1. Maximum Size Matchings & Input Queued Switches Sundar Iyer, Nick McKeown High Performance Networking Group, Stanford University, http://yuba.stanford.edu Allerton 2002 Wednesday, Oct 2 nd 2002

2. 2 Definition - 100% Throughput A switch gives 100% throughput if the expected size of the queues is finite for any admissible (no input or output is oversubscribed) load.

3. 3 A Characteristic Switch N=4 1 1 R R An input queued switch with a crossbar switching fabric Crossbar R R 1 N=4 1 VOQs

4. 4 Maximum Size Matching  Maximum Size Matching (MSM)  Choose a matching which maximizes the size  Contrary to intuition, MSM does not give 100% throughput Ref: [McKeown, Anantharam, Walrand - 1996], “Achieving 100% Throughput in an Input-Queued Switch“, IEEE Infocom '96.

5. 5 Contents 1. Background & Motivation 2. Non Pre-emptive Scheduling 3. Achieving 100% throughput with CMSM Bernoulli i.i.d. uniform traffic Bernoulli i.i.d. non-uniform traffic 4. Achieving 100% throughput with MSM Bernoulli i.i.d. uniform traffic 5. Conclusion

6. 6 An Example MSM does not give 100% throughput N=2 1 1 R R Crossbar R R  11 =0.49  12 =0.50  21 =0.50  22 =0.00 Ref: [Keslassy, Zhang, McKeown - 2002], “MSM is unstable for any input queued switch”, In Preparation. VOQs

7. 7 Motivation “To understand the conditions under which the class of MSMs give 100% throughput”

8. 8 Questions  Do all MSMs not achieve 100% throughput?  Is there a sub class of MSMs which achieve 100% throughput?  Do all MSMs achieve 100% throughput under uniform load?

9. 9 Contents 1. Background & Motivation 2. Non Pre-emptive Scheduling 3. Achieving 100% throughput with CMSM Bernoulli i.i.d. uniform traffic Bernoulli i.i.d. non-uniform traffic 4. Achieving 100% throughput with MSM Bernoulli i.i.d. uniform traffic 5. Conclusion

10. 10 Non Pre-emptive Scheduling … 1 Batch Scheduling  Main Idea  Scheduling cells in batches increases the choice for the matching and hence increases throughput  Allow the batch size to grow Ref: [Dolev, Kesselman - 2000], “Bounded latency scheduling scheme for ATM cells", Computer Networks, vol. 32(3) pp.325-331, 2000.

11. 11 Non Pre-emptive Scheduling … 2 Batch Scheduling N N 1 1 R R Priority-2 Crossbar R R 1 N 1 N Priority-1 Batch- (k+1) Batch- (k)

12. 12 Non Pre-emptive Scheduling … 2 Batch Scheduling N N 1 1 R R Priority-2 R R 1 N 1 N Priority-1 Crossbar Batch- (k+1) Batch- (k)

13. 13 Degree of a Batch 1 2 3 0 1 0 2 1 0 0 0 1 1 2 3 Batch Request Graph  Degree ( d v,k ):  The number of cells departing from (destined to) a vertex in batch k.  Maximum Degree ( D k )  The maximum degree amongst all inputs/outputs in batch k.

14. 14 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 1 2 3 0 1 0 2 1 0 0 0 1 1 2 3 Batch Request Graph with D k =3 2 3 1 2 3 1 Maximum Size Matching Why may MSM not give 100% throughput?

15. 15 Critical Maximum Size Matching A sub-class of MSM 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 1 2 3 0 1 0 2 1 0 0 0 1 1 2 3 Batch Request Graph with D k =3

16. 16 CMSM achieves 100% throughput under non pre- emptive scheduling, if the traffic is constrained to less than cells for any input/output in B timeslots.  This introduces deterministic constraints on the arrival traffic  We are interested in the traditional stochastic traffic Previous Results Ref: [Weller, Hajek - 1997], “Scheduling non-uniform traffic in a packet-switching system with small propagation delay,” IEEE/ACM Transactions on Networking 5(6): 813-823, 1997.

