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

June, 2004D. Raz, RAQFM, Sigmetrics04 2 Outline Motivation – The importance of fairness in queues – The physical properties of the problem – Related work The RAQFM approach Properties Analysis

June, 2004D. 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

June, 2004D. 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

June, 2004D. Raz, RAQFM, Sigmetrics04 5 Queues: The Physical Factors Size (Service Requirement) Resources Seniority (Arrival Time)

June, 2004D. Raz, RAQFM, Sigmetrics04 6 Fairness: Size and Seniority Size Seniority

June, 2004D. Raz, RAQFM, Sigmetrics04 7 The Size vs. Seniority Dilemma Short vs. Long Is it more fair to serve Short ahead of Long? What is the intuition? Long Short

June, 2004D. 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

June, 2004D. 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

June, 2004D. 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

June, 2004D. Raz, RAQFM, Sigmetrics04 11 RAQFM Philosophy Equal Share of Resources  Fairness

June, 2004D. 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:

June, 2004D. 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

June, 2004D. 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

June, 2004D. Raz, RAQFM, Sigmetrics04 15 Short vs. Long Revisited Seniority (Arrival Time) Difference Size (Service Requirement) Difference Is it more fair to serve Short ahead of Long?

June, 2004D. Raz, RAQFM, Sigmetrics04 16 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 Short vs. Long Revisited

June, 2004D. Raz, RAQFM, Sigmetrics04 17 Property 2: Bounds Individual discrimination is bound by – How good – How bad

June, 2004D. 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

June, 2004D. 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

June, 2004D. 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) –

June, 2004D. Raz, RAQFM, Sigmetrics04 21 Fairness of Service Policies Not Discriminating For Size as function of load Empty system: everyon e is alone LCFS: Severe seniority violation FCFS: no seniority violation PS: Absolut e Fairness

June, 2004D. Raz, RAQFM, Sigmetrics04 22 Is FCFS Always More Fair Than LCFS? FCFS = 0.9 Preemptive LCFS = 0.15

June, 2004D. 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

June, 2004D. 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.

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 Thank You

June, 2004D. Raz, RAQFM, Sigmetrics04 26 Property 4: RAQFM is Analyzable RAQFM yields to analysis via standard queueing theory techniques We compute – E[D|k] - discrimination given number seen on arrival – Var[D] - system unfairness Conducted analysis for M/M/1 type: Variety of service orders (FCFS, LCFS, ROS, more…)

June, 2004D. Raz, RAQFM, Sigmetrics04 27 Sensitivity to Size and Seniority: Mr. Short vs. Mrs. Long Revisited Observe a simple scenario: – C 0 in the system, being served – C 1 and C 2 arrive – Compare the fairness depending on the order of service. The difference in fairness is:

June, 2004D. Raz, RAQFM, Sigmetrics04 28 Analysis of RAQFM RAQFM yields to analysis via standard queueing theory techniques We compute for M/M/1 – E[D|k] – Discrimination given number seen on arrival – Var[D] - system unfairness

June, 2004D. Raz, RAQFM, Sigmetrics04 29 FCFS Analysis Time can be slotted by arrivals and departures, slot size is independent State description: a customers ahead of C, b customers behind C. If we assume C is in state (a,b) in slot i – customer arrives – probability λ̃, state (a,b+1) – customer departs - probability μ̃, state (a-1,b)

June, 2004D. Raz, RAQFM, Sigmetrics04 30 FCFS Analysis d(a,b) is the discrimination of a walk starting in (a,b), with moment d(a,b) c(a,b) is the momentary discrimination at (a,b) t the slot size

June, 2004D. Raz, RAQFM, Sigmetrics04 31 FCFS Analysis Solving the recursive equation: – Choose a maximum value of b, say N – Run through the matrix iteratively, column by column – O(N 2 ) time, O(N) space

June, 2004D. Raz, RAQFM, Sigmetrics04 32 Individual discrimination for FCFS k – number seen on arrival Bingo – clear road at rush hour Traffic Jam at 4AM

June, 2004D. Raz, RAQFM, Sigmetrics04 33 Short vs. Long Revisited C1C1 C2C2 C0C0 Seniority (Arrival Time) Difference Size (Service Requirement) Difference

June, 2004D. Raz, RAQFM, Sigmetrics04 34 Sensitivity to Size and Seniority: Mr. Short vs. Mrs. Long Revisited Size (Service Requirement) Difference

June, 2004D. Raz, RAQFM, Sigmetrics04 35 Sensitivity to Size and Seniority: Mr. Short vs. Mrs. Long Revisited C1C1 C2C2 C1C1 C2C2 C2C2 C1C1 A B

June, 2004D. Raz, RAQFM, Sigmetrics04 36 RAQFM Agrees With Intuition II Bounds Individual discrimination is bound by – How good – How bad (Conveniently, System unfairness is bound by Bounds provide a scale for comparison RAQFM is Sensitive to Size and Seniority

June, 2004D. Raz, RAQFM, Sigmetrics04 37 Property 1: PS is the most fair Theorem: Processor Sharing (PS) is the most fair policy PS - each customer gets 1/N(t) ⇒ PS is the most fair policy