Chapter 5 Little's Law

1 Chapter5 little ' serviceinaservicesystemareimportantmeasu rementsfora ' ,TranscendentalTastings, rackina bottleofwineat bigger, goodpartytherackis seemstobeabout2/3rdsfull, ,Carolinestartstowonderhowlong,onaverage , fewmonthsofwineinvoicesfromTranscendenta landestimatesthatshehasbought,onaverage, 'tknowwhenshedrankwhichbottleandsotheres eemstobenowayshecanfindout,evenapproxima tely, a goodtaskforLittle' (eds.) ,<9 SpringerScience+ BusinessMedia, , 'sLawDealswithQueuingSystemsA"queuingsys tem"consistsofdiscreteobjectsweshallcall "items"that"arrive"atsomeratetothe"syste m.

2 "Withinthesystemtheitemsmayformoneormore queuesandeventuallyreceive"service" ~queuingsystem:itemsin1-Departuresqueue& queuingsystemWhileitemsareinthesystem,th eymaybeinqueues, ,wesaythata bottle(an"item")arrivestothesystemwhenit is 'weviewthewinerackasa singlechannelserver,theservicetimeis is interestingtonote,however,thatwedonotkno wwhichbottleCarolinewillpickandthereis noparticularreasonto believethatshewillpickaccordingto a first-in,first-out(FIFO) ,todealwiththeaveragenumberofbottlesinth ecellaroraveragetimespentbya bottleinthecellar, 'sLawsaysthat,understeadystateconditions ,theaveragenumberofitemsina ,W= averagewaitingtimeinthesystemforanitem,a ndA=averagenumberofitemsarrivingperunitt ime,thelawisL=AW(1) ,butit ,whethereachserverhasitsownqueueora singlequeuefeedsallservers,whattheservic etimedistributionsare,orwhatthedistribut ionofinter-arrivaltimesis,orwhatis theorderofserviceofitems, ,theequation(1) isespeciallyhandyfor"backoftheenvelope" thattwoofthetermsin(1)

3 ' little 'sLaw83 ThusforCaroline,theaveragenumberofbottle sinthesystemisL=(240)*(2/3)=160bottlesan dtheaveragearrivalrateisA=(12)*(8)=96 ,shecancalculatetheaverageamountoftimea bottlestaysinhercellarasW=(160)/(96)== ' biggerrackandmorepatience,or,alternative ly, ' 'sLawwitha heuristicargumentforLittle' ( t)=thenumberofitemsinthequeuingsystematt imet;T=a longperiodoftime;A(T)=theareaunderthecur ven( t)overthetimeperiodT;N(T)=thenumberofarr ivalsinthetimeperiodT.'"Ontheonehand,ani teminthequeuingsystemis (t).ItsaveragevalueoverTis theintegralofn(t)overT( ,A(T)) ,attimeteachoftheitemsis waitingandsois (t)overthetimeperiodT,weobtaina cumulativemeasureofthewaitingtime,againe qualtoA(T).

4 Furthermore,thearrivalsarecountabletoo,a ndgivenbyN(T).Therefore,inspectingthefig ure,wedefinen(t)1 , (T)=N(T)/T=arrivalrateduringtimeperiodT, L(T)=A(T)/T=averagequeuelengthduringtime periodT,W(T)=A(T)/N(T)= (T)=A(T)W(T).Allofthesequantitieswigglea rounda (T) ,L(T)andA(T) , ,asT-increases,thesestochastic"wiggles"i nL(T),A(T),and'W(T)becomesmallerandsmall erpercentagesoftheireventualvaluessothat L(T),A(T),andW(T)eachgotoa ,usingtheobvioussymbolsforthelimits,weha ve:limL(T)=L;T~oolimA(T)=A;T~oolimW(T):T ~oofromwhichwegetthedesiredresult(1).It is ,inhisoriginalpaper(Little1961),notingth attherelationship(1)heldforeachevolution ofthetimeseriesofa ,if wewatcha specificcaseorrealizationdesignated,say, byOJ,asit developsovertime,thenwewillfindthat5 little 'sLaw85L(m)=A(m)W(my, particularsystemgives(1),butit isa usefulinsighttoknowthattheformulaholdsfo reachevolvingtimeseriesasit isobservedovera (1)iscommonlycalledLittle' ,aspointedoutbyvariouspeople,includingLi ttle(1992),Eq.)

5 (1)is a mathematicaltheoremandthereforea 'rnsouttobeusefulinpractice, physicallawsuchasNewton' 'sequationhastobemeasuredandit mathematicaltheorem,iftheassumptionsares atis-fiedbytheapplication, mathematicaltheorema ' , ,000wafersperday,onaverage; 'remainedfairlystableoverthepast9 (WIP) ,000and50,000wafers;theaverageWIPis 45, , :L=45, thesystemisW= manufacturingcontext,weoftenrefertothisa stheflowtime,thetimebetweenwhena jobstartsandfinishesina ,if wethinkofonewaferasbeinga job,thenit takesthefactoryonaverage45daystoprocessi t,thatis,toconvertit froma blankwaferintoa , criticalforplanningandschedul-ingthefact ory, ' :Managingoure-mailisa commonand, ishardtokeepupwiththevolumeofmessages, 'sLawtogeta.

