and MGI1PS queues(4)

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

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

gwe(x)=1=(1 x)2,a= sandb= inTheorem2, havethat

(x)

dsx x¿1

d xs (x)d x:s sRewriting x

∞ s (x)d xass f(x)dxweget,

(x)

ds

x x¿1

∞

d xss f(x)dx x:(3)sConversely,cal argumentwhenwehave (x)isdecreasing,usinganidenti-

(x)

d s

xx 61

∞

d xss f(x)dx sx:Now,bounds.wecansimplyevaluatetheintegral Eqs.(x)((theWe2)and(decreasingcasewillconsideronlythecaseoftoobtainincreasingour3followsidentically).UsingE[T]FB¿1 )

∞1 s

∞

0ss f(x)dx

×

d x

dsxs =∞

1 s11

0

sF

(s)

sds=∞

1 s s

0sF

(s)sds

=∞

F

(s)0

ds=E[X]

=E[T]PS

Thiscompletestheÿnaltwocasesoftheproof.References

[1]E.G.Co man,P.[2]Prentice-Hall,L.Kleinrock,EnglewoodDenning,OperatingSystemTheory,

QueueingSystems,Cli s,NJ,1973.

[3]Applications,I.schedulingforRai,G.Urvoy-Keller,Wiley,NewYork,1976.

Vol.II,Computer

ProceedingsjobsizedistributionsE.Biersack,withAnalysishighvariance,ofLAS

[4]andL.E.ModelingofofACMComputerSigmetricsSystems,ConferenceSanDonMeasurementin:remainingprocessingtimeSchrage,L.W.Miller,Thediscipline,queueM/G/1iegOper.witho,CA,2003.Res.theshortest

[5]670–684.

14(1966)D.Stoyan,D.J.[6]OtherR.W.StochasticDaley,Models,ComparisonWiley,NewMethodsYork,1983.

forQueuesand

Prentice-Hall,Wol ,StochasticEnglewoodModelingandCli s,NJ,1989.

theTheoryofQueues,

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