Distributed Scheduling Algorithms for Switching Systems Shunyuan Ye, Yanming Shen, Shivendra Panwar 2015-7-161.

Distributed Scheduling Algorithms for Switching Systems

Shunyuan Ye, Yanming Shen, Shivendra Panwar

23/4/19 1

Overview• Background– Problem definition, related work

• A randomized scheduling algorithm– Algorithm, example, proof sketch

• Applications– Buffered crossbar switch: DISQUO– Optoelectronic switch: HELIOS

23/4/19 2

Scheduling Problem

• Objective: Find a scheduling algorithm that can sustain 100% capacity

Input 1

Output 1

VOQs Switching Fabric

Related Work (1)• Maximum Weight Matching (MWM, Tassiulas ’92)

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Inputs Outputs10

15510

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Inputs Outputs

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Centralized O(N3) computations

Related Work (2)• Randomized Scheduling Algorithm (Tassiulas ’98)

CentralizedO(N) computations

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Inputs Outputs

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Inputs Outputs

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Poor Delay Performance

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

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

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

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)

Randomized Scheduling Algorithm• S(n) is generated following the rules:• a) For (i, j) not in H(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 S(n-1) H(n)

Stay the same

Randomized Scheduling Algorithm• Example

S(n) H(n+1)

• For (1, 3): none of its neighbors was active

• S13(n+1) = 1, with P13

• S13(n+1) = 0, with 1-P13

• S13(n+1) = 1, in the example

• For (2, 1): it was in S(n-1)

• S21(n+1) = 1, with P21

• S21(n+1) = 0, with 1-P21

• S21(n+1) = 1, in the example

• For (3, 2): the same as (1, 3)

• S32(n+1) = 0, in the example

S(n+1)

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.

Intuitive Explanation• Example

• A higher probability that the schedule is {(1,2), (2, 1)}

Q11 = 1Q12 = 10

Q21 = 8

Q22 = 2pij= log(Qij) / [1+ log(Qij)]

With p11 = 0, S11 = 1

With p22 = 0.4, S22 = 1

With p12 = 0.7, S12 = 1

With p21 = 0.8, S21 = 1

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.

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

Buffered Crossbar Switch

• Input scheduler and output scheduler can be independent, and thus distributed.

Output N

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N

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Input 2

Input N

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Output 1 Output 2

Input 1

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CBijVOQij

DISQUO Scheduling Algorithm

• Distributed Implementation Example

n = m+n = m_

• 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

DISQUO Scheduling Algorithm

• Distributed Implementation Example

n = (m+1)+n = (m+1)_

Inputs and outputs can learn each other’s decisions by observing the crosspoint buffer, so that they can keep the consistency of the schedule

• For input 1 and 2, they have to decide whether to keep (1, 2) and (2, 1) active based on P12 and P21.

• In the example, they both decide to become inactive.

• For input 3, it has to decide whether to make (3, 2) active with a probability P33

• In the example, it decides to become active.

Simulations

• Uniform traffic

Simulations

• Non-uniform traffic– Throughput of RR-RR under hotspot traffic is 85%.

Simulations

• Impact of switch size– Delay is almost independent of switch size.

Simulations

• Impact of buffer size– K=1 is sufficient

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

Architecture

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

Architecture

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

Architecture

• The optical fabric is a broadcast-and-select fabric.

The Linecard Model

• λ-monitor is used to sense the channels, so that the inputs know which wavelengths are being used.

Implementation Example

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)

Simulation

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

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

Thank you!

Q&A