BystudyingthespectralpropertiesoftheunderlyingoperatorcorrespondingtotheM/G/1queueingmodelwithoptionalsecondserviceweobtainthatthetime-dependentsolutionofthemodelstronglyconvergestoitssteady-statesolution.Wealsoshowthatthetime-dependentqueueingsizeatthedeparturepointconvergestothecorrespondingsteadystatequeueingsizeatthedeparturepoint.