Variability Basics. God does not play dice with the universe. Variability Makes a Difference!
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1 Vaiabiliy Bai God do no play di wih h univ. Alb Einin Sop lling God wha o do. Nil Boh Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 1 Vaiabiliy Mak a Diffn! Lil Law: TH WIP/CT, o a houghpu an b obaind wih lag WIP, long CT o all WIP, ho CT. Th diffn? Vaiabiliy! Pnny Fab On: ahiv full TH (.5 j/h) a WIPW 4 job if i bhav lik B Ca, bu qui WIP7 job o ahiv 95% of apaiy if i bhav lik h Paial Wo Ca. Why? Vaiabiliy! Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 1
2 Toi and Ha Exapl Two ahin: ubj o a wokload: 69 job/day (.875 job/h) ubj o unpdiabl ouag (availabiliy 75%) Ha X19: long, bu infqun ouag Tooi : ho, bu o fqun ouag Pfoan: Ha X19 i ubanially wo on all au han Tooi. Why? Vaiabiliy! Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 3 Vaiabiliy Viw Vaiabiliy: Any dpau fo unifoiy Rando vu onollabl vaiaion Randon: Enial aliy? Aifa of inopl knowldg? Managn ipliaion: obun i ky Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 4
3 Pobabilii Inuiion U of Inuiion: diving a a howing a ball aing h ok ak Fi Mon Eff: Thoughpu ina wih ahin pd Thoughpu ina wih availabiliy Invnoy ina wih lo iz Ou inuiion i good fo fi on g Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 5 Pobabilii Inuiion (on.) Sond Mon Eff: Whih i o vaiabl poing i of pa o bah? Whih a o diupiv long, infqun failu o ho fqun on? Ou inuiion i l u fo ond on Miinpaion.g., gion o h an Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 6 3
4 Vaiabiliy Dfiniion: Vaiabiliy i anyhing ha au h y o dpa fo gula, pdiabl bhavio. Sou of Vaiabiliy: up ahin failu aial hoag yild lo wok opao unavailabiliy wokpa vaiaion diffnial kill lvl ngining hang od uo od podu diffniaion aial handling Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 7 Mauing Po Vaiabiliy an po i of a job ó andad dviaion of po i σ offiin of vaiaion, CV No: w ofn u h quad offiin of vaiaion (SCV), Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 8 4
5 Vaiabiliy Cla in Faoy Phyi Low vaiabiliy (LV) Moda vaiabiliy (MV) High vaiabiliy (HV) Effiv Po Ti: aual po i a gnally LV ffiv po i inlud up, failu ouag,. HV, LV, and MV a all poibl in ffiv po i Rlaion o Pfoan Ca: Fo baland y MV Paial Wo Ca LV bwn B Ca and Paial Wo Ca HV bwn Paial Wo Ca and Wo Ca Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 9 Mauing Po Vaiabiliy Exapl Tial Mahin 1 Mahin Mahin Cla LV MV HV Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 1 5
6 Naual Vaiabiliy Dfiniion: vaiabiliy wihou xpliily analyzd au Sou: opao pa aial fluuaion podu yp (if no xpliily onidd) podu qualiy Obvaion: naual po vaiabiliy i uually in h LV agoy. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 11 Down Ti Man Eff Dfiniion: f ba po i ba po ioffiin of 1 ba apaiy (a,.g.,pa/h) an i o failu an i o pai offin of vaiabiliy vaiabiliy of pai i ( σ / ) Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 1 6
7 Down Ti Man Eff (on.) Availabiliy: Faion of i ahin i up f A + Effiv Poing Ti and Ra: A f / A Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 13 Tooi and Ha - Availabiliy Ha X19: 15 in σ 3.35 in σ / 3.35/15.5 f 1.4 h (744 in) h (48 in) 1. Tooi: 15 in σ 3.35 in σ / 3.35/15.5 f 1.9 h (114 in).633 h (38 in) 1. Availabiliy: A f f A f f No diffn bwn ahin in of availabiliy. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 14 7
