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Distributed Scheduling Algorithms for Switching Systems

Distributed Scheduling Algorithms for Switching Systems. Shunyuan Ye, Yanming Shen, Shivendra Panwar. Overview. Background Problem definition, related work A randomized scheduling algorithm Algorithm, example, proof sketch Applications Buffered crossbar switch: DISQUO

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Distributed Scheduling Algorithms for Switching Systems

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  1. Distributed Scheduling Algorithms for Switching Systems Shunyuan Ye, Yanming Shen, Shivendra Panwar

  2. Overview • Background • Problem definition, related work • A randomized scheduling algorithm • Algorithm, example, proof sketch • Applications • Buffered crossbar switch: DISQUO • Optoelectronic switch: HELIOS

  3. Scheduling Problem Objective: Find a scheduling algorithm that can sustain 100% capacity Input 1 Switching Fabric VOQs Output 1

  4. Related Work (1) Maximum Weight Matching (MWM, Tassiulas ’92) Inputs Outputs Inputs Outputs 10 1 1 1 1 15 15 5 10 10 2 2 2 2 2 6 3 8 3 3 3 3 12 12 Centralized O(N3) computations

  5. Related Work (2) Randomized Scheduling Algorithm (Tassiulas ’98) Inputs Outputs Inputs Outputs 1 1 1 1 12 12 6 2 2 2 2 5 8 8 10 4 4 3 3 3 3 O(N) computations Poor Delay Performance Centralized

  6. Related Work (3) iSLIP (McKoewn, ’98) Distributed, but cannot guarantee 100% throughput LAURA (Giaccone et al., ’02) Merge R(n) and S(n-1) Complexity is O(NlogN) EMHW (Li et al., ’04) Using exhaustive service matching, complexity is O(logN) Glauber dynamics work of Walrand et al., Srikant et al., Shah

  7. Question? Can we have a scheduling algorithm which satisfies all the conditions: Guaranteed 100% throughput Low computation complexity, i.e., O(1) Easy to implement in a distributed way

  8. Randomized Scheduling Algorithm Notation Neighbors: N(i, j) = {(i, j’) or (i’, j)} Feasible schedule: If Sij(n) = 1, for any (k,l) in N(i,j), Skl(n) = 0 Sij(n) = 1 Skl(n) = 0

  9. Randomized Scheduling Algorithm S(n-1) is the schedule at time n-1 Randomly generate a feasible schedule H(n): Pre-determined Hamiltonian walk: It can be implemented in a distributed manner with a time complexity of O(1) S(n-1) H(n)

  10. Randomized Scheduling Algorithm S(n) is generated following the rules: a) For (i, j) not inH(n), Sij(n) = Sij(n-1) b) For any (i, j) in H(n): If (i, j) in S(n-1): Sij(n)=1, with probability pij Sij(n)=0, with 1-pij (pij is a concave function of Qij) If (i, j) not in S(n-1): If for any (k, l) in N(i, j), (k, l) was free Sij(n)=1, with probability pij Sij(n)=0, with 1-pij Else, Sij(n) = 0 Stay the same S(n-1) H(n)

  11. Randomized Scheduling Algorithm Example • For (3, 2): the same as (1, 3) • For (2, 1): it was in S(n-1) • For (1, 3): none of its neighbors was active • S21(n+1) = 1, with P21 • S21(n+1) = 0, with 1-P21 • S32(n+1) = 0, in the example • S13(n+1) = 1, with P13 • S13(n+1) = 0, with 1-P13 • S21(n+1) = 1, in the example • S13(n+1) = 1, in the example S(n) S(n+1) H(n+1)

  12. Intuitive Explanation When (i, j) is picked by H(n), and none of its neighbors was active in the previous slot, (i, j) can decide to be active or not with a probability. If (i, j) becomes active, all of its neighbors are blocked from being active. If we define the probability as a concave function of Qij, longer queues have a higher probability to become active (and a lower probability to be blocked by short queues). The weight of active VOQs will be very close to the maximum after the system converges.

  13. Intuitive Explanation Example A higher probability that the schedule is {(1,2), (2, 1)} Q11 = 1 With p11 = 0, S11 = 1 With p12 = 0.7, S12 = 1 Q12 = 10 With p22 = 0.4, S22 = 1 With p21 = 0.8, S21 = 1 Q21 = 8 Q22 = 2 pij= log(Qij) / [1+ log(Qij)]

  14. System Stability Sketch of proof of system stability Define the state of the system as the schedule S(n) S(n-1), S(n), S(n+1) is a Markov chain, and it is time reversible, which implies a product-form stationary distribution. For any admissible Bernoulli arrival traffic, the weight of S(n) is always close to the maximum weight S*(n), after the system converges. System can be proved to be stable.

  15. DISQUO Scheduling Algorithm DISQUO is a distributed implementation for a buffered crossbar switch Advantages: Totally distributed without message passing Delay performance is very good Drawback: N2 crosspoint buffers are needed

  16. Buffered Crossbar Switch Input scheduler and output scheduler can be independent, and thus distributed. VOQij CBij 1 2 … Input 1 N Input 2 … Input N … Output 1 Output 2 Output N

  17. DISQUO Scheduling Algorithm _ n = m Distributed Implementation Example • If crosspoint (i, j) is active, input i and output j have to serve this crosspoint buffer. • Otherwise, they can randomly pick one to serve n = m+

  18. DISQUO Scheduling Algorithm _ n = (m+1) Distributed Implementation Example • For input 1 and 2, they have to decide whether to keep (1, 2) and (2, 1) active based on P12 and P21. • For input 3, it has to decide whether to make (3, 2) active with a probability P33 Inputs and outputs can learn each other’s decisions by observing the crosspoint buffer, so that they can keep the consistency of the schedule • In the example, it decides to become active. • In the example, they both decide to become inactive. n = (m+1)+

  19. Simulations Uniform traffic

  20. Simulations Non-uniform traffic Throughput of RR-RR under hotspot traffic is 85%.

  21. Simulations Impact of switch size Delay is almost independent of switch size.

  22. Simulations Impact of buffer size K=1 is sufficient

  23. HELIOS Scheduling Algorithm HELIOS is a distributed algorithm for a hybrid optical/electrical switch. Advantages: Easy implementation (DWDM optical fiber) Totally distributed without message passing Uses an optical fabric to reduce power consumption Guarantees 100% throughput for any admissible traffic

  24. Architecture Each input is equipped with a fast tunable laser as the transmitter, which can tune to different wavelengths.

  25. Architecture Each output has a fixed wavelength receiver operating in a specific WDM channel.

  26. Architecture The optical fabric is a broadcast-and-select fabric.

  27. The Linecard Model λ-monitor is used to sense the channels, so that the inputs know which wavelengths are being used.

  28. Implementation Example

  29. Simulation Under Bernoulli i.i.d. traffic, the delay performance is poor compared to MWM. But if one slot time is only a few nanoseconds, the delay is still acceptable (i.e. < 10μs)

  30. Simulation Under On-Off bursty traffic, with Pareto distribution (larger α means longer burst length). The delay performance is closer to MWM.

  31. Summary We proposed a scheduling algorithm with a very low computation complexity The algorithm can be easily implemented is a distributed way for different switching architectures It can guarantee 100% throughput for any admissible traffic, and for some architectures it can provide very good delay performance

  32. Thank you!Q&A

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