1. Urtzi Ayesta (LAAS-CNRS)
Recent sojourn time results for Multilevel Processor-Sharing scheduling disciplines
Samuli Aalto (TKK)
in cooperation with
Urtzi Ayesta (LAAS-CNRS)
Eeva Nyberg-Oksanen

2. Which one is better: PS or PS+PS?
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:
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

5. Service time distribution classes
IHR = Increasing Hazard Rate
DMRL = Decreasing Mean Residual Lifetime
NBUE = New Better than Used in Expectation
DHR = Decreasing Hazard Rate
IMRL = Increasing Mean Residual Lifetime
NWUE = New Worse than Used in Expectation
NWUE
IMRL
DHR
NBUE
DMRL
IHR

6. Scheduling/queueing/service disciplines
Anticipating:
SRPT = Shortest-Remaining-Processing-Time
strict priority according to the remaining service
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

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,
FCFS is optimal for NBUE service times; Righter, Shanthikumar and Yamazaki (1990)
FB is optimal for DHR service times; Yashkov (1987); Righter and Shanthikumar (1989)
NWUE
IMRL
DHR
DMRL
IHR
NBUE

8. 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

9. 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
NWUE
IMRL
DHR
NBUE
DMRL
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
NWUE
IMRL
DHR
NBUE
DMRL
IHR

12. Tool: Unfinished truncated work Ux(t)
Customers with attained service less than x:
Unfinished truncated work with truncation threshold x:
Unfinished work:

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 Ux(t) for any x and t in each sample path
Xi(t)
Ux(t)
FCFS FB s s x x t t

15. Idea of the mean delay comparison
Kleinrock (1976):
For all non-anticipating service disciplines p
so that (by applying integration by parts)
Thus,
Consequence:
among non-anticipating service disciplines, FB minimizes the mean delay for DHR service times

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 FB
FB/PS PS
FB/PS
FB/PS

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’
FB/PS
PS/FCFS

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 discipline
MLPS
MLPS’
FCFS
FCFS
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’ PS PS PS

20. Idea of the mean slowdown ratio comparison
Feng and Misra (2003):
For all non-anticipating service disciplines p
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

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
NWUE
IMRL
DHR
NBUE
DMRL
IHR

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:

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 Vx(t) (in any sense)
Xi(t)
Vx(t)
FCFS
FB not
optimal FB s s x x t t

26. Idea of the mean delay comparison
Righter, Shanthikumar and Yamazaki (1990):
For all non-anticipating service disciplines p
so that
Thus,

27. MLPS vs PS
Aalto (2006):
Any number of levels with FB and PS allowed as internal disciplines
Consequence:
FB/PS PS
FB/PS
FB/PS

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 e > 0 such that FB FB
FCFS

29. 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) ³ H(0) for all x < k and
h(x) is decreasing for all x > k
Examples:
Pareto (starting from k > 0), Exp+Pareto, Uniform+Pareto
NWUE
IMRL
DHR
NBUE
DMRL
IHR
NBUE+DHR

31. Tool: Gittins index Gittins (1989): J-function:
Gittins index for a customer with attained service a:
Optimal quota:

32. Example: Gittins index and optimal quota
Pareto service time distribution
k = 1
D*(0) = 3.732

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) ³ G(0) for all a
If service times are NBUE+DHR(k), then
D*(0) > k
G(a) ³ G(0) for all a < D*(0) and
G(a) is decreasing for all a > k
G(D*(0)) £ G(0) (if D*(0) < ¥)
NWUE
IMRL
DHR
DMRL
IHR
NBUE+DHR
NBUE

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(D*(0)) is optimal for NBUE+DHR service times
NWUE
IMRL
DHR
DMRL
IHR
NBUE+DHR
NBUE

35. Outline Introduction DHR service times IMRL service times
NBUE+DHR service times
Summary

36. 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
NWUE
IMRL
DHR
NBUE
DMRL
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