1. aalto_inria2.pptINRIA Sophia Antipolis, France, 24.3.2009 1 On the Gittins index in the M/G/1 queue Samuli Aalto (TKK) in cooperation with Urtzi Ayesta (LAAS-CNRS) Rhonda Righter (UC Berkeley)

2. 2 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?

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

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

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

6. 6 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)  [  ] H(x) It is known that –IHR  DMRL  NBUE and DHR  IMRL  NWUE NWUE IMRL DHR NBUE DMRL IHR

7. 7 NWUE IMRL DHR NBUE DMRL IHR 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

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

9. 9 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

10. 10 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) IHR DMRL NWUE IMRL DHR NBUE

11. 11 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 … For that, we need Our objective Yes, the conditions are necessary. The Gittins Index

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

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

14. 14 Example Pareto service time distribution starting from 1 k   *   

15. 15 Basic properties (1) Partial derivative w.r.t. to  : Lemma: –If  * (a)   and h(x) is continuous, then

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

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

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

19. 19 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)  (ii): Corollary in slide 18 –(ii)  (i): Corollary in slide 16

20. 20 IMRL [DMRL] and NWUE [NBUE] service times Lemma: Proof: Corollaries: –The service times are IMRL [DMRL] if and only if J(a,  )  [  ] J(a,  ) for all a, . –The service times are NWUE [NBUE] if and only if J(0,  )  [  ] J(0,  ) for all .

21. 21 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)  G(0) for all a if and only if (iii) G(0)  H(0). Proof: –(i)  (iii)  (ii): Corollary in slide 20 –(ii)  (i): Corollary/Lemma in slide 16

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

23. 23 Gittins index policy Definition [Gittins (1989)]: –Gittins index policy gives service to the job i with the highest Gittins index G i (a i ). 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)  G(0) for all a.

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

25. 25 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.

26. 26 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 h i (a i ). Theorem: –MAS minimizes the mean queue length if and only if the service times are DMRL.

27. 27 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 h i (a i ). 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

28. 28 THE END