6 Thenthisis thearrivalrate:A=50 ,supposethatSueremovesa , :L= 'sLawweimmediatelyhaveanestimateofhowlon git takesSuetoanswera message,onaverage:W= 3 :Wewishtodeterminethesizeandstaffingleve lsforthematernitywardfora daysbeforegoinghomewithchild;howeverocca sionally, months,wefindthat90%ofthebirthshaveresul tedin2-daystays;fortheremaining10%ofthec ases,theaveragelengthoftimeinthematernit ywardis7 ,onaverage, 2 + 7= ' ;thearrivalrateisA=5 :W= ,theexpectedqueuelengthornumberinthesyst emisL= ( ,beds) ,thelawonlyprovidestheaveragerequirement s,andonewouldneedto , , little 'sLawprovidesastartingpointforthi sinvestigation, :TheTedWilliamsTunneltravelsundertheBost onharbor, day,about50,000vehiclesgothroughthetolls at little 'sLaw87 TransitAuthority(MTA)triestomodulatethen umberoftollboothsthatareopenatanypointin timesothattheaveragenumberofvehicleswait ingatthetolls(includingthoseatthebooths) ,allsixboothsareopenduringthepeaktimeint hemorningfrom6.

7 00 AMto10 ,thetunnelhandlesupto4,000carsperhour,an dtheMTAestimatesthattheaveragenumberofve hicleswaitingat anypointoftimeis relativelystablerateoverthemorningrushho ur,wecanthenuseLittle' ,600vehiclesperhour(orI vehiclepersecond),andtheexpectednumberof vehiclesinthesystemisL= ,onaverage,thetimea vehiclespendsat thetollboothsisW= 20/3,600h = :Thelocalrealestateagentinyourcommunitye stimatesthatit takes120daysonaveragetosella house;whereasthisnumberchangessomewithth eeconomyandseason,it , ' is ,whena house"completesitsservice"anddepartsfrom themarket, ,namely,W= 120daysandL= ,wecanestimatethearrivalratetothesystem, :Fromyourdailymorningtriptothedoughnutsh op,youknowtheyhavea healthybusiness, franchise, ;youvisittheshopat randomtimesbetween6:00 AMand9:00AM.

8 Youobservethatthequeueaveragesabout10cus tomers,andthatit takesyouabout3 ,thenyoucanapplyLittle' , , $5pervisit,thenwiththeassumptionthatyoua rea typicalcustomer,wehavea roughestimateoftheshop'srevenueduringthe semorninghours, ,$1, , 'sLawinCertainSystemsSofarwehavedevelope danddiscussedLittle'sLawasa ,stable, ,inthecaseofthematernityward,weassumetha ttheaveragearrivalrateofmothershasbeen: steadyatfiveperdayforsometime, ,wehaveregardedtheserviceprocessas,being stationary;forinstance,wereadandprocesso ure-mailat roughlythesameaveragerate,dayinanddayout , ,wehavefocusedonanintervaloftime, , ,intheseinstancesduetothehugevolumeofarr ivals,wecontendthatthesystembehavioris virtuallyequivalenttothatofa toshowthegreatrobustnessandgeneralityofL ittle' ' ' ,whichopenseverydayat7:00 AMandcloses16h laterat11 AM, closesat11PM, ,customersarrivetothestore, , PM,andtendtohavefairlylengthyshoppingfor aysastheystockupforaweekata ,likefirstthinginthemorning,andwillalsob efairlyleisurelyintheirshopping,takingup toanhourtocompletea.

9 Theireveningvisitsareoftentorunin,grabso methingandrunout.'WeproposetomodelS&Sasa ,fromtheabovedis-cussion,weseethatthisis anythingbuta neverina ,customersarriveat varyingrates,andthenatureoftheirshopping tripsalsovariesovertheday, ,wewillshownextthatLittle' little 'sLaw89 AnAnalyticInterludeLetNdenotethenumberof customersthatshopona , tocorrespondtotheopeningat7 AM,andtimet= 16tobethestoreclosingat11PM,16h (t) ,aswestartthedaywithzerocustomers,wehave N(O)=0;asweassumea totalofNcustomersarriveduringtheday,weha veN(16)= stair-casefashion, ,over0 <t< (t) ,wehaveD(O)=0,D(16)=N,andthecumulativede parturesincreaseina stair-casefashionoverthetimeinterval0 <t<16; (t)~D(t), ,thedifferencebetweenthetwocumulativepro cessesis thenumberofcustomersinthesupermarketat timet:L(t)=N(t)-D(t).

10 :JZQ5E0en:J0s)S3S4c) system,forexample, , (N(t)-D(t)) (2)Tomodeltheaveragetimeinthesupermarket foreachcustomerisa {Sl,S2' ..SN} tobethesequenceofarrivalorstarttimesfort heNcustomers, {Cj,C2'..CN}tobethesequenceofdepartureor completiontimesfortheNcustomers,wherec. , :1(NN)W= (3)Tocompute(3) {CI,C2,..CN}tobethesequenceofdepar-tureo rcompletiontimesfortheNcustomers, , ,thesequenceI2N"..J{c, C,..,C} ISJusta permutatIOnorreordenngofthesequence{cI ' C2,..CN} ,wecandefinethejthwaittimeasW=d- S., :1N1(NN)-xLWj= (3),sinceLCj= (NN)W=-xLwj=-xLcj-LSj'Nj=lNj=lj=](4)Noww eneedtorelateourexpressionforL,givenby(2 )toourexpressionforW,givenby(4).