Recent sojourn time results for Multilevel Processor-Sharing scheduling disciplines

1aaltoeurandom.ppt Eurandom, Eindhoven, The Netherlands, 26.28.8.2008

Recent sojourn time results forMultilevel Processor-Sharing

scheduling disciplines

Samuli Aalto (TKK)in cooperation with

Urtzi Ayesta (LAAS-CNRS)

2

In the beginning was ...

• Eeva (Nyberg, currently Nyberg-Oksanen) ... • who went to Saint Petersburg in January 2002 and ... • met there Konstantin (Avrachenkov) ...• who invited her to Sophia Antipolis ...• where she met Urtzi (Ayesta).

• After a while, they asked:

Which one is better: PS or PS+PS?

3

Outline

• Introduction• DHR service times• IMRL service times• NBUE+DHR service times• Summary

4

Queueing context

• Model: M/G/1– Poisson arrivals– IID service times with a general distribution– single server

• Notation:

– At) = arrivals up to time t

– Si = service time of customer i

– Xit) = attained service (= age) of customer i at time t

– SiXit) = remaining service of customer i at time t

– Ti = sojourn time (= delay) of customer i

– Ri = Ti Si = slowdown ratio of customer i

5

NWUE

IMRL

DHR

NBUE

DMRL

IHR

Service time distribution classes

• DHR = Decreasing Hazard Rate• IMRL = Increasing Mean Residual Lifetime• NWUE = New Worse than Used in Expectation

• IHR = Increasing Hazard Rate• DMRL = Decreasing Mean Residual Lifetime• NBUE = New Better than Used in Expectation

6

Scheduling/queueing/service disciplines

• Non-anticipating:– FCFS = First-Come-First-Served

• service in the arrival order– PS = Processor-Sharing

• fair sharing of the service capacity– FB = Foreground-Background

• strict priority according to the attained service• a.k.a. LAS = Least-Attained-Service

– MLPS = Multilevel Processor-Sharing• multilevel priority according to the attained service

• Anticipating:– SRPT = Shortest-Remaining-Processing-Time

• strict priority according to the remaining service

7

Optimality results for M/G/1

• Among all scheduling disciplines, – SRPT is optimal (minimizing the mean delay);

Schrage (1968)

• Among non-anticipating scheduling disciplines, – FB is optimal for DHR service times;

Yashkov (1987); Righter and Shanthikumar (1989)– FCFS is optimal for NBUE service times;

Righter, Shanthikumar and Yamazaki (1990)

NWUEIMRL

DHRDMRL

IHR

NBUE

8

Multilevel Processor-Sharing (MLPS) disciplines

• Definition: Kleinrock (1976), vol. 2, Sect. 4.7– based on the attained service times

– N1 levels defined by N thresholds a1 … aN

– between levels, a strict priority is applied– within a level, an internal discipline is applied

(FB, PS, or FCFS)

a

FCFS+FB(a)Xi(t)

t

FB

FCFS

9

• We compare MLPS disciplines in terms of the mean delay:

– MLPS vs MLPS– MLPS vs PS– MLPS vs FB– Optimality of MLPS disciplines

• We consider the following service time distribution classes:

– DHR – IMRL– NBUE+DHR

Our objective

NWUEIMRL

DHR

NBUEDMRL

IHR

NBUE+DHR

10

Outline


11

Class: DHR service times

• Service time distribution:

• Density function:

• Hazard rate:

• Definition: – Service times are DHR if

h(x) is decreasing• Examples:

– Pareto (starting from 0) and hyperexponential

)(1)( },{)( xFxFxSPxF

}{)( dxSPxf

x dyyf

xfxFxfxh

)(

)()()()(

NWUEIMRL

DHR

NBUEDMRL

IHR

12

Tool: Unfinished truncated work Ux(t)

• Customers with attained service less than x:

• Unfinished truncated work with truncation threshold x:

• Unfinished work:

}},min{)(|)({)( xStXtAitN iix

)( ))(},(min{)( tNi iix xtXxStU

)( ))(()()( tNi ii tXStUtU

13

Example: Mean unfinished truncated work

bounded Pareto service time distribution

14

Optimality of FB w.r.t. Ux(t)

