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

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)