On the rate of convergence and limiting characteristics for one quasi-birth–death process

A queuing system with one server and different repair and failure options is considered, the number of requirements in which is described by a quasi-birth-death process. To reasonably find the limiting probabilistic characteristics of the system, the rate of convergence to them is studied (that is, the rate at which the initial conditions of the system are “forgotten”). To study the rate of convergence to the limiting regime, a recently developed version of the approach based on the concept of the logarithmic norm of the operator function corresponding to the estimate of the norm of the Cauchy matrix as well as a modernized special transformation of the forward Kolmogorov system was applied. A numerical example is considered for which the estimation of the rate of convergence is shown in detail as well as the construction of some limiting characteristics of the model based on these estimates.