On the gittins index in the m g 1 queue
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On the Gittins index in the M/G/1 queue. Samuli Aalto (TKK) in cooperation with Urtzi Ayesta (LAAS-CNRS) Rhonda Righter (UC Berkeley). Fundamental question. It is well known that … … in the M/G/1 queue … among the non-anticipating scheduling disciplines … the optimal discipline is

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On the gittins index in the m g 1 queue

On the Gittins indexin the M/G/1 queue

Samuli Aalto (TKK)

in cooperation with

Urtzi Ayesta (LAAS-CNRS)

Rhonda Righter (UC Berkeley)


Fundamental question

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?


Outline

Outline

  • Service time distribution classes

  • Known optimality results

  • Gittins index

  • Gittins index and service time distribution classes

  • Gittins index policy

  • New optimality results


Queueing model 1

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:


Queueing model 2

Queueing model (2)

  • Remaining service time distribution:

  • Mean remaining service time:

  • H-function:


Service time distribution classes 1

NBUE

NWUE

DMRL

IMRL

IHR

DHR

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


Service time distribution classes 2

NBUE

NWUE

DMRL

IMRL

IHR

DHR

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


Outline1

Outline

  • Service time distribution classes

  • Known optimality results

  • Gittins index

  • Gittins index and service time distribution classes

  • Gittins index policy

  • New optimality results


Scheduling queueing service disciplines

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


Known optimality results

NWUE

NBUE

DMRL

IMRL

IHR

DHR

Known optimality results

  • 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)


Our objective

Our objective

  • 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


Outline2

Outline

  • Service time distribution classes

  • Known optimality results

  • Gittins index

  • Gittins index and service time distribution classes

  • Gittins index policy

  • New optimality results


Gittins index

Gittins index

  • Efficiency function (J-function):

  • Gittins index for a customer with attained service a:

  • Optimal (individual) service quota:


Example

Example

Pareto service time distribution starting from 1

k= 1

D*(0)= 3.732


Basic properties 1

Basic properties (1)

  • Partial derivative w.r.t. to D:

  • Lemma:

    • If D*(a) <¥ and h(x) is continuous, then


Basic properties 2

Basic properties (2)

  • Lemma:

  • Corollary:

  • Lemma:

  • Corollary:


Outline3

Outline

  • Service time distribution classes

  • Known optimality results

  • Gittins index

  • Gittins index and service time distribution classes

  • Gittins index policy

  • New optimality results


Dhr ihr service times

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.


Dhr service times

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


Imrl dmrl and nwue nbue service times

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.


Dmrl and nbue service times

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)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


Outline4

Outline

  • Service time distribution classes

  • Known optimality results

  • Gittins index

  • Gittins index and service time distribution classes

  • Gittins index policy

  • New optimality results


Gittins index policy

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)³G(0) for all a.


Outline5

Outline

  • Service time distribution classes

  • Known optimality results

  • Gittins index

  • Gittins index and service time distribution classes

  • Gittins index policy

  • New optimality results


Single job class 1

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.


Single job class 2

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.


Multiple job classes

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


The end

THE END


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