raqfm – a resource allocation queueing fairness measure david raz school of computer science, tel...
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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
June, 2004 D. Raz, RAQFM, Sigmetrics04220 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
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