On One Approach to Obtaining Estimates of the Rate of Convergence to the Limiting Regime of Markov Chains

Abstract

We revisit the problem of the computation of the limiting characteristics of (in)homogeneous continuous-time Markov chains with the finite state space. In general, it can be performed only numerically. The common rule of thumb is to interrupt calculations after quite some time, hoping that the values at some distant time interval will represent the sought-after solution. Convergence or ergodicity bounds, when available, can be used to answer such questions more accurately; i.e., they can indicate how to choose the position and the length of that distant time interval. The logarithmic norm method is a general technique that may allow one to obtain such bounds. Although it can handle continuous-time Markov chains with both finite and countable state spaces, its downside is the need to guess the proper similarity transformations, which may not exist. In this paper, we introduce a new technique, which broadens the scope of the logarithmic norm method. This is achieved by firstly splitting the generator of a Markov chain and then merging the convergence bounds of each block into a single bound. The proof of concept is illustrated by simple examples of the queueing theory.

Keywords

• inhomogeneous continuous-time Markov chains
• weak ergodicity
• logarithmic norm
• QBD process