048866: Packet Switch Architectures

1. 048866: Packet Switch Architectures MSM (Maximum Size Matching) and MWM (Maximum Weight Matching) Dr. Isaac Keslassy Electrical Engineering, Technion isaac@ee.technion.ac.il http://comnet.technion.ac.il/~isaac/

2. Achieving 100% throughput • 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) • Faster scheduling algorithms • Technique: Speedup (maximal matchings) • Technique: Memory and randomization (Tassiulas) • Technique: Twist architecture (buffered crossbar) • Accelerate scheduling algorithm • Technique: Pipelining • Technique: Envelopes • Technique: Slicing • No scheduling algorithm • Technique: Load-balanced router 048866 – Packet Switch Architectures

3. Unknown Traffic Matrix • We want to maximize throughput • Traffic matrix unknown  cannot use BvN • Idea: maximize instantaneousthroughput • In other words: transfer as many packets as possible at each time-slot • Maximum Size Matching (MSM) algorithm 048866 – Packet Switch Architectures

4. Maximum Size Matching (MSM) MSM maximizes instantaneous throughput MSM algorithm: among all maximum size matches, pick a random one Q11(n)>0 Maximum Size Match QN1(n)>0 Bipartite Match Request Graph 048866 – Packet Switch Architectures

5. Question • Is the intuition right? • Answer: No, there is a counter-example for which, in a given VOQ (i,j), ij < ij 048866 – Packet Switch Architectures

6. Counter-example Consider the following non-uniform traffic pattern, with Bernoulli IID arrivals: Three possible matches, S(n): 048866 – Packet Switch Architectures

7. Simulation of simple 3x3 example 048866 – Packet Switch Architectures

8. Idea: Use Lyapunov 048866 – Packet Switch Architectures

9. Some definitions 048866 – Packet Switch Architectures

10. Some definitions 048866 – Packet Switch Architectures

11. Problem 048866 – Packet Switch Architectures

12. Maximum Weight Matching (MWM) S*(n) Q11(n) A11(n) A1(n) D1(n) 1 1 A1N(n) “LQF” MWM Algorithm AN1(n) AN(n) DN(n) ANN(n) N N QNN(n) Q11(n) Maximum Weight Match QN1(n) Bipartite Match “Request” Graph 048866 – Packet Switch Architectures

13. Outline of Proof Note: proof based on paper by McKeown et al. 048866 – Packet Switch Architectures

14. Proof • Let’s prove: • First, we’ll work with the approximate Lyapunov function • We’ll assume that there exists some  such that: • For all i, j ij≤ 1- • For all j, i ij≤ 1- • In other words, · (1-) m with m doubly stochastic 048866 – Packet Switch Architectures

15. Proof 048866 – Packet Switch Architectures

16. Proof 048866 – Packet Switch Architectures

17. Proof • We worked with the approximate Lyapunov function • Now, let’s work with the real Lyapunov function, and show that 048866 – Packet Switch Architectures

18. End of Proof 048866 – Packet Switch Architectures

19. Review of Proof 048866 – Packet Switch Architectures

20. Review of Proof 048866 – Packet Switch Architectures

21. LQF (Longest Queue First) • LQF is the name given to the maximum weight matching, where weight wij(n) = Lij(n). • But the name is so bad that people keep the name “MWM”! • LQF doesn’t necessarily serve the longest queue. • LQF can leave a short queue unserved indefinitely. • However, MWM-LQF is very important theoretically: most (if not all) scheduling algorithms that provide 100% throughput for unknown traffic matrices are variants of MWM! 048866 – Packet Switch Architectures

22. LQF (Longest Queue First) • Question: what if or • What if weight wij(n) = Wij(n) (waiting time)? • Preference is given to cells that have waited a long-time. • Is it stable? • We call the algorithm OCF (Oldest Cell First). • Remember that it doesn’t guarantee to serve the oldest cell! 048866 – Packet Switch Architectures

23. OCF (Oldest Cell First) Cij(n) Cij(n+l) Wij(n) n n+l tij(n) Cij(n) Cij(n+l) 048866 – Packet Switch Architectures

24. Rough outline of proof Expectation given W(n) Note: full proof in paper by McKeown et al. 048866 – Packet Switch Architectures

