Maximum Size Matchings & Input Queued Switches

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

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.

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

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.

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

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

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

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?

Non Pre-emptive Scheduling … 1Batch 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.

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

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

Degree of a Batch Batch Request Graph • Degree (dv,k): • The number of cells departing from (destined to) a vertex in batch k. • Maximum Degree (Dk) • The maximum degree amongst all inputs/outputs in batch k. 0 1 1 1 0 0 2 2 2 1 0 3 0 3 1

1 1 2 2 3 3 1 1 Maximum Size MatchingWhy may MSM not give 100% throughput? 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3 Batch Request Graph with Dk =3 0 1 1 1 0 0 2 2 2 1 0 3 0 3 1

1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 Critical Maximum Size MatchingA sub-class of MSM Batch Request Graph with Dk =3 0 1 1 1 0 0 2 2 2 1 0 3 0 3 1

Previous Results 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 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.

Arrival Traffic

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 Tk be the time to schedule batch k • Then for batch k+1 we buffer new arrivals for time Tk • We expect about  Tkpackets at every input/output • Hence, the maximum degree of batch k +1, i.e. Dk+1  Tk • Hence for a CMSM, Tk+1 = Dk+1  Tk < Tk • Hence Tk is bounded in mean.

Observe that Formal ArgumentsOutline • We are going to show that • Alternatively we will first show that

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

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

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

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

Formal Arguments …5Finishing Off.. • Hence, there is a constant Tcsuch that • Formally, using a linear Lyapunov function V(Tk) = Tk, we can say that Tk (averaged over the batch index) is bounded in mean.

Formal Arguments …6Some Final Points.. • In the paper we use a quadratic Lyapunov function V(Tk) = (Tk)2 , and show that Tk2(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.

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.

1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 Example of a Uniform Graph Batch Request Graph with Dk =3 1 1 1 1 1 1 2 1 2 1 1 3 1 3 1

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

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.

Maximum Size Matchings & Input Queued Switches