On eigenvalues of the generator of a \(C_0\)-semigroup appearing in queueing theory. (English) Zbl 1473.47015

Summary: 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.


47D06 One-parameter semigroups and linear evolution equations
90B22 Queues and service in operations research
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