Optimal Response Time Tail in Gittins Policy

1. YEQT 2021 WHEN DOES THE GITTINS POLICY HAVE OPTIMAL RESPONSE TIME TAIL? Ziv Scully (CMU) & Lucas van Kreveld (UvA)

2. The M/G/1-queue Objective: Choose service policy to minimize the response time ? (sojourn time) ? ◦ Minimize mean response time ?(?) ◦ Minimize response time tail ℙ ? > ? • Focus on ? → ∞ ??(?)

3. Size-aware policies ? ◦ Idea: serve the job that is closest to finishing ◦ Shortest Remaining Processing Time policy (SRPT) ◦ SRPT • Mean response time: optimal • (Asymptotic) response time tail: “Optimal” under heavy-tailed job sizes “Pessimal” under light-tailed job sizes ??(?) Remaining processing time Attained service

4. Size-blind policies ? ◦ Foreground-Background (FB): serves the job that has received the least amount of service ◦ Gittins policy: uses distribution of the job sizes to determine which job is “most likely to finish soon” Priority (lower is better) Attained service lim light-tailed job sizes Optimal ??(?) ?→∞ℙ(? > ?) for ?→∞ℙ(? > ?) for lim heavy-tailed job sizes Pessimal Optimal ?(?) FCFS FB Generally not optimal Generally not optimal Pessimal Optimal assuming condition on hazard rate Optimal if Gittins=FCFS, pessimal if Gittins=FB Gittins Optimal

5. SOAP-class of service policies ◦ Age = attained service so far ◦ SOAP-scheduling: Based on its age ?, assign to each job a rank ?(?), and serve the job with smallest rank rank ◦ A SOAP-policy is characterized by its rank function ?(?) age

6. Gittins policy ? = job size ◦ The Gittins policy is the SOAP-policy with rank function ?ℙ ? > ? ?? ? ℙ ? ∈ ?,? ? ? = inf ?>? 2 3 1 3 10 ?.?. ◦ Job size distribution ? ∼ rank 30 ?.?. ?(?) 20 15 10 30 age

7. Outline of results ◦ Heavy-tailed case: • A SOAP-policy is tail-optimal if its rank function satisfies a certain condition • The Gittins policy satisfies this condition ◦ Light-tailed case: • The tail behavior of a SOAP-policy is determined by the age of the global maximum of the rank function • The Gittins policy can be tail-optimal, tail-pessimal or in between ◦ Tail-pessimality of Gittins can sometimes be remedied by tweaking the rank function ?→∞ℙ(? > ?) for ?→∞ℙ(? > ?) for lim light-tailed job sizes Optimal lim heavy-tailed job sizes Pessimal Optimal ?(?) FCFS FB Generally not optimal Generally not optimal Pessimal Optimal assuming condition on hazard rate Optimal if Gittins=FCFS, pessimal if Gittins=FB Gittins Optimal

8. Tail characterizations ? = response time ? = job size ◦ General goal: minimize ℙ(? > ?) for large ? Heavy-tailed job sizes • A non-negative random variable ? is heavy-tailed if ℙ ? > ? = Θ ?−?for some ? > 1 • A service policy ? is tail-optimal if ℙ ??> ? = Θ ?−? Light-tailed job sizes • A non-negative random variable ? is light-tailed if ? ???< ∞ for some ? > 0 • With ? ? = lim tail-optimal if ? maximizes ?(??) tail-pessimal if ? minimizes ?(??) tail-intermediate otherwise ?→∞−1 ?logℙ(? > ?), a service policy ? is

9. Heavy-tail results o Let ℙ ? > ? = Θ(?−?) for some ? > 1 rank o Let ??be the highest rank of a size ? job ?? o Condition: There exist ?,?,? ∈ [0,∞] such that for any job size ? and for any interval (?,?) with rank below ?? with ? ≥ ?, we have ? − ? = ?(????) • ? = ?(??) • ?2 ?1 ? ?1 ?2 age ???) ???) ?(?1 ?(?2 Theorem A SOAP-policy is tail-optimal if ?(??) Theorem 1 − ?+ ? <? − 1 ? + ? − 1+− ????????= 0 and ????????= 1 ?

10. Light-tail results Suppose there exists ? > 0 such that ? ???< ∞ Let ?∗= inf ? ≥ 0 ? ? ≥ ? ? ∀? ∈ ℝ+} be the age at which the rank function has its (first) global maximum 1 2 1 2 ???(1) ?.?. Theorem A SOAP-policy is • tail-optimal if ?∗= 0 • tail-intermediate if 0 < ?∗< ∞ • tail-pessimal if ?∗= ∞ ? ∼ 1 ??? ?.?. FCFS 100 ↑ Gittins rank Theorem Gittins policy can be tail-optimal, tail intermediate or tail-pessimal Age →

11. Light-tail results Theorem Let ? be a SOAP-policy such that 1 ? ≥ 0. Then ?[??] ≤ ?2?[????????] ??? ?????????≤ ? for all q≤ Theorem Suppose a job’s Gittins rank is uniformly bounded at all ages. Then for all ? > 0, there exists a tail- intermediate policy that has within a factor 1 + ? of optimal mean response time. ↑ Gittins rank Age →

12. Conclusions ?→∞ℙ(? > ?) for ?→∞ℙ(? > ?) for lim light-tailed job sizes Optimal lim heavy-tailed job sizes Pessimal Optimal ?(?) FCFS FB Generally not optimal Generally not optimal Pessimal Optimal assuming condition on hazard rate Optimal if Gittins=FCFS, pessimal if Gittins=FB or pessimal Optimal, intermediate Gittins Optimal Optimal ◦ Under some conditions, an approximation of Gittins policy guarantees intermediate tail with almost optimal mean response time