• Feng and Misra (2003); Aalto, Ayesta and Nyberg-Oksanen (2004):

– FB minimizes the unfinished truncated work Uxt) for

any x and t in each sample path

s

FCFSXi(t)

t

x

FB

t

Ux(t)

sx

15

Idea of the mean delay comparison

• Kleinrock (1976):

– For all non-anticipating service disciplines

– so that (by applying integration by parts)

• Thus,

• Consequence: – among non-anticipating service disciplines,

FB minimizes the mean delay for DHR service times

0

'1' )]([)( xhdUUTT xx

'' & DHR TTxUU xx

0

1 ][ )(

xUdxhT

16

MLPS vs PS

• Aalto, Ayesta and Nyberg-Oksanen (2004):– Two levels with FB and PS allowed as internal

disciplines

• Aalto, Ayesta and Nyberg-Oksanen (2005): – Any number of levels with FB and PS allowed as

internal disciplines

PSPSPSPSFBFB DHR TTTT

PSMLPSFB DHR TTT

FB/PS

FB/PS

FB/PS

PSFB

17

MLPS vs MLPS: changing internal disciplines

• Aalto and Ayesta (2006a):– Any number of levels with all internal disciplines

allowed– MLPS derived from MLPS’ by changing an internal

discipline from PS to FB (or from FCFS to PS)MLPS'MLPS DHR TT

FB/PS PS/FCFS

MLPS’MLPS

18

MLPS vs MLPS: splitting FCFS levels

• Aalto and Ayesta (2006a):– Any number of levels with all internal disciplines

allowed– MLPS derived from MLPS’ by splitting any FCFS level

and copying the internal disciplineMLPS'MLPS DHR TT

FCFSFCFS

MLPS’MLPS

FCFS

19

MLPS vs MLPS: splitting PS levels

• Aalto and Ayesta (2006a): – Any number of levels with all internal disciplines

allowed– The internal discipline of the lowest level is PS– MLPS derived from MLPS’ by splitting the lowest level

and copying the internal discipline

• Splitting any higher PS level is still an open problem!

MLPS'MLPS DHR TT

PSPS

MLPS’MLPS

PS

20

Idea of the mean slowdown ratio comparison

• Feng and Misra (2003):

– For all non-anticipating service disciplines

– so that

• Thus,

• Consequence: – Previous optimality (FB) and comparison (MLPS vs PS,

MLPS vs MLPS) results are also valid when the criterion is based on the mean slowdown ratio

0

)('1' ][)(xxh

xx dUURR

'' & DHR RRxUU xx

0

)(1 ][

xx

xh UdR

21

Outline


22

Class: IMRL service times

• Recall: Service time distribution:

• H-function:

• Mean residual lifetime (MRL):

• Definition: – Service times are IMRL if

H(x) is decreasing• Examples:

– all DHR service time distributions, Exp+Pareto

}{)( ),(1)( },{)( dxSPxfxFxFxSPxF

xx

x

dyyF

xF

dyyF

dyyfxH

)(

)(

)(

)()(

NWUEIMRL

DHR

NBUEDMRL

IHR

)(1

)(

)(]|[

xHxF

dyyFxxSxSE

23

Tool: Level-x workload Vx(t)

• Customers with attained service less than x:

• Unfinished truncated work with truncation threshold x:

• Level-x workload:

• Workload = unfinished work:

}},min{)(|)({)( xStXtAitN iix

)( ))(},(min{)( tNi iix xtXxStU

)())(()()( )( tUtXStVtV tNi ii

)( ))(()( tNi iix xtXStV

24

Example: Mean level-x workload

bounded Pareto service time distribution

25

Non-optimality of FB w.r.t. Vx(t)

• Aalto and Ayesta (2006b):

– FB does not minimize the level-x workload Vxt) (in any sense)

s

FCFSXi(t)

t

x

FB

t

Vx(t)

sx

FB notoptimal

26

Idea of the mean delay comparison

• Righter, Shanthikumar and Yamazaki (1990): – For all non-anticipating service disciplines

