1. DJW. Infocom 2006 optimal scheduling algorithms for input-queued switches Devavrat Shah, MIT Damon Wischik, UCL Note. The animations in these slides have been removed. For the full set of slides, including animations, download http://www.cs.ucl.ac.uk/staff/D.Wischik/Talks/switch-infocom.zip http://www.cs.ucl.ac.uk/staff/D.Wischik/Talks/switch-infocom.zip

2. DJW. Infocom 2006 Input-queued crossbar switch The matching (or scheduling) algorithm decides which inputs to match with which outputs What is a good scheduling algorithm? What is the quality of service / queueing performance? What is the relationship between scheduling and performance?

3. DJW. Infocom 2006 Example: Serve the longer queue first queue size second queue size

4. DJW. Infocom 2006 Example: Serve the longer queue first queue size second queue size The system is basically one-dimensional + noise If we keep track of the total queue size W, we can deduce the individual queue sizes Q 1 ≈ Q 1 ≈ W /2 This relationship does not depend on the arrival rates (so long as the system is stable)

5. DJW. Infocom 2006 Example: Serve the longer queue first queue size second queue size first queue second queue total workload Another way to visualize this relationship is to plot both — the actual queue sizes — the queue sizes estimated from the workload W, Q 1 = Q 1 = W /2 These agree almost perfectly

6. DJW. Infocom 2006 Terminology State space collapse (SSC) –the fact that the vector of queue sizes Q can be written as a function of the workload W, Q =Δ W Workload –a carefully chosen sum of queue sizes Lifting map –Δ Collapsed (or invariant) space –{ (Q 1, Q 2 ) : Q 1 = Q 2 } –more generally, the set of achievable Q =Δ W, as W varies

7. DJW. Infocom 2006 Input-queued switches have state space collapse! For a n×n switch there are n 2 queues to keep track of The workload vector lists the total queue size for each input port and for each output port, and it has dimension 2n-1 The lifting map Δ depends on the scheduling algorithm Here I’ve illustrated the maximum weight matching scheduler: –at each time step, choose a set of queues to serve so that the sum of their queue sizes is as large as possible output 1output 2output 3output 4 input workload input 1 input 2 input 3 input 4 output workloads measured queue sizes, from a simulation queue sizes inferred from the measured workloads

8. DJW. Infocom 2006 Technical details Method –write down differential equations describing the system, i.e. a fluid model –use combinatorial techniques to prove convergence, and to characterize the fixed points –use heavy traffic queueing theory to show that as the load on the switch increases, fluctuations in Q away from Δ W become negligible compared to Q Lifting map: for arrival rate matrix λ, q =Δ w is the unique solution to w 1 w 2 w 3 w 4 w1w1 w2w2 w3w3 w4w4

9. DJW. Infocom 2006 Why is this useful? (I) We’ve shown that Q =Δ W Once we’ve found Δ it’s easy to work out if any queues are persistently small, i.e. guaranteed good quality of service We can also see if giving priority to some queues has a negative impact on others

10. DJW. Infocom 2006 Why is this useful? (II) We’ve shown that Q =Δ W W is easier to reason about than Q –In those states where every queue is non-zero, the switch is work-conserving (i.e. no service is wasted) –It’s therefore useful to look at the space {W : Δ W >0} and to choose the scheduling algorithm to make this as big as possible We’ve used this to conjecture an optimal scheduling algorithm –i.e. one which never wastes service because of poor scheduling (in a stochastic limit sense) –At each timestep, it considers all max- size matchings, and chooses one with maximum log-weight

11. DJW. Infocom 2006 Conclusion We have reduced the problem of analysing scheduling algorithms to questions about the algebra of Δ It’s hard algebra! So far we only know Δ for a small class of algorithms, variations on maximum-weight matching The same approach works for any generalized switch, e.g. schedulers in wireless base-stations