Attainability for Markov and Semi-Markov Chains

Abstract

When studying Markov chain models and semi-Markov chain models, it is useful to know which state vectors n, where each component ni represents the number of entities in the state Si, can be maintained or attained. This question leads to the definitions of maintainability and attainability for (time-homogeneous) Markov chain models. Recently, the definition of maintainability was extended to the concept of state reunion maintainability (SR-maintainability) for semi-Markov chains. Within the framework of semi-Markov chains, the states are subdivided further into seniority-based states. State reunion maintainability assesses the maintainability of the distribution across states. Following this idea, we introduce the concept of state reunion attainability, which encompasses the potential of a system to attain a specific distribution across the states after uniting the seniority-based states into the underlying states. In this paper, we start by extending the concept of attainability for constant-sized Markov chain models to systems that are subject to growth or contraction. Afterwards, we introduce the concepts of attainability and state reunion attainability for semi-Markov chain models, using SR-maintainability as a starting point. The attainable region, as well as the state reunion attainable region, are described as the convex hull of their respective vertices, and properties of these regions are investigated.