249ModelinginProbabilityandStatisticsAndreA.NapoliScenario1:TheCommoOfficerSupposeyouarethecommunicationsofficerforanarmoredbattalion.YourbattalionTOCmustmaintaincommunicationswiththebrigadeTOCatalltimeswheninacombatenvironment.Inpreparationforadeploymentthatwilllastamonth,yourbattalionXOwantsyoutotellhimthenumberofradiostheunitshouldbringinordertohavea99%reliablelinkfrombattaliontobrigade.Youlearnthatoneoftheseradiostypicallyfailsevery300hours.Problem1:Modelthissituationasastandbyredundantsystem(asshowninFigure1)anddeterminethenumberofradiosrequiredtoachieve99%reliability.Figure1.BattalionCommunicationsSystemSolution1:WeassumethatastandbyredundantsystemmodelsradiofailuresasaPoissonprocess.Therearethreeassumptionsrequiredofsuchaprocess–stationarity,rarityandindependence.•Stationarity–theexpectednumberofradiofailuresinanysubintervalisproportionaltothelengthofthesubinterval.Aconsequenceofthisisthattheradiofailureratestaysconstantthroughouttheentireinterval.•Rarity–theprobabilitythatoneradiofailsinaverysmalltimeintervalisverysmall,andtheprobabilitythattwofailinthesameintervalisevensmaller,essentiallyzero.•Independence–thenumberofradiofailuresinonetimeintervalisindependentofthenumberinanyothernon-overlappingsubinterval.Aretheseassumptionsrealistic?Theymightoversimplifyanotherwisecomplexsystem,butshouldsufficeforaroughestimate.AsketchofthissystemispresentedinFigure1.The“DS”noderepresentsadecisionswitch–whichessentially“replaces”componentsastheyfail.250LetXbearandomvariablerepresentingthenumberofradiosworkingoutofnradiosinthesystem.WeknowthatundercertainconditionsX~Poisson(λ)whereλisthemeannumberoffailuresinatimeperiod,t.Inthisproblem,X~Poisson(6.3=λ).Then,anexpressionrepresentingthereliabilityofthesystemis99.0!)1()720(10≥=−≤=∑−=−nxxxenXPRλλ.Notethelowerboundonthesystemreliabilityof0.99.ThesmallestvalueofnforwhichthiscumulativePoissonissatisfiedwithmean3.6isn=9.So,weconcludethatthebattalionshoulddeploywithnineradios.DISCUSSION:Weassumethatonceara...
