Recent sojourn time results for
Multilevel Processor-Sharing
scheduling disciplines
Samuli Aalto (TKK)
in cooperation with
Urtzi Ayesta (LAAS-CNRS)
aaltoeurandom.ppt
Eurandom, Eindhoven, The Netherlands, 26.-28.8.2008
1
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?
2
Outline
•
•
•
•
•
Introduction
DHR service times
IMRL service times
NBUE+DHR service times
Summary
3
Queueing context
• Model: M/G/1
– Poisson arrivals
– IID service times with a general distribution
– single server
• Notation:
– A(t) = arrivals up to time t
– Si = service time of customer i
– Xi(t) = attained service (= age) of customer i at time t
– Si - Xi(t) = remaining service of customer i at time t
– Ti = sojourn time (= delay) of customer i
– Ri = Ti / Si = slowdown ratio of customer i
4
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
NBUE
NWUE
DMRL
IMRL
IHR
DHR
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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
6
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)
NBUE
DMRL
IHR
NWUE
IMRL
DHR
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Multilevel Processor-Sharing (MLPS) disciplines
• Definition: Kleinrock (1976), vol. 2, Sect. 4.7
– based on the attained service times
– N+1 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)
Xi(t)
FCFS+FB(a)
FB
FCFS
a
t
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Our objective
• 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
NBUE+DHRNWUE
NBUE
DMRL
IMRL
IHR
DHR
9
Outline
•
•
•
•
•
Introduction
DHR service times
IMRL service times
NBUE+DHR service times
Summary
10
Class: DHR service times
• Service time distribution:
F ( x)  P{S  x}, F ( x)  1 - F ( x)
• Density function:
f ( x)  P{S  dx}
• Hazard rate:
f ( x)
h( x )  F ( x ) 
f ( x)

x f ( y ) dy
• Definition:
NBUE
DMRL
IHR
– Service times are DHR if
h(x) is decreasing
• Examples:
– Pareto (starting from 0) and hyperexponential
NWUE
IMRL
DHR
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Tool: Unfinished truncated work Ux(t)
• Customers with attained service less than x:
N x (t )  {i  A(t ) | X i (t ) < min{ Si , x}}
• Unfinished truncated work with truncation threshold x:
U x (t )  iN  (t ) (min{ Si , x} - X i (t ))
x
• Unfinished work:

U  (t )  U 
(t )  iN  (t ) ( Si - X i (t ))
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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 Ux(t) for
any x and t in each sample path
Xi(t)
FCFS
Ux(t)
FB
s
x
s
x
t
t
14
Idea of the mean delay comparison
• Kleinrock (1976):
– For all non-anticipating service disciplines


1
T   0 h( x) d [U x ]
– so that (by applying integration by parts)


T  - T  '  1 0 (U x - U x ' )d [-h( x)]
• Thus,
DHR & U x  U x ' x  T   T  '
• Consequence:
– among non-anticipating service disciplines,
FB minimizes the mean delay for DHR service times
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MLPS vs PS
• Aalto, Ayesta and Nyberg-Oksanen (2004):
– Two levels with FB and PS allowed as internal disciplines
DHR  T FB  T FB+ PS  T PS+ PS  T PS
• Aalto, Ayesta and Nyberg-Oksanen (2005):
– Any number of levels with FB and PS allowed as internal
disciplines
DHR  T FB  T MLPS  T PS
FB/PS
FB
FB/PS
PS
FB/PS
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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)
DHR  T MLPS  T MLPS'
MLPS
MLPS’
FB/PS
PS/FCFS
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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 discipline
DHR  T MLPS  T MLPS'
MLPS
FCFS
FCFS
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
DHR  T MLPS  T MLPS'
• Splitting any higher PS level is still an open problem!
MLPS
PS
PS
MLPS’
PS
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Idea of the mean slowdown ratio comparison
• Feng and Misra (2003):
– For all non-anticipating service disciplines
R


 h( x )
1
  0 x d [U x ]
– so that

R  - R  '  1 0 (U x - U x ' )d [- x ]
h( x )
• Thus,
DHR & U x  U x ' x  R   R  '
• 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
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Outline
•
•
•
•
•
Introduction
DHR service times
IMRL service times
NBUE+DHR service times
Summary
21
Class: IMRL service times
• Recall: Service time distribution:
F ( x)  P{S  x}, F ( x)  1 - F ( x), f ( x)  P{S  dx}
• H-function:
H ( x) 

x f ( y ) dy

x F ( y ) dy
• Mean residual lifetime (MRL):
E[ S - x | S  x] 

