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On the Gittins index in the M/G/1 queue

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On the Gittins indexin the M/G/1 queue

Samuli Aalto (TKK)

in cooperation with

Urtzi Ayesta (LAAS-CNRS)

Rhonda Righter (UC Berkeley)

- 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?

- Service time distribution classes
- Known optimality results
- Gittins index
- Gittins index and service time distribution classes
- Gittins index policy
- New optimality results

- 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:

- Remaining service time distribution:
- Mean remaining service time:
- H-function:

NBUE

NWUE

DMRL

IMRL

IHR

DHR

- 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

IMRL

IHR

DHR

- 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

- Service time distribution classes
- Known optimality results
- Gittins index
- Gittins index and service time distribution classes
- Gittins index policy
- New optimality results

- Anticipating:
- SRPT = Shortest-Remaining-Processing-Time
- strict priority according to the remaining service

- SRPT = Shortest-Remaining-Processing-Time
- 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

- FCFS = First-Come-First-Served

NWUE

NBUE

DMRL

IMRL

IHR

DHR

- Among all scheduling disciplines,
- SRPT is optimal(minimizing the queue length pathwise); Schrage (1968)

- Among the non-anticipatingscheduling disciplines,
- FCFS isoptimal for NBUE service times (minimizing the mean queue length); Righter, Shanthikumar and Yamazaki (1990)
- FB isoptimal for DHR service times (minimizing the queue length stochastically); Righter and Shanthikumar (1989)

- We will show that …
- … among the non-anticipatingscheduling disciplines
- FCFS isoptimal only for NBUE service times
- FB isoptimal only for DHR service times

- In other words, we will show that …
- For that, we need

Yes, the conditions are necessary.

The Gittins Index

- Service time distribution classes
- Known optimality results
- Gittins index
- Gittins index and service time distribution classes
- Gittins index policy
- New optimality results

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

Pareto service time distribution starting from 1

k= 1

D*(0)= 3.732

- Partial derivative w.r.t. to D:
- Lemma:
- If D*(a) <¥ and h(x) is continuous, then

- Lemma:
- Corollary:
- Lemma:
- Corollary:

- Service time distribution classes
- Known optimality results
- Gittins index
- Gittins index and service time distribution classes
- Gittins index policy
- New optimality results

- 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.

- 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

- 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.

- 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)³G(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

- Service time distribution classes
- Known optimality results
- Gittins index
- Gittins index and service time distribution classes
- Gittins index policy
- New optimality results

- 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)³G(0) for all a.

- Service time distribution classes
- Known optimality results
- Gittins index
- Gittins index and service time distribution classes
- Gittins index policy
- New optimality results

- 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.

- 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.

- 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