Dynamic Analysis of the M/G/1 Queueing Model with Single Working Vacation

Abstract

By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with single working vocation has a unique nonnegative time-dependent solution. When the service completion rate are constant, by studying spectral properties of the operator corresponding to the model we study asymptotic behavior of its time-dependent solution. Fist of all, through studying the resolvent set of the adjoint operator of the operator we obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator. Next, we prove that zero is eigenvalue of the underlying operator and its adjoint operator. Therefore, by combining these results we deduce that the time-dependent solution of the model strongly converges to its steady-state solution.