and MGI1PS queues(2)

www.elsevier.com/locate/dsw A note on comparingresponse times in the M/GI/1/FB

74A.Wiermanetal./OperationsResearchLetters32(2004)73–76

thepolicyexpected[6]or[P2.responsetimeofajobofsizexunder]Thenforatheprooffollowingclassicofthese):

resultsexist(seeE[T(x)]PS=x

;

E[T(x)]FB=

xt2

f(t)dt+ x2F (x)+x xx

x

= tF (t)dtx+x

x;

where

x

x=

tf(t)dt+xF

x

F

0(x)= 0

(t)dt:Noticethat xcanbethoughtofastheloadofjobs

frommeanWeservicebrie ydistributiondiscusssomeXxdef

=min(x;X).

underresponsesackM/GI/1/PStimes.Rai,underM/GI/1/FBpriorworkwithcomparingthoseEare[T]FB[3]provethatforUrvoy-Keller,anyservicedistributionandBier-distributionsconcerned6(2 with )=(2understandingfor 2 )E[T]PS.Inourwhichpaper,serviceweningconsiderE[T]FB6E[T]PS.Co manthefollowingrelationexactly[this1,p.question188–189]:andhypothesizeandDen-E[T]FB¡E[T]PSwhenC¿1;E[T]PS¿E[T]FBwhenC¡1;

whereC2def

=Var(X)=E[X]2ofmakesvariationoftheservicedistribution.isthesquared(Notecoe cientthat[1thisformulationthestatementisequivalent.)

intermsofwaitingtimes,butthat]isslightnotItturnsalwaysouttruethat(seeCo manExampleandDenning’s1hypothesissuchareÿnement.reÿnement.

Ourfollowingmainbelow)theoremandneedsgivesaTheoremdef

of1.Let (x) (x)bethehazardrateandthePSservicerelatedistribution=f(asfollows:

.xIn)=F

anM/GI/1systemFB1.2.If (x)3.IfIf ((xx)is)isdecreasingisconstantincreasing,E,,[EET[[]TFB]FB6E[T]PS.T]FB=¿E[ET[]PST].PS.

Co manObservefollowingwell-knownandthatDenning’sthistheoremhypothesisisareÿnementof

relatesthehazardrateandlemmathecoe cient[5,p.because16–19],ofofvariation.whichtheLemma (x)isincreasing1.When C(x6)is1.

decreasing,C¿1andwhendistributionsNoticethatTheorem1doesingforuessomewhosehazardratenotisbothsaystrictlyanythingaboutincreas-thisofdecreasing)situationx.Ourx(whereexampleandstrictlydecreasingforotherval-hazardbelowrateshowsisboththatitisexactlyCo man,Denninghypothesis.

whichleadstoacounterexampleincreasingandtotheExampledistribution1.whereThefollowingexampleC2PS

givesajobsize

sider¿1butE[T]¡E[T

thediscretedistribution

]FB.Con-X=

1withprobability ;6withprobability+ :Itforisanyeasy ¿to0verify,butbyE[Tsimple]PS¡Ecalculation[T]FBforsmallthatC2 ¿¿0.1andExample1distributionDenning,isisneitheralwaysbelongsandcountermoreovertothetoaclassobservehypothesisofCo manwherethatthethishazardjobsizeratethehazardintuitionprovingTheoremdecreasingnorbehindthetheorem.1,italwaysincreasing.2BeforeIntuitively,isusefultowhendescribetheyoungjobsrateandThus,oldjobsareofthearelikelyservicelikelytodistributionisdecreasingtohavehavesmallhighremainingremainingtimestimes.jobsthethenumberwithFBissmallmimickingSRPTbygivingpreferencetoofremainingtimes,jobsinthesystem,andandthusequivalentlyminimizinghazardoverallyoungjobsratemeanoftheresponsetime.Likewise,whentheinarelikelyservicetohavedistributionlargerremainingtimes,isincreasing,system.whichIncasetheFBcasemaximizesofconstantthehazardnumberrate,ofajobsjob’sinage

the2

butionStrictlyspeaking,thehazarddistributionisdiscrete.approaching0.consistingofHowever,Gaussianswecanratesapproximateareundeÿnedasthedistri-atx=1andby6acontinuousthesisdoesnotItholdiseasyfortothisseecontinuousthatCo mandistributionandDenningwithvarianceeither.

’shypo-

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