1. Guaranteed Smooth Scheduling in Packet Switches Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis (Bell-Labs)

2. Outline 1.Frame Scheduling 2.GLJD Algorithm 3.GLJD Guarantees 4.Simulations 5.Conclusion

3. Crossbar Scheduler inputs outputs 1 N 1N.......... Packet Switch Scheduling

4. Frame-Based Scheduling Example:

5. Frame-Based Scheduling Traffic Demand Matrix: –Given a frame size of F, and an NxN integer demand matrix R of row and column sum equal to F –Can we send R ij amount of traffic from i to j during any frame? Birkhoff-von Neumann (BvN) Decomposition: We can decompose R as a sum of K < N 2 weighted permutations, with the weight sum equal to F. Frame-Scheduling: Apply BvN decomposition, then cycle through the permutations (C.S Chang et al., 2000).

6. Problems in BvN Decomposition Storage –N=512  Nlog 2 N = 4.6 Kbits per permutation  N 3 log 2 N = 1.1 Gbits total –Too many permutations to store on a chip Speed –Time complexity in O(N 4.5 ) (when F ≥ N 2 ) Jitter (variable delay)

7. What is jitter? BvN (naive implementation): Smooth scheduling (low-jitter):

8. Why smooth scheduling? Low-jitter guaranteed-bandwidth traffic –For instance Expedited Forwarding in Diffserv –Typically, 10% of the traffic Less burstiness –Bursty TCP traffic results in multiple losses Increased short-term fairness Less buffering (or delay lines) for smoothly arriving flows

9. Outline 1.Frame Scheduling 2.GLJD Algorithm 3.GLJD Guarantees 4.Simulations 5.Conclusion

10. Smooth scheduling idea 1.Decomposition: find a decomposition of R into matches such that each entry of R appears in at most one match. 2.Scheduling: use a scheduling algorithm to smoothly schedule the matches (matches are independent). Our algorithm Known method

11. Smooth scheduling example 1.Decomposition: 2.Scheduling: Note: BvN could yield:

12. Optimal Smooth Decomposition Theorem: Optimal decomposition is NP-hard  Need to find a provably good approximation algorithm Formal Problem:

13. Smooth Decomposition Example Idea: group together close coefficients

14. GLJD (Greedy Low-Jitter Decomposition)

15. Outline 1.Frame Scheduling 2.GLJD Algorithm 3.GLJD Guarantees 4.Simulations 5.Conclusion

16. Theorem 1 (matrices): GLJD needs at most K=2N-1 matrices Proof outline: Consider the union of a row i and a column j. It has at most 2N-1 non-zero elements. At each iteration, at least one of these elements is scheduled and removed from further consideration. GLJD guarantees

17. Theorem 2 (upper bound): Assume R of sum 1. Both D and GLJD need a bandwidth  2H N -1, i.e. O(ln N) Proof outline: upper-bound each matrix weight and sum the bounds Theorem 3 (lower bound): Both D and GLJD need a bandwidth of  (ln N) Proof outline: use a specific matrix as a counter-example GLJD guarantees

18. Theorem 4 (approximation ratio): GLJD is a (2-1/N) bandwidth approximation algorithm to D Proof outline: upper-bound bandwidth needed by GLJD and lower-bound D based on matrix structure GLJD guarantees

19. Outline 1.Frame Scheduling 2.GLJD Algorithm 3.GLJD Guarantees 4.Simulations 5.Conclusion

20. GLJD Simulation Summary (N=64) Gain with respect to BvN: –Smoothness –Storage GLJD needs 10 times less matrices than BvN –Complexity: GLJD is 100 times faster than BvN Trade-off with: –Bandwidth Efficiency Bandwidth ratio guarantee = 2H N -1 = 8.5 Simulation ~ 1.55

21. Simulations: jitter

22. Conclusion BvN decomposition is an optimal but impractical algorithm Practical smooth decomposition with –Lower storage requirements –Lower complexity –  (ln N) bandwidth approximation ratio guarantee