Well-posedness and asymptotic behavior of the time-dependent solution of an M/G/1 queueing model

Abstract

By using the Hille–Yosida theorem, Phillips theorem and Fattorini theorem we prove that the M/G/1 queueing model with vacations and multiple phases of operation, which is described by infinitely many partial differential equations with integral boundary conditions, has a unique positive time-dependent solution that satisfies the probability condition. Next, by studying the spectrum of the operator, which corresponds to the model, on the imaginary axis we prove that the time-dependent solution of the model strongly converges to its steady-state solution.