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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
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