– so that

• Thus,

0

'1' )]([)( xHdVVTT xx

'' & IMRL TTxVV xx

0

1 ][ )(

xVdxHT

27

MLPS vs PS

• Aalto (2006): – Any number of levels with FB and PS allowed as

internal disciplines

• Consequence:

PSMLPS IMRL TT

FB/PS

FB/PS

FB/PS

PS

PSFB IMRL TT

28

Non-optimality of FB

• Aalto and Ayesta (2006b):– FB does not necessarily minimize the mean delay for

IMRL service times• Counter-example:

– Exp+Pareto is IMRL but not DHR (for 1 c e):

– There is 0 such that

,

0 ,)(

cxx

cxcxF

c

x

FB)(FBFCFS TT c

FB

FCFSFB

29

Outline


30

Class: NBUE+DHR service times

• Recall: Hazard rate

• Recall: H-function:

• Definition:– Service times are NBUE+DHR(k) if

• H(x) H(0) for all xk and

• h(x) is decreasing for all xk • Examples:

– Pareto (starting from k 0), Exp+Pareto, Uniform+Pareto

xx

x

dyyF

xF

dyyF

dyyfxH

)(

)(

)(

)()(

x dyyf

xfxFxfxh

)(

)()()()(

NWUEIMRL

DHR

NBUEDMRL

IHR

NBUE+DHR

31

Tool: Gittins index

• Gittins (1989):– J-function:

– Gittins index for a customer with attained service a:

– Optimal quota:

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

)(aHaJahaJaJ

aa

aa

dyyF

dyyf

),(sup)( 0 aJaG

)}(),(|0sup{)(* aGaJa

32

Example: Gittins index and optimal quota

Pareto service time distribution

k *

33

Properties

• Aalto and Ayesta (2007), Aalto and Ayesta (2008):– If service times are DHR, then

• G(a) is decreasing for all a – If service times are NBUE, then

• G(a) G(0) for all a– If service times are NBUE+DHR(k), then

• *(0) k

• G(a) G(0) for all a*(0) and • G(a) is decreasing for all a k

• G(*(0)) G(0) (if *(0) )

NWUEIMRL

DHRDMRL

IHR

NBUE+DHRNBUE

34

Optimality of the Gittins discipline

• Definition:– Gittins discipline serves the customer with highest

index

• Gittins (1989); Yashkov (1992):– Gittins discipline minimizes the mean delay in M/G/1

(among the non-anticipating disciplines)

• Consequences: – FB is optimal for DHR service times

– FCFS is optimal for NBUE service times– FCFS+FB(*(0)) is optimal for NBUE+DHR service

times

NWUEIMRL

DHRDMRL

IHR

NBUE+DHRNBUE

35

Outline


36

• We compared MLPS disciplines in terms of the mean delay:

– MLPS vs MLPS– MLPS vs PS– MLPS vs FB– Optimality of MLPS disciplines

• We considered the following service time distribution classes:

– DHR – IMRL– NBUE+DHR

Summary

NWUEIMRL

DHR

NBUEDMRL

IHR

NBUE+DHR

37

Our references

• Avrachenkov, Ayesta, Brown and Nyberg (2004)– IEEE INFOCOM 2004

• Aalto, Ayesta and Nyberg-Oksanen (2004)– ACM SIGMETRICS –

PERFORMANCE 2004

• Aalto, Ayesta and Nyberg-Oksanen (2005)– Operations Research

Letters, vol. 33

• Aalto and Ayesta (2006a)– IEEE INFOCOM 2006

• Aalto and Ayesta (2006b)– Journal of Applied

Probability, vol. 43• Aalto (2006)

– Mathematical Methods of Operations Research, vol. 64

• Aalto and Ayesta (2007)– ACM SIGMETRICS 2007

• Aalto and Ayesta (2008)– ValueTools 2008

38

THE END

Recent sojourn time results for Multilevel Processor-Sharing scheduling disciplines

Documents