Markov Regenerative Process in Sharpe

Markov regenerative processes (MRGPs) constitute a more general class of stochastic processes than traditional Markov processes. Markovian dependency, the first-order dependency, is the simplest and most important dependency in stochastic processes. Past history of a Markov chain is summarized in the current state and the behavior of the system thereafter only depends on the current state. Sojourn time of a homogeneous continuous time Markov chain (CTMC) is exponentially distributed. However, non-exponentially distributed transitions abound in real life systems. Semi-Markov processes (SMPs) incorporate generally distributed sojourn times, but lack the ability to capture local behaviors during the intervals between successive regenerative points. MRGPs provide a natural generalization of semi-Markov processes with local behavior accounted. This thesis is devoted to studying a class of MRGPs and implementing the numerical MRGP solver integrated as a cornerstone in the versatile reliability, availability and performance modeling software package, SHARPE. In the class of MRGPs we study, at most one generally distributed transition is allowed in any state. This restriction, however, assures the subordinated stochastic process is a CTMC that is amenable for automated numerical analysis. We set forth by providing theoretical background of MRGPs and attempting to clarify the evolving threads from conventional stochastic processes to MRGPs. We then present important theorems fundamental to a steady-state solver. We turn to the issues in implementing the solver following the MRGP mathematical overview. After outlining the important common data structures in SHARPE, we describe key modules of the solver. MRGP syntax in SHARPE language is addressed and its usage is illustrated by examples. MRGPs are considered as the underlying stochastic processes of a class of stochas-