On the Gittins index in the MG1 queue

On the Gittins index in the M/G/1 queue Samuli Aalto (TKK) in cooperation with Urtzi Ayesta (LAAS-CNRS) Rhonda Righter (UC Berkeley) aalto_inria 2. ppt INRIA Sophia Antipolis, France, 24. 3. 2009 1

Fundamental question • • It is well known that … … in the M/G/1 queue … among the non-anticipating scheduling disciplines … the optimal discipline is – FCFS if the service times are NBUE – FB if the service times are DHR • So, these conditions are sufficient for the optimality of FCFS and FB, respectively. But, … Are the conditions necessary? 2

Outline • • • Service time distribution classes Known optimality results Gittins index and service time distribution classes Gittins index policy New optimality results 3

Queueing model (1) • M/G/1 queue – Poisson arrivals with rate l – IID service times S with a general distribution – single server • Service time distribution: • Density function: • Hazard rate: 4

Queueing model (2) • Remaining service time distribution: • Mean remaining service time: • H-function: 5

Service time distribution classes (1) • Service times are – IHR [DHR] if h(x) is increasing [decreasing] – DMRL [IMRL] if H(x) is increasing [decreasing] – NBUE [NWUE] if H(0) £ [³] H(x) • It is known that – IHR Ì DMRL Ì NBUE and DHR Ì IMRL Ì NWUE NBUE NWUE DMRL IHR DHR 6

Service time distribution classes (2) • 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 NBUE NWUE DMRL IHR DHR 7

Outline • • • Service time distribution classes Known optimality results Gittins index and service time distribution classes Gittins index policy New optimality results 8

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 – FB = Foreground-Background • strict priority according to the attained service • a. k. a. LAS = Least-Attained-Service 9

Known optimality results • Among all scheduling disciplines, – SRPT is optimal (minimizing the queue length pathwise); Schrage (1968) • Among the non-anticipating scheduling disciplines, – FCFS is optimal for NBUE service times (minimizing the mean queue length); Righter, Shanthikumar and Yamazaki (1990) – FB is optimal for DHR service times (minimizing the queue length stochastically); Righter and Shanthikumar (1989) NBUE DMRL IHR NWUE IMRL DHR 10

Our objective • We will show that … • … among the non-anticipating scheduling disciplines – FCFS is optimal only for NBUE service times – FB is optimal only for DHR service times • In other words, we will show that … Yes, the conditions are necessary. • For that, we need The Gittins Index 11

Outline • • • Service time distribution classes Known optimality results Gittins index and service time distribution classes Gittins index policy New optimality results 12

Gittins index • Efficiency function (J-function): • Gittins index for a customer with attained service a: • Optimal (individual) service quota: 13

Example Pareto service time distribution starting from 1 k=1 D*(0) = 3. 732 14

Basic properties (1) • Partial derivative w. r. t. to D: • Lemma: – If D*(a) < ¥ and h(x) is continuous, then 15

Basic properties (2) • Lemma: • Corollary: 16

Outline • • • Service time distribution classes Known optimality results Gittins index and service time distribution classes Gittins index policy New optimality results 17

DHR [IHR] service times • Lemma: • Proof: • Corollary: – If the service times are DHR [IHR], then J(a, D) is decreasing [increasing] w. r. t. to D for all a, D. • Corollary: – If the service times are DHR [IHR], then G(a) = h(a) [H(a)] for all a. 18

DHR service times • Proposition: – (i) The service times are DHR if and only if (ii) G(a) is decreasing for all a. – In this case, G(a) = h(a) for all a. • Proof: – (i) Þ (ii): Corollary in slide 18 – (ii) Þ (i): Corollary in slide 16 19

IMRL [DMRL] and NWUE [NBUE] service times • Lemma: • Proof: • Corollaries: – The service times are IMRL [DMRL] if and only if J(a, ¥) £ [³] J(a, D) for all a, D. – The service times are NWUE [NBUE] if and only if J(0, ¥) £ [³] J(0, D) for all D. 20

DMRL and NBUE service times • • Proposition: – (i) The service times are DMRL if and only if (ii) G(a) is increasing for all a if and only if (iii) G(a) = H(a) for all a. – (i) The service times are NBUE if and only if (ii) G(a) ³ G(0) for all a if and only if (iii) G(0) = H(0). Proof: – (i) Û (iii) Þ (ii): Corollary in slide 20 – (ii) Þ (i): Corollary/Lemma in slide 16 21

Outline • • • Service time distribution classes Known optimality results Gittins index and service time distribution classes Gittins index policy New optimality results 22

Gittins index policy • Definition [Gittins (1989)]: – Gittins index policy gives service to the job i with the highest Gittins index Gi(ai). • Theorem [Gittins (1989), Yashkov (1992)]: – Among the non-anticipating disciplines, Gittins index policy minimizes the mean queue length in the M/G/1 queue (with possibly multiple job classes) • Observations: – FB is a Gittins index policy if and only if G(a) is decreasing for all a. – FCFS (or any other non-preemptive policy) is a Gittins index policy if and only if G(a) ³ G(0) for all a. 23

Outline • • • Service time distribution classes Known optimality results Gittins index and service time distribution classes Gittins index policy New optimality results 24

Single job class (1) • Theorem: – FB minimizes stochastically the queue length if and only if the service times are DHR. • Proof: – Theorem in slide 23 and Proposition in slide 19 together with Righter, Shanthikumar and Yamazaki (1990). • Theorem: – FCFS minimizes the mean queue length if and only if the service times are NBUE. • Proof: – Theorem in slide 23 and Proposition in slide 21. 25

Single job class (2) • Additional assumption: – arriving jobs have already attained a random amount of service elsewhere • Theorem: – FB = LAS minimizes the mean queue length if and only if the service times are DHR. • Definition: – MAS (Most-Attained-Service) gives service to the job i with the highest hazard rate hi(ai). • Theorem: – MAS minimizes the mean queue length if and only if the service times are DMRL. 26

Multiple job classes • Additional assumption: – arriving jobs have already attained a random amount of service elsewhere • Definition: – HHR (Highest-Hazard-Rate) gives service to the job i with the highest hazard rate hi(ai). • Theorem: – If all service time distributions are DHR, then HHR minimizes the mean queue length • Theorem: – If all service time distributions are DMRL, then SERPT minimizes the mean queue length 27

THE END 28