and MGI1PS queues(3)

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),

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说教育文库and MGI1PS queues(3)在线全文阅读。