  Consideration is given to the three different analytical methods for the
computation of upper bounds for the rate of convergence to the limiting regime
of one specific class of (in)homogeneous continuous-time Markov chains. This
class is particularly suited to describe evolutions of the total number of
customers in (in)homogeneous $M/M/S$ queueing systems with possibly
state-dependent arrival and service intensities, batch arrivals and services.
One of the methods is based on the logarithmic norm of a linear operator
function; the other two rely on Lyapunov functions and differential
inequalities, respectively. Less restrictive conditions (compared to those
known from the literature) under which the methods are applicable, are being
formulated. Two numerical examples are given. It is also shown that for
homogeneous birth-death Markov processes defined on a finite state space with
all transition rates being positive, all methods yield the same sharp upper
bound.
