Asymptotic Behavior of the Time-Dependent Solution of the M^[X]/G/1 Queuing Model with Feedback and Optional Server Vacations Based on a Single Vacation Policy

By using the $C_0$-semigroup theory, we study the asymptotic behavior of the time-dependent solution and the time-dependent indices of the ${\mathrm{M}}^{\left[\mathrm{X}\right]}/\mathrm{G}/1$ queuing model with feedback and optional server vacations based on a single vacation policy. This queuing model is described by infinitely many partial differential equations with integral boundary conditions in an unbounded interval. Under certain conditions, by studying spectrum of the underlying operator of this queuing model on the imaginary axis, we prove that the time-dependent solution of this queuing model strongly converges to its steady-state solution. Next, we prove that the time-dependent queuing length of this queuing system converges to its steady-state queuing length and the time-dependent waiting time of this queuing system converges to its steady-state waiting time as time tends to infinity. Our results extend the steady-state results of this queuing system.