We describe the point spectrum of the generator of a -semigroup associated with the M/M/1 queueing model that is governed by an infinite system of partial differential equations with integral boundary conditions. Our results imply that the essential growth bound of the -semigroup is 0 and, therefore, that the semigroup is not quasi-compact. Moreover, our result also shows that it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.
"On Eigenvalues of the Generator of a -Semigroup Appearing in Queueing Theory." Abstr. Appl. Anal. 2014 (SI62) 1 - 9, 2014. https://doi.org/10.1155/2014/896342