SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors

Janardhan Kulkarni, Duke University

Sungjin Im (UC Merced ), Kamesh Munagala (Duke), Kirk Pruhs (UPitt)

1

2

M

ONLINE UNRELATED MACHINE MODEL

3- processing length

- rate of processing

- weight

1

2

M

ONLINE UNRELATED MACHINE

3

100

(rate = 1, length=100)

(rate = 0.5, length 200)

(rate = 2, length =50)

Time

- processing length

- rate of processing

(rate = 0, length = infinity)

1

2

M

NON-CLAIRVOYANCE

3

(rate = 1, length=100)

(rate = 0, infinity)

(rate = 0.5, length = 200)

(rate = 2, length =50)

- processing length

- rate of processing

- weight

Time

FLOW-TIME (DELAY OR RESPONSE TIME)

ANOTHER FORM

SPEED-AUGMENTATION: - Online scheduler runs at higher speed

PREMPTION: - Stop a job in middle of processing and continue later.

MIGRATION: - Change the assignment of jobs to machines at no cost.

NO PARALLEL PROCESSING

- No job is simultaneously processed on multiple machines.

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2

OK

NO PARALLEL PROCESSING:

- No job is simultaneously processed on multiple machines.

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2

NOT ALLOWED

How to schedule jobs to minimize sum of weighted flow-time

for unrelated machines in non-clairvoyant model?

It was very difficult

Motivation

• Heterogeneous architectures are ubiquitous

• Non-clairvoyance is very important in practice

Motivation

SELFISHMIGRATE is -speed -competitive for minimizing

weighted flow-time on unrelated machines.

Result also extends to energy + weighted flow-time.

THE MAIN THEOREM

CLAIRVOYANT ALGORITHM

Anand-Garg-Kumar, SODA ‘12Chadha-Garg-Kumar-Muralidhara, STOC ‘09

SJF

SJF

ALGORITHM

53 70

101 10020

2020

Increase B = 28 + 20

Increase A = 31 + 20 NEW

For all jobs

For all machines

THE DUAL

For all jobs machines and time instants

The whole frame work breaks down for non-clairvoyant settings…

1. We cannot measure the increase in the objective

2. Migrating jobs is necessary

3. Migration of jobs makes calculating the flow-times difficult and complicates how jobs interfere with each other

SELFISHMIGRATE

KEY IDEA I

Focus on speed at which jobs are getting processed and monotonicity of speed

Speed of a job =

fraction of the job processed at time

If is monotone, then

If is monotone, then

Dual Constraint:

If is monotone, then

Dual Constraint:

Fraction of processor allocated =

Round Robin or Proportional Sharing

If is monotone, then

Dual Constraints:

If is monotone, then

Dual Constraints:

But which machine we should assign the job ?

If is monotone, then

Dual Constraints:

But which machine we should assign the job ? ANS IS MINIMUM

Dual constraints need us…

For all jobs machines and time instants

1. BEST MACHINE- which maximizes speed.

2. The speed MONOTONICALLY increases.

KEY IDEA II: VIRTUAL QUEUES, VIRTUAL SPEED

RR

j

VIRTUAL QUEUES, VIRTUAL SPEEDS

RR

Inverse of instantaneous delay the job induces on the other jobs ahead of its virtual queue.

KEY IDEA II: VIRTUAL QUEUES, VIRTUAL SPEED

RR

Total delay induced by job if it stays on the machine till its completion is proportional to

KEY IDEA III: SELFISHMIGRATE

RULE 1: NO JUMPING (not even virtual ones!)

Virtual speed of other jobs does not decrease if a job enters the machine!

RULE 1: NO JUMPING

RULE 2: BE SELFISH!

Increases virtual speed… migrate! Any time!

RULE 2: BE SELFISH!

Remains Same

Remains Same Increases!

Jobs migrate whenever they can increase their virtual speed...

Jobs migrate whenever they can increase their virtual speed…..

Jobs migrate whenever they can increase their virtual speed…

Jobs migrate whenever they can increase their virtual speed…..

This cannot go on forever… as virtual speeds are finite

We get …MONOTONICITY

Virtual speeds are monotone non-decreasing!

BEST MACHINE

each job chooses the best machineon arrival

BEST MACHINE (Nash Equilibrium on Virtual Speed)

SELFISHMIGRATE ensures this istrue at all times by design!

BEST MACHINE

Entire process can be thought asCoordination ‘game’ on virtual speeds

DUAL SETTINGS

The total delay job causes on the jobs ahead of its virtual queue over its entire sojourn

DUAL SETTINGS

The total delay job j causes on the jobs ahead of its virtual queue over its entire sojourn + its total processing time.

The flow-time of all jobs (Cost of the algorithm)

The total number of jobs in the queue at time t.

The total flow-time of all jobs (Cost of the algorithm)

Dual setting

=

=

DUAL OBJECTIVE

What About Constraints?

For all jobs machines and time instants

DELAY LEMMAS

(Follows from a property of Round Robin)

Total delay a job induced on jobs ahead of it over the interval

Total delay a job induced on jobs ahead of it in

DELAY LEMMA

DELAY LEMMA

MINIMUM

(BEST MACHINE)

We gave a new framework to analyze non-clairvoyant scheduling problems

CONCLUSIONS and OPEN PROBLEMS

- norms of weighted flow-time

THANKS