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-