1 Scheduling Reserved Traffic in Input-Queued Switches: New Delay Bounds via Probabilistic Techniques Matthew Andrews and Milan Vojnović Bell Labs, Lucent Technologies; EPFL, Switzerland Infocom 2003, SF, March 30-April 3, 2003

2 Introduction: Input-Queued Switch input portsoutput ports II crossbar At any point in time, connectivity restricted to permutation matrices

3 Some Existing Approaches for Crossbar Scheduling maximum-weight matching (McKeown ‘96, many others) decomposition-based scheduling (Chang et al, 2000) fluid-tracking (Tabatabaee et al, ToN ’01)

4 Decomposition-Based Scheduling Given: M, a I x I doubly sub-stochastic rate demand matrix Constraint: crossbar Objective: Find an ordered sequence of permutation matrices, which if used as a schedule provides lower bound M to the service in the long-run Desired Property: we want a schedule to be “smooth”

5 Decomposition-Based Schedulers 3x3 example A schedule: A smoother schedule:

6 A Decomposition: Birkoff/von Neumann M k a permutation matrix  k > 0 intensity of the k-th permutation matrix Theorem by Birkoff/von Neumann Applied to the switch problem by Chang et al 2000 M, a I x I doubly stochastic

7 The Problem that We Study Given: M 1, M 2, …, M K a sequence of permutation matrices Find: schedules with a guarantee on their smoothness “smooth” quantified through the concept of latency defined shortly Why is the Problem Important diffserv EF (Expedited Forwarding), MPLS, Connection-Reservation-Table for switch control traffic Rate and delay-jitter guarantees for

8 Smooth per-permutation matrix may not mean smooth per input- output port An input-output port pair may be scheduled by more than one permutation matrix Aggregate of subset of permutation matrices may be not smoothly scheduled, even though the schedule of permutation matrices is smooth If each input-output port pair would have 1 in exactly 1 perm. matrix, then classical polling

9 Latency of a Schedule Number of slots offered to the ij-th port pair in [0,m) m

10 Rest of the Talk Reminder on schedule and latency of Chang et al Our schedulers and their latencies –Random-Permutation –Random-Phase –Random-Distortion –Poisson Competition (not displayed in the slides) A numerical example Conclusion

11 Scheduler by Chang et al Superposition: M1:M1: M2:M2: M3:M3: PGPS token arrivalstokens placed back as new arrivals Initialization: token of type k arrive at Schedule:

12 Scheduler by Chang et al (cont’d) The bound of Chang et al is almost tight Example 3x3:

13 Random Permutation Scheduler Superposition: M1:M1: copy from [0,1)... copy from [0,1) M2:M2: M3:M3:

14 Random-Phase Scheduler Superposition: M1:M1: M2:M2: M3:M3:

15 Random-Distortion Scheduler Schedule: M1:M1: M2:M2: M3:M3:

16 Results: Latencies Random permutation: Random phase: Random distortion:

17 Numerical Example: varying switch size Ob.: except for small switch sizes, the random-phase bound is tighter than PGPS; the random-distortion bound is tightest

18 Conclusion By randomizing the point processes one can obtain less pessimistic bounds on latency that hold in probability One can derandomize and obtain latencies that hold with probability 1 Approach of the point processes may be used to construct other schedulers Worth to try to obtain sharper results A question remains: what is the best possible latency for load larger than 1/4