Abstract:
We discuss a queueing model, namely, in the usual notation, the $M/M^B/1$ model, and two reliability models, by viewing them as abstract Cauchy problems. In all three cases, our procedure is as follows: We show well-posedness of the problem and use the theory of $C_0$-semigroups and an analysis of the spectrum of the respective generator to determine the asymptotic behaviour of the solutions. We obtain convergence to a steady-state solution in all three cases. In two of these cases, the semigroup is irreducible and the convergence is in norm. In the thrid case, the situation is less favourable and we only obtain convergence in the almost-weak sense.