulation
λ11 = 0.2, λ21 = 0.3, λ32 = 0.15, λ33 = 0.35

9. Conclusions λmax = 0.75 for “L” traffic pattern
Found throughput formula that matches simulation for any non-uniform Bernoulli traffic pattern for a 3x3 switch
We believe that λmax = 0.75 is true for any non-uniform Bernoulli traffic pattern for any switch size