Spectral distribution and semigroup properties of a queueing model with exceptional service time

In this paper, we studied the spectrum and semigroup properties of the M/G/1 queueing model with exceptional service time for the first customer in each busy period. First, we described the point spectrum of the system operator that corresponds to the model, and we prove that the system operator has an uncountable infinite number of eigenvalues on the left-half complex plane. Second, by using the spectrum analysis and semigroup theory, we obtained that the spectrum-determined growth (SDG) condition holds and the semigroup is not asymptotically stable, compact, eventually compact, or even quasi-compact. Finally, in order to clarify the results of spectral distribution, some numerical analysis were conducted.