  The paper is devoted to the estimation of the rate of of exponential
convergence of nonhomogeneous queues exhibiting different types of ergodicity.
  The main tool of our study is the method, which was proposed by the second
author in the late 1980-s and was subsequently extended and developed in
different directions in a series of joint papers by the authors of the present
paper. The method originated from the idea of Gnedenko and Makarov to employ
the logarithmic norm of a matrix to the study of the problem of stability of
nonhomogeneous Markov chains.
  In the present paper, we apply the method to a class of Markov queues with a
special form of nonhomogenuity that is common in applications.