8 Down Ti Vaiabiliy Eff Effiv Vaiabiliy: / A A σ ( + σ )( 1 ) Vaiabiliy σ + A A dpnd on A A pai i σ + ( 1+ ) ( 1 ) in addiion o availabiliy Conluion: Failu infla an, vaian, and CV of ffiv po i Man ( ) ina popoionally wih 1/A SCV ( ) ina popoionally wih SCV ( ) ina popoionally in Fo onan availabiliy (A), long infqun ouag ina SCV o han ho fqun on Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 15 Tooi and Ha - Vaiabiliy Ha X19: Tooi A (1 + in ) A(1 A).5 + (1 + 1).75(1.75) high vaiabiliy A (1 + in ) A(1 A) (1 + 1).75(1.75) oda vaiabiliy Ha X19 i uh o vaiabil han Tooi! Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 16 8
9 Sup Man and Vaiabiliy Eff Analyi: N avag no. avag up duaion σ d.dv.of up i σ + N σ N 1 ó σ + + N σ job bwn up N Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 17 Sup Man and Vaiabiliy Eff (on.) Obvaion: Sup ina an and vaian of poing i. Vaiabiliy duion i on bnfi of flxibl ahin. Howv, h inaion i oplx. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 18 9
10 Sup Exapl Daa: Fa, inflxibl ahin h up vy 1 job 1h N 1 job/up h + 1/ / N 1+ /1 1. h 1/(1 + /1).8333 job/h Slow, flxibl ahin no up 1. h 1/ 1/ job/h Tadiional Analyi: No diffn! Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 19 Sup Exapl (on.) Faoy Phyi Appoah: Copa an and vaian Fa, inflxibl ahin h up vy 1 job N + / N 1+ / 1 1. h 1 h job / up h / 1 /( 1 + / 1) job / h σ N 1 σ σ + + N N Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 1
11 Sup Exapl (on.) Slow, flxibl ahin no up 1. h.5 1/ 1/ job/h.5 Conluion: Flxibiliy an du vaiabiliy. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 1 Sup Exapl (on.) Nw Mahin: Conid a hid ahin a a pviou ahin wih up, bu wih ho, o fqun up Analyi: Conluion: N 5 job/up 1h 1 / 1 /( / 5). 833 job / h N 1 σ σ σ N N 16. Sho, o fqun up indu l vaiabiliy. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 11
12 Oh Po Vaiabiliy Inflao Sou: opao unavailabiliy yl bahing aial unavailabiliy a, a, a Eff: infla infla Conqun: Effiv po vaiabiliy an b LV, MV,o HV. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 3 Illuaing Flow Vaiabiliy Low vaiabiliy aival ooh! High vaiabiliy aival buy! Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 4 1
13 Mauing Flow Vaiabiliy a an i bwn aival a 1 a aival a σ a andad dviaion of ibwn aival a σ a a offiin of vaiaion of inaival i Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 5 Popagaion of Vaiabiliy a (i) (i) d (i) a (i+1) i i+1 Singl Mahin Saion: ( 1 u ) a wh u i h aion uilizaion givn by u a Muli-Mahin Saion: u d 1+ (1 u )( a 1) + ( 1) a wh i h nub of (idnial) ahin and u Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o d u + dpau va dpnd on aival va and po va 6 13
14 Popagaion of Vaiabiliy High Uilizaion Saion High Po Va Low Flow Va High Flow Va Low Uilizaion Saion High Po Va Low Flow Va Low Flow Va Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 7 Popagaion of Vaiabiliy High Uilizaion Saion Low Po Va High Flow Va Low Flow Va Low Uilizaion Saion Low Po Va High Flow Va High Flow Va Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 8 14
15 Vaiabiliy Inaion Ipoan of Quuing: anufauing plan a quuing nwok quuing and waiing i opi ajoiy of yl i Sy Chaaii: Aival po Svi po Nub of v Maxiu quu iz (bloking) Svi diiplin (FCFS, LCFS, EDD, SPT,.) Balking Rouing Many o Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 9 Kndall' Claifiaion A/B/C A: aival po B: vi po C: nub of ahin A B M: xponnial (Makovian) diibuion G: oplly gnal diibuion D: onan (dinii) diibuion. Quu Sv C Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 3 15
