Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method

The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose the position and the length of the &ldquo
distant time interval&rdquo
(in the periodic case) on which the solution has to be computed. They can also be helpful whenever the state space truncation is required. In this paper one such analytic method&mdash
the logarithmic norm method&mdash
is being reviewed. Its applicability is shown within the queueing theory context with three examples: the classical time-varying M/M/2 queue
the time-varying single-server Markovian system with bulk arrivals, queue skipping policy and catastrophes
and the time-varying Markovian bulk-arrival and bulk-service system with state-dependent control. In each case it is shown whether and how the bounds on the rate of convergence can be obtained. Numerical examples are provided.