An Approach to Obtain Upper Ergodicity Bounds for Some QBDs with Countable State Space

Usually, when the computation of limiting distributions of (in)homogeneous (in)finite continuous-time Markov chains (CTMC) has to be performed numerically, the algorithm has to be told when to stop the computation. Such an instruction can be constructed based on available ergodicity bounds. One of the analytical methods to obtain ergodicity bounds for CTMCs is the logarithmic norm method. It can be applied to any CTMC; however, since the method requires a guessing step (search for proper Lyapunov functions), which may not be successful, the obtained bounds are not always meaningful. Moreover, the guessing step in the method cannot be eliminated or automated and has to be performed in each new use-case, i.e., for each new structure of the infinitesimal matrix. However, the simplicity of the method makes attempts to expand its scope tempting. In this paper, such an attempt is made. We present a new technique that allows one to apply, in one unified way, the logarithmic norm method to QBDs with countable state spaces. The technique involves the preprocessing of the infinitesimal matrix of the QBD, finding bounding for its blocks, and then merging them into the single explicit upper bound. The applicability of the technique is demonstrated through a series of examples.