16 Quuing Paa a h a of aival in uo (job) p uni i ( a 1/ a h avag i bwn aival). a h CV of in-aival i. h nub of ahin. h a of h aion in job p uni i /. h CV of ffiv po i. No: a aion an b dibd wih 5 paa. u uilizaion of aion a /. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 31 Quuing Mau Mau: CT q h xpd waiing i pn in quu. CT h xpd i pn a h po n, i.., quu i plu po i. WIP h avag WIP lvl (in job) a h aion. WIP q h xpd WIP (in job) in quu. Rlaionhip: CT CT q + WIP a CT WIP q a CT q Rul: If w know CT q, w an opu WIP, WIP q, CT. Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 3 16
17 Th G/G/1 Quu Foula: CT q V U a + u 1 u Obvaion: Uful odl of ingl ahin wokaion Spaa fo vaiabiliy, uilizaion, po i. CT q (and oh au) ina wih a and Flow vaiabiliy, po vaiabiliy, o boh an obin o infla quu i. Vaiabiliy au ongion! Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 33 Th G/G/ Quu Foula: CT q V U a + ( + 1) 1 u ( 1 u) Obvaion: Uful odl of uli-ahin wokaion Exly gnal. Fa and aua. Eaily iplnd in a padh (o pakag lik MPX). Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 34 17
18 bai daa failu up yild au VUT Spadh MEASURE: STATION: Aival Ra (pa/h) a Aival CV a Naual Po Ti (h) Naual Po CV Nub of Mahin MTTF (h) f MTTR (h) Availabiliy A Effiv Po Ti (failu only) ' Eff Po CV (failu only) ' Job Bwn Sup N Sup Ti (h) Sup Ti CV Eff Po Ti (failu+up) Eff Po Ti Va (failu+up) σ Eff Po CV (failu+up) Dpau CV d Yild y Final Dpau Ra a*y Final Dpau CV yd +(1-y) Uilizaion u Thoughpu TH Quu Ti (h) CTq Cyl Ti (h) CTq Cuulaiv Cyl Ti (h) Σi(CTq(i)+(i)) WIP in Quu (job) actq WIP (job) act Cuulaiv WIP (job) Σia(CT(i)) Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 35 Sking Ou Vaiabiliy Gnal Sagi: look fo long quu (Lil' law) fou on high uilizaion ou onid boh flow and po vaiabiliy ak why fiv i Spifi Tag: quipn failu up wok opao paing anyhing ha pvn gula aival and po i Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 36 18
19 Vaiabiliy Pooling Bai Ida: h CV of a u of indpndn ando vaiabl da wih h nub of ando vaiabl. Exapl (Ti o po a bah of pa): i o po ingl pa ( bah) n σ ( bah) nσ σ andad dviaion of i o po ingl pa σ σ ( bah) nσ ( bah) ( bah) n CV of i o po ingl pa σ n? n Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o ( bah) n 37 Safy Sok Pooling Exapl PC oni of 6 oponn (CPU, HD, CD ROM, RAM, ovabl oag dvi, kyboad) 3 hoi of ah oponn: diffn PC Eah oponn o $15 ($9 aial o p PC) Dand fo all odl i noally diibud wih an 1 p ya, andad dviaion 1 p ya Rplnihn lad i i 3 onh, o avag dand duing LT i θ 5 fo opu and θ 5(79/3) 675 fo oponn U ba ok poliy wih fill a of 99% Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 38 19
20 Pooling Exapl - Sok PC Ba Sok Lvl fo Eah PC: yl ok afy ok R θ + z σ ( 5) 37 On-Hand Invnoy fo Eah PC: I(R) R - θ + B(R) R - θ z σ uni Toal (Appoxia) On-Hand Invnoy : 1 79 $9 $7,873, Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 39 Pooling Exapl - Sok Coponn Nay Svi fo Eah Coponn: S (.99) 1/ z.93 Ba Sok Lvl fo Eah Coponn: yl ok R θ + z σ ( 675) 633 afy ok On-Hand Invnoy Lvl fo Eah Coponn: I(R) R - θ + B(R) R - θ z σ uni Toal Safy Sok: A 9% duion! 8 18 $15 $615,6 Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 4
21 Bai Vaiabiliy Takaway Vaiabiliy Mau: CV of ffiv po i CV of inaival i Coponn of Po Vaiabiliy failu up any oh - dfla apaiy and infla vaiabiliy long infqun diupion wo han ho fqun on Conqun of Vaiabiliy: vaiabiliy au ongion (i.., WIP/CT inflaion) vaiabiliy popoga vaiabiliy and uilizaion ina poold vaiabiliy l duiv han individual vaiabiliy Walla J. Hopp, Mak L. Spaan, 1996, hp://faoy-phyi.o 41 1
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