On Eigenvalues of the Generator of a C 0 -Semigroup Appearing in Queueing Theory

Abstract

We describe the point spectrum of the generator of a $C_0$-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 $C_0$-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.