JIANetal.:BETA/M/1MODELFORMACHINETYPECOMMUNICATION587
out.Numericalanalysisshowsthatifthearrivalpattern
followsBetadistribution,itcouldpotentiallyincreasethe
meansojourntimeandmeanwaitingtimeofthesystem.And
increasedmeanserverrateordecreasedserverutilizationmay
helptosolvetheproblemthoughthisisnotwhattheoperators
want.Beyond3GPPsproposals,ifwewanttochoosean
appropriateBetadistributionfordifferentMTCapplications,it
isbettertoletα<βanddetermineareasonablegapbetween
αandβ.
APPENDIXA
BETADISTRIBUTION
TheprobabilitydensityfunctionoftheBetadistribution,for
x∈[0,1]andshapeparametersα>0andβ>0,isapower
functionofthevariablexanditsre?ection1?x:
Fig.3.NumericalsolutionofBforgeneralBeta/M/1model.
Γ(α+β)
α?1β?1
f(x)=x(1?x)(9)
A
Γ(α)Γ(β)
whereΓ(z)istheGamafunction.Themeanandvarianceof
variancetoestimateBetadistribution’sshapeparameters,these
xis:
conclusionscanbeusedasapreliminaryguidetochoosethe
ααβ
appropriatemodelfordifferentMTCapplicationscenariosbe-
E[x]=,Var[x]=(10)
2
α+β(α+β)(α+β+1)
yond3GPPTR37.868andpromptlyjudgetheeffectivenessof
thechosenmodel.AsBeta(1,1)equalsuniformdistribution
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theperformanceanalysisofBeta/M/1modelcanbecarried |
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