RAQFM – a Resource Allocation Queueing Fairness Measure

1. RAQFM – a Resource Allocation Queueing Fairness Measure David Raz School of Computer Science, Tel Aviv University Jointly with Hanoch Levy, Tel Aviv University Benjamin Avi-Itzhak, RUTGERS University Sigmetrics – Performance, June 2004

2. Outline • Motivation • The importance of fairness in queues • The physical properties of the problem • Related work • The RAQFM approach • Properties • Analysis D. Raz, RAQFM, Sigmetrics04

3. Why is Fairness in Queues Important? • Fundamental reason: • Isn’t fairness why we have queues in the first place? • Scientific evidence, recent studies, Rafaeli et. al. [2003] (experimental psychology): • Experiments on humans in multi-queue and single queue • Fairness in queue is very important to people • Perhaps even more than the delay itself D. Raz, RAQFM, Sigmetrics04

4. Queueing Theory and Fairness • Decades of research on: • Queueing structures / policies • Focused on efficiency – delay distribution/moments, utilization, etc. • The issue of fairness is discussed, but not quantified • Larson (1988), Palm (1953), Mann (1969), Whitt (1984) etc. • Existing measures for streams (WFQ) • Little analysis on job fairness • Morris & Wang (1985) • Avi-Itzhak & Levy (2004) • Wierman & Harchol-Balter (Sigmetrics 2003)  We know very little about job fairness D. Raz, RAQFM, Sigmetrics04

5. Size (Service Requirement) Seniority (Arrival Time) Resources Queues: The Physical Factors D. Raz, RAQFM, Sigmetrics04

6. Fairness: Size and Seniority • Size • Seniority D. Raz, RAQFM, Sigmetrics04

7. Short Long The Size vs. Seniority Dilemma • Short vs. Long • Is it more fair to serve Short ahead of Long? What is the intuition? D. Raz, RAQFM, Sigmetrics04

8. Recent Related Work(1) • Avi-Itzhak & Levy (2004): • Axiomatic approach to fairness • Departure point & emphasis: Seniority (Order of service) • Some results (for equal service times) • FCFS is the most fair (among non preemptive) • LCFS is the most unfair (among non preemptive) • Takes Seniority into account D. Raz, RAQFM, Sigmetrics04

9. Recent Related Work(2) • Wierman & Harchol-Balter (2003): • Propose a Fairness Criterion • Slowdown: for job of size x compute E[T(x)/x]. If it is bounded the system is FAIR. • Some results: • FCFS is “Always UNFAIR” • LCFS (preemptive) is FAIR • Takes Size into account D. Raz, RAQFM, Sigmetrics04

10. Requirements From a Job Fairness Measure • Reacts well to both Seniority and Size • Aim for standard  have a consistent view and agree with Intuition build confidence • Yields to Analysis RAQFM – A Resource Allocation Queueing Fairness Measure D. Raz, RAQFM, Sigmetrics04

11. RAQFM Philosophy Equal Share of Resources  Fairness D. Raz, RAQFM, Sigmetrics04

12. RAQFM - How to Apply the Philosophy: Individual Discrimination • At every epoch t with N(t) customers in the system, each customer should get 1/N(t) • Warranted service: • Granted service: • Compare the warranted service with the granted service : discrimination: D. Raz, RAQFM, Sigmetrics04

13. Basic Properties of the Discrimination • E[D]=0(every sample path, every policy)Proof sketch: The momentary rate of discrimination is”zero sum” • For PS individual discrimination is zero D. Raz, RAQFM, Sigmetrics04

14. RAQFM – System Fairness System Unfairness: differences in customer treatment (customer discrimination)  We measure the unfairness in a system using Var[D] (D is r.v.) Var[D] ≥ 0  Property 1: PS is the most fair policy D. Raz, RAQFM, Sigmetrics04

15. Seniority (Arrival Time) Difference Size (Service Requirement) Difference Short vs. Long Revisited Is it more fair to serve Short ahead of Long? D. Raz, RAQFM, Sigmetrics04

16. Short vs. Long Revisited • The difference in unfairness resulting from serving short ahead of long is • If size difference is small – serve by order of arrival • If seniority difference is small – serve by order of service requirement RAQFM Agrees With Intuition D. Raz, RAQFM, Sigmetrics04

17. Property 2: Bounds • Individual discrimination is bound by • How good • How bad D. Raz, RAQFM, Sigmetrics04

18. Property 3: Reaction to Seniority • Theorem: If customers have equal service requirements • For each pair of customers, it is more fair to serve the senior first ⇒FCFS is the most fair ⇒LCFS is the least fair (Proof sketch: compare scenarios) RAQFM Reacts Well to Seniority D. Raz, RAQFM, Sigmetrics04

19. Property 3: Reaction to Size • Theorem: If customers arrive together • For each pair of customers, it is more fair to serve the shorter first ⇒SJF is the most fair ⇒LJF is the least fair (Proof sketch – prove ) RAQFM Reacts Well to Size D. Raz, RAQFM, Sigmetrics04

20. Property 4: RAQFM is Analyzable • Example: analysis of M/M/1 FCFS • Markov chain state is (a,b)=(# ahead, # behind) • c(a,b)=momentary discrimination rate at state (a,b) • d(a,b)=expected discrimination of a walk starting at (a,b) • Similarly for the second moment d(2)(a,b) D. Raz, RAQFM, Sigmetrics04

21. Fairness of Service Policies Not Discriminating For Sizeas function of load LCFS: Severe seniority violation Empty system: everyone is alone FCFS: no seniority violation PS: Absolute Fairness D. Raz, RAQFM, Sigmetrics04

22. Is FCFS Always More Fair Than LCFS? FCFS = 0.9 Preemptive LCFS = 0.15 D. Raz, RAQFM, Sigmetrics04

23. Summary • Fairness is extremely important, yet there are little quantification methods for jobs • RAQFM • Philosophy: fair share of service • Agrees with intuition • Reacts well to seniority and size • Yields to analysis D. Raz, RAQFM, Sigmetrics04

24. Future Work • More service policies – SRPT, SJF, FB etc. • Generalized service requirement • Different architectures – multi-class, multi-server, multi-queue, polling etc. D. Raz, RAQFM, Sigmetrics04

25. Thank You RAQFM – a Resource Allocation Queueing Fairness Measure David Raz School of Computer Science, Tel Aviv University Jointly with Hanoch Levy, Tel Aviv University Benjamin Avi-Itzhak, RUTGERS University Sigmetrics – Performance, June 2004 http://www.cs.tau.ac.il/~davidraz