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

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?

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

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

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

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

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

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

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

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

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

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

Example: Mean unfinished truncated work bounded Pareto service time distribution

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

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

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

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

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

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

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

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

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

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:

Example: Mean level-x workload bounded Pareto service time distribution

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

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

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

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

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

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

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

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

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

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

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

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

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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