F ( x)

x F ( y ) dy

x F ( y ) dy
F ( x)
 H 1( x )
• Definition:
NBUE
DMRL
IHR
– Service times are IMRL if
H(x) is decreasing
• Examples:
– all DHR service time distributions, Exp+Pareto
NWUE
IMRL
DHR
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Tool: Level-x workload Vx(t)
• Customers with attained service less than x:
N x (t )  {i  A(t ) | X i (t ) < min{ Si , x}}
• Unfinished truncated work with truncation threshold x:
U x (t )  iN  (t ) (min{ Si , x} - X i (t ))
x
• Level-x workload:
Vx (t )  iN  (t ) ( Si - X i (t ))
x
• Workload = unfinished work:
V  (t )  V (t )  iN  (t ) ( Si - X i (t ))  U  (t )
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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 Vx(t)
(in any sense)
Xi(t)
FCFS
Vx(t)
FB
s
x
FB not
optimal
s
x
t
t
25
Idea of the mean delay comparison
• Righter, Shanthikumar and Yamazaki (1990):
– For all non-anticipating service disciplines 

T   1 0 H ( x) d [Vx ]
– so that

T  - T  '  1 0 (Vx - Vx ' )d [- H ( x)]
• Thus,
IMRL & Vx  Vx ' x  T   T  '
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MLPS vs PS
• Aalto (2006):
– Any number of levels with FB and PS allowed as internal
disciplines
IMRL  T MLPS  T PS
• Consequence:
IMRL  T FB  T PS
FB/PS
FB/PS
PS
FB/PS
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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):
c - x , 0  x  c
F ( x)   -c
x , x  c
– There is e  0 such that
T FCFS + FB(c +e ) < T FB
FB
FCFS
FB
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Outline
•
•
•
•
•
Introduction
DHR service times
IMRL service times
NBUE+DHR service times
Summary
29
Class: NBUE+DHR service times
• Recall: Hazard rate
f ( x)
h( x )  F ( x ) 
• Recall: H-function:
H ( x) 

x f ( y ) dy

x F ( y ) dy
f ( x)

x f ( y ) dy

F ( x)

x F ( y ) dy
• Definition:
– Service times are NBUE+DHR(k) if
• H(x)  H(0) for all x < k and
NBUE+DHRNWUE
NBUE
DMRL
IMRL
IHR
DHR
• h(x) is decreasing for all x  k
• Examples:
– Pareto (starting from k  0), Exp+Pareto, Uniform+Pareto
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Tool: Gittins index
• Gittins (1989):
– J-function:
J ( a,  ) 
a+
a f ( y ) dy
,
a+
a F ( y ) dy
J (a,0)  h(a), J (a, )  H (a)
– Gittins index for a customer with attained service a:
G(a)  sup 0 J (a, )
– Optimal quota:
* (a)  sup{   0 | J (a, )  G (a)}
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Example: Gittins index and optimal quota
Pareto service time distribution
k1
*(0)  3.732
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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
NBUE
DMRL
IHR
NBUE+DHRNWUE
IMRL
DHR
• 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) < )
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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
NBUE
DMRL
IHR
NBUE+DHRNWUE
IMRL
DHR
– FCFS is optimal for NBUE service times
– FCFS+FB(*(0)) is optimal for NBUE+DHR service times
34
Outline
•
•
•
•
•
Introduction
DHR service times
IMRL service times
NBUE+DHR service times
Summary
35
Summary
• 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
NBUE+DHRNWUE
NBUE
DMRL
IMRL
IHR
DHR
36
Our references
•
Avrachenkov, Ayesta, Brown
and Nyberg (2004)
– IEEE INFOCOM 2004
•
Aalto, Ayesta and NybergOksanen (2004)
– ACM SIGMETRICS –
PERFORMANCE 2004
•
Aalto, Ayesta and NybergOksanen (2005)
– Operations Research Letters,
vol. 33
•
•
Aalto and Ayesta (2006b)
•
Aalto (2006)
•
Aalto and Ayesta (2007)
•
Aalto and Ayesta (2008)
– Journal of Applied
Probability, vol. 43
– Mathematical Methods of
Operations Research, vol. 64
– ACM SIGMETRICS 2007
– ValueTools 2008
Aalto and Ayesta (2006a)
– IEEE INFOCOM 2006
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THE END
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