1. FINDING THE OPTIMAL QUANTUM SIZE Revisiting the M/G/1 Round-Robin Queue VARUN GUPTA Carnegie Mellon University

2. 2 M/G/1/RR Jobs served for q units at a time Poisson arrivals Incomplete jobs

3. 3 M/G/1/RR Jobs served for q units at a time Poisson arrivals Incomplete jobs

4. 4 M/G/1/RR arrival rate = job sizes i.i.d. ~ S load Jobs served for q units at a time Completed jobs Incomplete jobs Poisson arrivals Squared coefficient of variability (SCV) of job sizes: C 2  0

5. 5 M/G/1/RR M/G/1/PS M/G/1/FCFS q q → 0 q =   Preemptions cause overhead (e.g. OS scheduling)  Variable job sizes cause long delays small qlarge q preemption overheads job size variability

6. 6 GOAL Optimal operating q for M/G/1/RR with overheads and high C 2 SUBGOAL Sensitivity Analysis: Effect of q and C 2 on M/G/1/RR performance (no switching overheads) PRIOR WORK Lots of exact analysis: [Wolff70], [Sakata et al.71], [Brown78]  No closed-form solutions/bounds  No simple expressions for interplay of q and C 2 effect of preemption overheads

7. 7 Outline  Effect of q and C 2 on mean response time  Approximate analysis  Bounds for M/G/1/RR  Choosing the optimal quantum size No preemption overheads Effect of preemption overheads

8. 8 Approximate sensitivity analysis of M/G/1/RR Approximation assumption 1: Service quantum ~ Exp(1/q) Approximation assumption 2: Job size distribution

9. 9 Approximate sensitivity analysis of M/G/1/RR Monotonic in q –Increases from E[T PS ] → E[T FCFS ] Monotonic in C 2 –Increases from E[T PS ] → E[T PS ](1+ q) For high C 2 : E[T RR* ] ≈ E[T PS ](1+ q)

10. 10 Outline  Effect of q and C 2 on mean response time Approximate analysis  Bounds for M/G/1/RR  Choosing the optimal quantum size

11. 11 THEOREM: M/G/1/RR bounds Assumption: job sizes  {0,q,…,Kq} Lower bound is TIGHT: E[ S ] = iq Upper bound is TIGHT within (1+  /K): Job sizes  {0,Kq}

12. 12 Outline Effect of q and C 2 on mean response time Approximate analysis Bounds for M/G/1/RR  Choosing the optimal quantum size

13. 13 Optimizing q Preemption overhead = h 1. 2.Minimize E[T RR ] upper bound from Theorem: Common case:

14. 14 service quantum (q) Mean response time Job size distribution: H 2 (balanced means)E[ S ] = 1, C 2 = 19,  = 0.8 h=0

15. 15 service quantum (q) Mean response time Job size distribution: H 2 (balanced means)E[ S ] = 1, C 2 = 19,  = 0.8 approximation q* optimum q 1% h=0

16. 16 service quantum (q) Mean response time Job size distribution: H 2 (balanced means)E[ S ] = 1, C 2 = 19,  = 0.8 1% 2.5% h=0 approximation q* optimum q

17. 17 service quantum (q) Mean response time Job size distribution: H 2 (balanced means)E[ S ] = 1, C 2 = 19,  = 0.8 1% 2.5% 5% h=0 approximation q* optimum q

18. 18 service quantum (q) Mean response time Job size distribution: H 2 (balanced means)E[ S ] = 1, C 2 = 19,  = 0.8 1% 2.5% 5% 7.5% h=0 approximation q* optimum q

19. 19 service quantum (q) Mean response time Job size distribution: H 2 (balanced means)E[ S ] = 1, C 2 = 19,  = 0.8 1% 2.5% 5% 7.5% 10% h=0 approximation q* optimum q

20. 20 Outline Effect of q and C 2 on mean response time Approximate analysis Bounds for M/G/1/RR Choosing the optimal quantum size

21. Conclusion/Contributions Simple approximation and bounds for M/G/1/RR Optimal quantum size for handling highly variable job sizes under preemption overheads q

22. 22 Bounds – Proof outline D i = mean delay for i th quantum of service D = [D 1 D 2... D K ] D is the fixed point of a monotone linear system: D T = A P D T +b D1D1 D2D2 f 1 =0 f 2 =0 D D’D’ D*D* Sufficient condition for lower bound: D ’ T  A P D ’ T +b for all P Sufficient condition for upper bound: D * T  A P D * T +b for all P

23. 23 Optimizing q Preemption overhead = h q’ = q+h, E[ S ]’ → E[ S ] (1+h/q),  ’ = E[ S ]’ Heavy traffic:Small overhead: