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Mean Delay Optimization for the M/G/1 Queue with Pareto Type Service Times

This study explores optimal scheduling disciplines for M/G/1 queues with Pareto service times, focusing on minimizing mean delays. Known results for M/G/1 queues are discussed, with emphasis on SRPT, FCFS, and FB optimality. The concepts of NBUE, DHR, and CDHR service times are introduced, along with the Gittins index approach for job prioritization. Numerical results for Type-2 Pareto distribution are presented, showing the impact of thresholds on mean delay ratios. The conclusion highlights the FCFS+FB(∆*(0)) discipline as optimal for CDHR service times and suggests future research directions.

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Mean Delay Optimization for the M/G/1 Queue with Pareto Type Service Times

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  1. Mean Delay Optimizationfor the M/G/1 Queuewith Pareto Type Service Times Samuli Aalto TKK Helsinki University of Technology, Finland Urtzi Ayesta LAAS-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! h(x) h(x)

  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: h(x) h(x) h(x)

  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)]

  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 Δ*(0) G(a) Δ*(a) G(0) k k Δ*(0)

  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: ρ = 0.5 max gain 18% ρ = 0.8 Δ*(0)

  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 h(x) G(a)

  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)

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