M a r y a m E l a h ib m e l a h i @ u c a l g a r y. c a

Fairness in Speed Scaling Design

J o i n t w o r k w i t h : C a re y W i l l i a m s o n a n d P h i l i p p Wo e l f e l

2

SCHEDULING

• Sharing resources• Maximize efficiency• Minimize cost

Datacenters

LocksDatabase

s

RoutersInternet

And more…

Requests

(Jobs)Queue

Arrival Rate (λ)

Shared Resource (server)

CPUs

Fixed Service Rates

3

THE DECISION

• Scheduler:• Which job to serve?• Preemptive• Non-preemptive

• Goal(s):• 1: Minimize response

time

Simple Model

?

M/M/1

4

SCHEDULERS

• FCFS: First-Come-First-Served

• PS: Processor Sharing

• SRPT: Shortest-Remaining-Processing-Time• ...

System Load0 0.25 0.5 0.75

Mean

Resp

on

se T

ime

PS

SRPT

FCFS

SRPT is proved optimal, but in practice often PS is used.

5

OTHER GOALS?

• Goal(s):• 1: Minimize response

time

Other QoS:

• 2: Fairness

• 3: Robustness• …

• Optimizes response time

• Is it fair to large jobs?

• What is fair?

SRPT

6

JUSTIFICATION

• Aristotle's notion of fairness• Treat like cases alike• Treated different cases differently, in proportion to their

differences

Response times should be proportional to job sizes

E[T(x)]/x should be constant

Normalized response time or slowdown

T(x): Response

Time of a job with size x

Policy P is fair if: [Wierman et al. 2003]

7

x

E[T

(x)]

/ x

FAIRNESS OF SCHEDULERS

• PS: Always Fair

• FCFS: Always Unfair

• SRPT: Sometimes Fair

• ...

• Fair and Optimal?

FCFS

SRPTPS

x

E[T

(x)]

/ x

SRPT

PS

Lig

ht

load

Hig

h e

nou

gh

load

8

FAIR AND OPTIMAL?

• FSP: Fair Sojourn Protocol [Friedman, et al. 2003]

• Implement SRPT on the PS remaining times

• Slowdown: never worse than PS

• Mean response time: close to that of SRPT

E[T

(x)]

/ x

SRPTPS

FSP

x

• Compute the completion time under virtual PS

• Order the jobs based on their virtual completion times

• Execute the job with the earliest PS completion time

9

THE DECISIONS

• Scheduler:• Which job to serve?

• What speed to use?• Dynamic speed scaling• Gated-static (shut-down)

• Goal(s):• 1: Minimize response time (T)

• 2: Minimize energy usage (E)

?

Adjustable Service Rates

Speed

Pow

er

P(s) = sα

10

THE TRADEOFF

• Combined Goal: Linear combination of the goals:

• Other objectives considered:• Minimize energy for jobs with

deadlines• Minimize response time

subject to energy budget• response-time x energy response-timeenergy 𝑇E

How much reduction in response time justifies using one extra joule

β: cost of energy

11

DYNAMIC SPEED SCALING

Which job to serve?

• Speed Scaling Policy

What speed?

• Job scheduling

SRPT

?

P(s) = sα

n: jobs in the system

+

[Bansal et al. 2009]: 3-competitive for arbitrary power function

12

FAIRNESS AND SPEED SCALING

FCFS

SRPT

PS

FSP

Biased towards big jobs

Biased towards small jobs

Treats all the same

?

[Andrew et al. 2010]: Jobs that run when the queue is big, run faster

13

FAIRNESS AND SPEED SCALING

• Is Slowdown of PS still the right criterion for fairness?

[Andrew et al. 2010]:

- For PS with speed scaling, stays constant.

- Dynamic Speed Scaling magnifies unfairness under SRPT and non-preemptive policies like FCFS.

14

SIMULATION STUDY

• Discrete event simulator written in C++• Scheduling policies: FCFS, PS, SRPT, FSP• Speed Scaling:

• Workload:• Poisson Arrival: rate λ• Exponentially distributed job sizes: rate μ

15

RESULTS: CONSTANT SPEEDLOAD: 0.5

16

RESULTS: CONSTANT SPEEDLOAD: 0.8

17

RESULTS: DYNAMIC SPEED SCALINGLOAD: 0.5

18

RESULTS: DYNAMIC SPEED SCALINGLOAD: 0.8

19

CONCLUSIONS

• FSP with speed scaling shows better fairness behavior in comparison to speed scaled SRPT

• The definition of fairness for scheduling with speed scaling requires further investigations

QUESTIONS?THANK YOU!