On a weak convergence of distribution of one-dimensional piecewise linear Markov process to the stationary one

The piecewise linear Markov processes (PLMPs) were invented by B.V.Gnedenko and I.N.Kovalenko in 1966 as one of the convenient devices for research of behavior of queueing systems. In queueing theory the research of convergence and limit distributions of PLMPs describing queueing systems has an important role; many researches are devoted to this. However the exact algorithm of calculation of constants in convergence rate estimation isn't always known (for various type of rate estimation and in various metrics). Usually, for the many queueing systems the convergence rate depends on stationary distribution. Possibly, for the first time stationary distribution of PLMP for queueing system with nonexponentially distributed service was described by B.A.Sevastjanov (1957), but still stationary distribution of PLMPs wasn't described in a general view. We’ll tell about necessary and sufficient conditions of week convergence of one-dimensional PLMPs to the stationary distribution, we’ll describe this stationary distribution, also we’ll give an example of use of the stationary distribution of PLMP for an estimate of convergence rate of distribution of some queueing system to the stationary one in a total variation metrics.