1aalto.ppt ACM Sigmetrics 2007, San Diego, CA, 12-16 June 2007

Mean Delay Optimizationfor the M/G/1 Queue

with Pareto Type Service Times

Samuli AaltoTKK Helsinki University of Technology, Finland

Urtzi AyestaLAAS-CNRS, France

2

Known optimality results for M/G/1

• Among all scheduling disciplines– SRPT (Shortest-Remaining-Processing-Time) optimal

• minimizing the queue length process; thus, also the mean delay (i.e. sojourn time)

• Among non-anticipating (i.e. blind) scheduling disciplines– FCFS (First-Come-First-Served) optimal for NBUE

(New-Better-than-Used-in-Expectation) service times• minimizing the mean delay

– FB (Foreground-Background) optimal for DHR (Decreasing-Hazard-Rate) service times

• minimizing the mean delay• Definitions:

– NBUE: E[S] ≥ E[S – x|S > x] for all x– DHR: hazard rate h(x) = f(x)/(1-F(x)) decreasing for all x

3

Pareto service times

• Pareto distribution– has a power-law (thus heavy) tail– has been used to model e.g. flow sizes in the Internet

• Definition (type-1):

– belongs to the class DHR– thus, FB optimal non-anticipating discipline

• Definition (type-2):

– does not belong to the class DHR– optimal non-anticipating discipline an open question

... until now!

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4

CDHR service times

• CDHR(k) distribution class (first-Constant-and-then-Decresing-Hazard-Rate)– includes type-2 Pareto distributions

• Definition:– A1: hazard rate h(x) constant for all x < k– A2: hazard rate h(x) decreasing for all x ≥ k– A3: h(0) < h(k)

• Examples:

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5

Gittins index

• Function J(a,∆) for a job of age a and service quota ∆:

– numerator: completion probability = ”payoff”– denominator: expected servicing time = ”investment”

• Gittins index G(a) for a job of age a:

• Original framework:– Multiarmed Bandit Problems [Gittins (1989)]

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6

Example: Pareto distribution

• Type-2 Pareto distribution with k = 1 and α = 2– Left: Gittins index G(a) as a function of age a– Right: Optimal quota ∆*(a) as a function of age a

• Note: – ∆*(0) > k– G(∆*(0)) = G(0)– G(a) = h(a) for all a > k

1 2 3 4 5Attained service

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7

Gittins discipline

• Gittins discipline:– Serve the job with the highest Gittins index;

if multiple, then PS among those jobs• Known result [Gittins (1989), Yashkov (1992)]:

– Gittins discipline optimal among non-anticipating scheduling disciplines

• minimizing the mean delay• Our New Result:

– For CDHR service times (satisfying A1-A3) the Gittins discipline (and thus optimal) is FCFS+FB(∆*(0))

• give priority for jobs younger than threshold ∆*(0) and apply FCFS among these priority jobs;

• if no priority jobs, serve the youngest job in the system (according to FB)

8

Numerical results: Pareto distribution

• Type-2 Pareto distribution with k = 1 and α = 2– Depicting the mean delay ratio

– Left: Mean delay ratio as a function of threshold θ– Right: Minimum mean delay ratio as a function of load ρ

– Note:

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9

Impact of an upper bound: Bounded Pareto

• Bounded Pareto distribution – lower bound k and upper bound p

• Definition:

– does not belong to the class CDHR

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10

Conclusion and future research

• Optimal non-anticipating scheduling studied for M/G/1 by applying the Gittins index approach

• Observation: – Gittins index monotone iff the hazard rate monotone

• Main result:– FCFS+FB(∆*(0)) optimal for CDHR service times

• Possible further directions: – To generalize the result for IDHR service times– To apply the Gittins index approch

• in multi-server systems or networks with the non-work-conserving property

• in wireless systems with randomly time-varing server capacity

• in G/G/1– To calculate performance metrics for a given G(a)