Some estimates of the convergence rate in stability theorems

We study the rate of convergence of the stationary distributions of the waiting time for one-channel queueing systems when the distributions of governing sequences converge. Estimates for the one-dimensional distributions are obtained in terms of Levy's and the uniform metrics. If the governing sequences are given on a common probability space, the estimates obtained in metrics $\rho(\xi,\eta)=\inf\{\varepsilon\colon\mathbf P(|\xi-\eta|>\varepsilon)<\varepsilon\}$ are best possible.