www.elsevier.com/locate/dsw A note on comparingresponse times in the M/GI/1/FB
A.Wiermanetal./OperationsResearchLetters32(2004)73–7675
iscaseindependentproveitwouldseemofitsthatremainingserviceFBschedulingshouldtime,innotwhichim-formulationTheuponproofPS.
ofTheorem1thefollowingLemma.ofresponsetimeswillunderrelyonFBanasalternativestatedinLemma2. x
E[T(x)]FB=
(1 s)dsx(1)
Proof.termsandTointerchangederivethisnewintegralsexpressionwecancombine asfollows:
x
E[T(x)]FB=x tF (t)dtx+
x=
x x x+ x
(t)dttF
x=
x x
sdsxx
=(1 s)dsx:
servingthatThethird equationx
followsfrom xthelinethenfollows0 sds=x x bywritingxas0 x
tF (tsecond)dt.Thebyÿnalob-01ds.
pleNoticethatEq.(1)givesusaparticularlyform,formfortheresponsetimeunderFB.Thissimplesim-ity(statedcombinedbelow),withwilltheallowChebyshevustoproveIntegralTheoremInequal-1.Theoremhtion(x)2(ChebyshevIntegralInequality).Letonbe[aa;non-negativeb].
,integrable,increasingfunc-1.Letfunctiong(x)onbea[a;non-negative,integrable,increasing bb].
b 2.ThenLet,(b a)g(x)dx¿b
ah(x)ah(x)dxafunctiong(x)beg(x)dx.ona[a;non-negative,integrable,decreasingThen,(b a) bb].
h(x)g(x)dx6 bx)dx b
aah(ag(x)dx.IntegralUsingLemmaInequality,2inwecombinationwillnowprovewiththeTheoremChebyshev1.
Proofwhereofthe service (Theoremx)1.Wewillstartwiththecasedistributionisconstant.isNoticeexponentialthatthiswithimpliessomethatratecipline,.RecallofwheretheMarkovthestatechaincorrespondsfortheM/M/1/FCFStodis-chainjobsinthesystem,andforalli¿0,thetheMarkovnumberthewithMarkovmovesfromstateitostatei+1withrate ,andrepresentedrate .chainNoticemovesthatfromstateitostatei 1isthethatbyijobswhenbytheexactthesameM/M/1/PSchain.ThedisciplinekeypointisinthethesystemMarkovarechainservedisinatstaterate =ii,,eachwhich,ofresultssuperpositioniIn 1.Ainofexponentialdistributions,againsimilaratotalargumenttransitionrateof fromstateitoprocessorstatei,somenumberofcanjobsbemadej6forM/M/1/FB.ofconservingpolicyjjobsreceivingevenly,and =jthusthetotalicompletionwillshareratethesizesthatservicedoesnotis depend.Infact,onanytheworkmeancanthequeueberepresentedlengthandthebythissamechain.Thus,jobthemakesamesinceuseforallworkconservingpoliciesmeansojourntimethataredoalsonotrivalwedidofjobnotsize.makeItanyisalsoassumptionsinterestingtonotethat,lengthprocessforofallworkandtheintheaboveargument,ingFBas
1,wecanwritethemeanresponsetimeunderE[T]FB
=
∞0
E[T(x)]f(x)dx
=
∞
x
(1 s)ds
0
f(x)dx
x=∞0(1 ∞
s)f(x)
s
xdxds:
f(Finally,x)= (xobservingthat)F (x),wegetd x=dx= F
(x)andthatE[T]FB
=1∞ 0(1
s) (x) d xdsxs:(2)Inequality.AtthispointisFirst,wewewillwillapplydealwiththeChebyshevthecasewhenIntegral (1=(1increasing. Notethat isincreasingandhencex)xx)2isincreasing.Thussettingh(x)= (x),