Recent sojourn time results for Multilevel Processor-Sharing scheduling disciplines

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

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: • 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. NBUE NWUE DMRL IMRL IHR DHR 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

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. NWUE NBUE DMRL IMRL IHR DHR Optimality results for M/G/1 • Among all scheduling disciplines, • SRPT is optimal(minimizing the mean delay); Schrage (1968) • Among non-anticipatingscheduling disciplines, • FCFS isoptimal for NBUE service times; Righter, Shanthikumar and Yamazaki (1990) • FB isoptimal for DHR service times; Yashkov (1987); Righter and Shanthikumar (1989)

8. Multilevel Processor-Sharing (MLPS) disciplines • Definition: Kleinrock (1976), vol. 2, Sect. 4.7 • based on the attained service times • N+1levels 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. NBUE+DHR NBUE NWUE DMRL IMRL IHR DHR 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

10. Outline • Introduction • DHR service times • IMRL service times • NBUE+DHR service times • Summary

11. NBUE NWUE DMRL IMRL IHR DHR 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

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. NBUE NWUE DMRL IMRL IHR DHR Class: IMRL service times • Recall: Service time distribution: • H-function: • Mean residual lifetime (MRL): • Definition: • Service times are IMRL ifH(x) is decreasing • Examples: • all DHR service time distributions, Exp+Pareto

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> 0such that FB FB FCFS

29. Outline • Introduction • DHR service times • IMRL service times • NBUE+DHR service times • Summary

30. NBUE+DHR NBUE NWUE DMRL IMRL IHR DHR Class: NBUE+DHR service times • Recall: Hazard rate • Recall: H-function: • Definition: • Service times are NBUE+DHR(k) if • H(x) ³H(0) for all x<k and • h(x) is decreasing for all x>k • Examples: • Pareto (starting from k >0), Exp+Pareto, Uniform+Pareto

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

34. NBUE+DHR NWUE NBUE DMRL IMRL IHR DHR 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

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

36. NBUE+DHR NBUE NWUE DMRL IMRL IHR DHR 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

37. 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 Our references

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