25. Implementing MSM • How can we find maximum size matches? • We do so by recasting the problem as a network flow problem 048866 – Packet Switch Architectures

26. Network Flows 10 10 10 1 10 1 10 1 a c Source s Sink t b d • Let G = [V,E] be a directed graph with capacity cap(v,w) on edge [v,w]. • A flow is an (integer) function, f, that is chosen for each edge so that • We wish to maximize the flow allocation. 048866 – Packet Switch Architectures

27. A maximum network flow exampleBy inspection 10 10 10 1 10 1 10 1 a c 10, 10 Source s Sink t 10, 10 1 10, 10 10 1 10 b d 1 Flow is of size 10 a c Source s Sink t b d Step 1: 048866 – Packet Switch Architectures

28. A maximum network flow example Not obvious Maximum flow: a c 10, 9 Source s Sink t 10, 10 1,1 10, 10 1,1 10, 2 b d 10, 2 1, 1 Flow is of size 10+2 = 12 Step 2: a c 10, 10 Source s Sink t 10, 10 1 10, 10 1 10, 1 b d 10, 1 1, 1 Flow is of size 10+1 = 11 048866 – Packet Switch Architectures

29. Ford-Fulkerson Method of Augmenting Paths Set f(v,w) = -f(w,v) on all edges. Define a Residual Graph, R, in which res(v,w) = cap(v,w) – f(v,w) Find paths from s to t for which there is positive residue. Increase the flow along the paths to augment them by the minimum residue along the path. Keep augmenting paths until there are no more to augment. 048866 – Packet Switch Architectures

30. Example of Residual Graph a c 10, 10 10, 10 1 10, 10 s t 10 1 10 b d 1 Flow is of size 10 Residual Graph, R res(v,w) = cap(v,w) – f(v,w) a c 10 10 10 1 s t 10 1 10 b d 1 Augmenting path 048866 – Packet Switch Architectures

31. Example of Residual Graph Step 2: a c 10, 10 s t 10, 10 1 10, 10 1 10, 1 b d 10, 1 1, 1 Flow is of size 10+1 = 11 Residual Graph a c 10 s t 10 10 1 1 1 1 b d 9 1 9 048866 – Packet Switch Architectures

32. Complexity of network flow problems In general, it is possible to find a solution by considering at most |V|.|E| paths, by picking shortest augmenting path first. There are many variations, such as picking most augmenting path first. 048866 – Packet Switch Architectures

33. Finding a maximum size match How do we find the maximum size match? A 1 2 B 3 C 4 D 5 E 6 F 048866 – Packet Switch Architectures

34. Network flows and bipartite matching Finding a maximum size bipartite matching is equivalent to solving a network flow problem with capacities and flows of size 1. A 1 2 B Sink t Source s 3 C 4 D 5 E 6 F 048866 – Packet Switch Architectures

35. Example: Maximum Size MatchingFord-Fulkerson method Residual Graph for first three paths: A 1 2 B t s 3 C 4 D 5 E 6 F 048866 – Packet Switch Architectures

36. Example: Maximum Size MatchingFord-Fulkerson method Residual Graph for next two paths: A 1 2 B t s 3 C 4 D 5 E 6 F 048866 – Packet Switch Architectures

37. Example: Maximum Size MatchingFord-Fulkerson method Residual Graph for augmenting path: A 1 2 B t s 3 C 4 D 5 E 6 F 048866 – Packet Switch Architectures

38. Example: Maximum Size MatchingFord-Fulkerson method Residual Graph for last augmenting path: A 1 2 B t s 3 C 4 D 5 E 6 F Note that the path augments the match: no input and output is removed from the match during the augmenting step. 048866 – Packet Switch Architectures

39. Example: Maximum Size MatchingFord-Fulkerson method Maximum flow graph: A 1 2 B t s 3 C 4 D 5 E 6 F 048866 – Packet Switch Architectures

40. Example: Maximum Size MatchingFord-Fulkerson method Maximum Size Matching: A 1 2 B 3 C 4 D 5 E 6 F 048866 – Packet Switch Architectures

41. LPF (Largest Port First) Note: full proof in paper by Mekkitikul and McKeown 048866 – Packet Switch Architectures

42. LPF 048866 – Packet Switch Architectures