On the Complex Spectrum of the Underlying Operator of a Reliability Model

We study the complex point spectral distribution of the underlying operator of the system consisting of a reliable machine, a storage buffer with infinite capacity and an unreliable machine. This system is described by infinitely many partial differential equations with integral boundary conditions. The known literature proved that all points in a set in the left half of the complex plane are eigenvalues of the underlying operator and indicated that all points outside of the set remain undetermined. In this paper, we study the spectrum outside of the set and, under certain conditions, prove that some points outside the set are eigenvalues of the underlying operator, whereas other points are not. By combining our result with the results in the existing literature, we give a description of the point spectral distribution of the underlying operator on the whole complex plane. Our idea and method are suitable for studying point spectral distribution of the underlying operators of some queueing models described by infinitely many partial differential equations with integral boundary conditions.