DJW. Infocom 2006

optimal scheduling algorithmsfor 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

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?

DJW. Infocom 2006

Example: Serve the longer queue

first queue size

second queue size

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 Q1 ≈ Q1 ≈ W /2

• This relationship does not depend on the arrival rates (so long as the system is stable)

DJW. Infocom 2006

Example: Serve the longer queue

first queue size

second queue size

firstqueue

secondqueue

totalworkload

• Another way to visualize this relationship is to plot both— the actual queue sizes

— the queue sizes estimated from the workload W, Q1 = Q1 = W /2

• These agree almost perfectly

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=ΔW

• Workload– a carefully chosen sum of queue sizes

• Lifting map– Δ

• Collapsed (or invariant) space– { (Q1, Q2) : Q1= Q2 }

– more generally, the set of achievable Q=ΔW, as W varies

DJW. Infocom 2006

Input-queued switches have state space collapse!

• For a n×n switch there are n2 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

D

X

output 1 output 2 output 3 output 4input

workload

input 1

input 2

input 3

input 4

output workloads

measured queue sizes, from a simulation

queue sizes inferred from the measured workloads

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 ΔW become negligible compared to Q

• Lifting map: for arrival rate matrix λ, q=Δw is the unique solution to

D

X

w1•

w2•

w3•

w4•

w•1 w•2 w•3 w•4

DJW. Infocom 2006

Why is this useful? (I)

• We’ve shown that Q=ΔW

• 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

DJW. Infocom 2006

Why is this useful? (II)• We’ve shown that Q=ΔW • 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 : ΔW >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

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