RAQFM – a Resource Allocation Queueing Fairness Measure

David RazSchool of Computer Science, Tel Aviv University

Jointly withHanoch Levy, Tel Aviv UniversityBenjamin Avi-Itzhak, RUTGERS University

Sigmetrics – Performance, June 2004

June, 2004 D. Raz, RAQFM, Sigmetrics042

Outline

Motivation– The importance of fairness in queues– The physical properties of the problem– Related work

The RAQFM approach Properties Analysis

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Why is Fairness in Queues Important?

Fundamental reason:– Isn’t fairness why we have queues in the first

place? Scientific evidence, recent studies, Rafaeli et.

al. [2003] (experimental psychology):– Experiments on humans in multi-queue and single

queue – Fairness in queue is very important to people– Perhaps even more than the delay itself

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Queueing Theory and Fairness

Decades of research on:– Queueing structures / policies– Focused on efficiency – delay distribution/moments,

utilization, etc. The issue of fairness is discussed, but not quantified

– Larson (1988), Palm (1953), Mann (1969), Whitt (1984) etc. Existing measures for streams (WFQ) Little analysis on job fairness

– Morris & Wang (1985)– Avi-Itzhak & Levy (2004)– Wierman & Harchol-Balter (Sigmetrics 2003)

We know very little about job fairness

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Queues: The Physical Factors

Size(Service Requirement)

ResourcesSeniority(Arrival Time)

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Fairness: Size and Seniority

Size

Seniority

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The Size vs. Seniority Dilemma

Short vs. Long

Is it more fair to serve Short ahead of Long? What is the intuition?

LongShort

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Recent Related Work(1)

Avi-Itzhak & Levy (2004):– Axiomatic approach to fairness– Departure point & emphasis: Seniority (Order of

service)

Some results (for equal service times)– FCFS is the most fair (among non preemptive)– LCFS is the most unfair (among non preemptive)

Takes Seniority into account

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Recent Related Work(2)

Wierman & Harchol-Balter (2003):– Propose a Fairness Criterion– Slowdown: for job of size x compute E[T(x)/x]. If it

is bounded the system is FAIR.

Some results:– FCFS is “Always UNFAIR”– LCFS (preemptive) is FAIR

Takes Size into account

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Requirements From a Job Fairness Measure

Reacts well to both Seniority and Size Aim for standard have a consistent view

and agree with Intuition build confidence Yields to Analysis

RAQFM – A Resource Allocation Queueing Fairness Measure

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RAQFM Philosophy

Equal Share of Resources

Fairness

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RAQFM - How to Apply the Philosophy: Individual Discrimination

At every epoch t with N(t) customers in the system, each customer should get 1/N(t)

Warranted service:

Granted service:

Compare the warranted service with the granted service : discrimination:

i

i

dttN

Ri

departure

arrival )(

1

iii sdttsGi

i

departure

arrival

)(

iii RGD

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Basic Properties of the Discrimination

E[D]=0 (every sample path, every policy)Proof sketch: The momentary rate of discrimination is”zero sum”

For PS individual discrimination is zero

0)(

1)(1

tNtN

0)(

1

)(

1

tNtNDi

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RAQFM – System Fairness

System Unfairness: differences in customer treatment (customer discrimination)

We measure the unfairness in a system using Var[D] (D is r.v.)

Var[D] ≥ 0

Property 1: PS is the most fair policy

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Short vs. Long Revisited

Seniority (Arrival Time)

Difference sen Size (Service Requirement)

Difference

sizeIs it more fair to serve Short ahead of Long?

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The difference in unfairness resulting from serving short ahead of long is

If size difference is small – serve by order of arrival If seniority difference is small – serve by order of

service requirement

)( sizesenc

RAQFM Agrees With Intuition

Short vs. Long Revisited

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Property 2: Bounds

Individual discrimination is bound by– How good– How bad

i iD s

/ 2i iD W

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Property 3: Reaction to Seniority

Theorem: If customers have equal service requirements– For each pair of customers, it is more fair to serve

the senior first

⇒FCFS is the most fair⇒LCFS is the least fair

(Proof sketch: compare scenarios)

RAQFM Reacts Well to Seniority

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Property 3: Reaction to Size

Theorem: If customers arrive together– For each pair of customers, it is more fair to serve

the shorter first

⇒SJF is the most fair⇒LJF is the least fair

(Proof sketch – prove )

RAQFM Reacts Well to Size

2

1

[ ] ( )1

N

ii

N iVar D S

N i

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Property 4: RAQFM is Analyzable

Example: analysis of M/M/1 FCFS– Markov chain state is (a,b)=(# ahead, # behind)– c(a,b)=momentary discrimination rate at state (a,b)– d(a,b)=expected discrimination of a walk starting

at (a,b)–

– Similarly for the second moment d(2)(a,b)–

( 1, ) 0( , ) ( , ) ( , 1)

0 0

d a b ad a b tc a b d a b

a

kk

PkdDE

0

)2(2 )0,(][

June, 2004 D. Raz, RAQFM, Sigmetrics04210 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

1

2

3

4

5

6

7

Var

[D]

Preemptive LCFSNon-Preemprive LCFSROSFCFSPS

Fairness of Service Policies Not Discriminating For Sizeas function of load

Empty system: everyon

e is alone

LCFS: Severe

seniority violation

FCFS: no seniority violation

PS:

Absolute

Fairness

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0

1

2

3

4

5

6

7

Var

[D]

Preemptive LCFSNon-Preemprive LCFSROSFCFSPS

Is FCFS Always More Fair Than LCFS?

Mice

Elephants

1.0/1,5.0

10/1,01.0

22

11

FCFS = 0.9

Preemptive LCFS = 0.15

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Summary

Fairness is extremely important, yet there are little quantification methods for jobs

RAQFM– Philosophy: fair share of service– Agrees with intuition– Reacts well to seniority and size– Yields to analysis

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Future Work

More service policies – SRPT, SJF, FB etc. Generalized service requirement Different architectures – multi-class, multi-

server, multi-queue, polling etc.

RAQFM – a Resource Allocation Queueing Fairness Measure

David RazSchool of Computer Science, Tel Aviv University

Jointly withHanoch Levy, Tel Aviv UniversityBenjamin Avi-Itzhak, RUTGERS University

Sigmetrics – Performance, June 2004

Thank You

http://www.cs.tau.ac.il/~davidraz