17. 17 Arrival Traffic

18. 18 Contents 1. Background & Motivation 2. Non Pre-emptive Scheduling 3. Achieving 100% throughput with CMSM Bernoulli i.i.d. uniform traffic Bernoulli i.i.d. non-uniform traffic 4. Achieving 100% throughput with MSM Bernoulli i.i.d. uniform traffic 5. Conclusion

19. 19 CMSM with Uniform Traffic  Theorem 1: CMSM gives 100% throughput under non pre-emptive scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform  Informal Arguments:  Let T k be the time to schedule batch k  Then for batch k+1 we buffer new arrivals for time T k  We expect about  T k packets at every input/output  Hence, the maximum degree of batch k +1, i.e. D k+1   T k  Hence for a CMSM, T k+1 = D k+1   T k < T k  Hence T k is bounded in mean.

20. 20  We are going to show that  Alternatively we will first show that  Observe that Formal Arguments Outline

21. 21  We shall use the Chernoff bound to get  If we want to bound D k+1, we require that all the 2N vertices are bounded Formal Arguments … 1 Bounding the degree of a batch

22. 22  Choose  > 0, such that.  Choose  such that  We get Formal Arguments … 2 Bounding the deviation of the service time of a batch

23. 23  Hence Formal Arguments … 3 Bounding the service time of a batch

24. 24  Choose  < (1-  ) /2,  This gives  Observe that  Q is now a function of T k only for a constant   We can make Q as close to 1, by choosing a large T k Formal Arguments …4 Tightening the bound

25. 25  Hence, there is a constant T c such that  Formally, using a linear Lyapunov function V(T k ) = T k, we can say that T k (averaged over the batch index) is bounded in mean. Formal Arguments …5 Finishing Off..

26. 26  In the paper we use a quadratic Lyapunov function V(T k ) = (T k ) 2, and show that T k 2 (averaged over the batch index) is bounded in mean.  There are a few technical steps after this to show that the queue size (averaged over time) is bounded in mean.  Then, it follows that CMSM gives 100% throughput for Bernoulli i.i.d. uniform traffic. Formal Arguments …6 Some Final Points..

27. 27 Contents 1. Background & Motivation 2. Non Pre-emptive Scheduling 3. Achieving 100% throughput with CMSM Bernoulli i.i.d. uniform traffic Bernoulli i.i.d. non-uniform traffic 4. Achieving 100% throughput with MSM Bernoulli i.i.d. uniform traffic 5. Conclusion

28. 28 CMSM with Non-Uniform Traffic  Theorem 2: CMSM achieves 100% throughput under non pre-emptive scheduling, if the input traffic is admissible and Bernoulli i.i.d.

29. 29 Contents 1. Background & Motivation 2. Non Pre-emptive Scheduling 3. Achieving 100% throughput with CMSM Bernoulli i.i.d. uniform traffic Bernoulli i.i.d. non-uniform traffic 4. Achieving 100% throughput with MSM Bernoulli i.i.d. uniform traffic 5. Conclusion

30. 30 Example of a Uniform Graph 1 2 3 1 1 1 1 1 1 1 1 1 1 2 3 Batch Request Graph with D k =3 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1

31. 31 MSM with Non-Uniform Traffic  Theorem 3: MSM achieves 100% throughput under non pre-emptive scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform

32. 32 Contents 1. Background & Motivation 2. Non Pre-emptive Scheduling 3. Achieving 100% throughput with CMSM Bernoulli i.i.d. uniform traffic Bernoulli i.i.d. non-uniform traffic 4. Achieving 100% throughput with MSM Bernoulli i.i.d. uniform traffic 5. Conclusion

33. 33 Conclusions  We have used the more traditional stochastic arrivals and shown using batch scheduling that  CMSM gives 100% throughput for Bernoulli i.i.d. traffic  MSM gives 100% throughput for Bernoulli i.i.d. uniform traffic  It would be nice to understand the stability of MSM with uniform load with continuous scheduling.