On The Point Spectrum Of The Operator Corresponding To The M/M/1 Queueing Model With Optional Second Service

In this thesis we study point spectrum of the operator associated with the M/M/1 queueing model with optional second service that was established by G.Choudhury in 2003 and prove that if the arrival rate of customers ?.the service rate of the first essential service?1,the service rate of the optional second service?2,the probability that a customer chooses the second op-tional service ? satisfy ?1(1-?)>?,?2>?,then the point spectrum of the operator contains the open interval(1-?,0)and the geometric multiplicity of each point in the interval is equal to 1.From this we verify that the C0-semigroup generated by the operator is not compact,not eventually compact,even not quasi-compact and it is impossible that the time-dependent solution of the model exponentially(uniformly)converges to its steady state solution.