Bounds and Error bounds for unsolvable CT Markov chains

An approach is presented to compare two Markov Chains, particularly Continuous-Time Markov Chains (CTMC) such as to model Queueing Networks (QN). Here one may typically think of one CTMC or QN to be a solvable modification (e.g. a product form QN) of the other one, say the original, which is of practical interest but unsolvable. The approach is essentially based upon evaluating performance measures by cumulative reward structures and analytically bounding so-called bias-terms, also known as relative gains or fundamental matrix elements. A general comparison and error bound result will be provided. The approach, referred to as Markov Reward approach, is related to Stochastic Dynamic programming and
• may lead to analytic error bounds for the discrepancy, and
• may still apply while stochastic comparison fails
To motivate and illustrate the approach, the presentation will contain an instructive finite tandem queue example and a practical result for a real-life application of an Operation Theater-Intensive care unit system. Some remaining questions for research will be addressed briefly.