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Raz, RAQFM, Sigmetrics04O  =FnRAQFM  a Resource Allocation Queueing Fairness Measure 8 jDavid Raz School of Computer Science, Tel Aviv University Jointly with Hanoch Levy, Tel Aviv University Benjamin Avi-Itzhak, RUTGERS University Sigmetrics  Performance, June 2004PZH  $        Outline   Motivation The importance of fairness in queues The physical properties of the problem Related work The RAQFM approach Properties Analysis 6 Y( Y(  $Why is Fairness in Queues Important?%%  % ZFundamental 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 itselfTZ6ZWZZ6W&o    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) PeP7P;PDPZPe   ;DY    tH           8!Queues: The Physical Factors    Fairness: Size and Seniority2 f f  Size Seniority*   The Size vs. Seniority Dilemma:f f   XShort vs. Long Is it more fair to serve Short ahead of Long? What is the intuition?YZY Y A&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 Y'\; f'\ f     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!q2!q2f 4     (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 AnalysisZ ff' f f f  9"RAQFM Philosophy   >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:TZ%0ZdI0Z<a a 3a1   J(&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 zeroa!(   a  .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] e" 0 fI/    @     ;#Short vs. Long Revisited  !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   P)Property 2: Bounds  7Individual discrimination is bound by How good How bad 6&& 8 #!Property 3: Reaction to Seniority" f " 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)`6F/ "6F/a" c   S+Property 3: Reaction to Sizef  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 )`&G- 8&G-a8 c   U,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  a aaaa)a a  a  a  a 0$$(a(%,,0a00i00a00a00a044&  2 (LFairness of Service Policies Not Discriminating For Size as function of loadM9  M  d0#Is FCFS Always More Fair Than LCFS?$#  $  *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 &\o\o  a. Future Work  4More service policies  SRPT, SJF, FB etc. Generalized service requirement Different architectures  multi-class, multi-server, multi-queue, polling etc.   b/nRAQFM  a Resource Allocation Queueing Fairness Measure 8 jDavid Raz School of Computer Science, Tel Aviv University Jointly with Hanoch Levy, Tel Aviv University Benjamin Avi-Itzhak, RUTGERS University Sigmetrics  Performance, June 2004PZH  $       &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& )RMZPZVZMIV N  .   ,      DSensitivity to Size and Seniority: Mr. Short vs. Mrs. Long RevisitedDE    $  E Observe a simple scenario: C0 in the system, being served C1 and C2 arrive Compare the fairness depending on the order of service. The difference in fairness is:h_  0Analysis of RAQFM  NRAQFM 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 XWPW4 X  -     1 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 , state (a,b+1) customer departs - probability , state (a-1,b) b \cL /  4 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 @-E  5 FCFS Analysis  Solving the recursive equation: Choose a maximum value of b, say N Run through the matrix iteratively, column by column O(N2) time, O(N) spaced o 8  'zIndividual discrimination for FCFS k  number seen on arrival>#  >  `-Short vs. Long Revisited  <$DSensitivity to Size and Seniority: Mr. Short vs. Mrs. Long RevisitedDE    $  E  =%DSensitivity to Size and Seniority: Mr. Short vs. Mrs. Long RevisitedDE    $  E  Q*%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 comparison6&V&V  Property 1: PS is the most fair  Theorem: Processor Sharing (PS) is the most fair policy PS - each customer gets 1/N(t) ! 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