1. aalto.pptACM Sigmetrics 2007, San Diego, CA, 12-16 June 2007 1 Mean Delay Optimization for the M/G/1 Queue with Pareto Type Service Times Samuli Aalto TKK Helsinki University of Technology, Finland Urtzi Ayesta LAAS-CNRS, France

2. 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. 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)

4. 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)

5. 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. 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 G(a) Δ*(a) G(0) Δ*(0) kk

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

9. 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. 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)