Recent sojourn time results for Multilevel Processor-Sharing scheduling disciplines
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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
• Introduction• DHR service times• IMRL service times• NBUE+DHR service times• Summary
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
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Example: Mean unfinished truncated work
bounded Pareto service time distribution
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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
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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
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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
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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
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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
• Introduction• DHR service times• IMRL service times• NBUE+DHR service times• Summary
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
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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
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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
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Outline
• Introduction• DHR service times• IMRL service times• NBUE+DHR service times• Summary
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
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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
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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
• Introduction• DHR service times• IMRL service times• NBUE+DHR service times• Summary
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
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THE END
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