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
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