On the Derivative Counting Processes of First- and Second-order Aggregated Semi-Markov Systems

In this paper, first- and second-order discrete-time semi-Markov systems are considered with their finite state space divided into three subsets as perfect functioning states, imperfect functioning states and failure states, respectively. The counting processes for one-step increasing transitions, one-step equivalent transitions and one-step decreasing transitions in working/failure periods are defined and investigated in detail. Formulas for related distributions, joint distributions, expectations, generating functions and joint generation functions are derived in their Z-transforms. Numerical examples are presented to illustrate the results established. Extended discussions on related reliability measures are also considered. Finally, some concluding remarks and discussions are presented. Applications of the results presented here can be found in different fields such as seismology, reliability, biology and finance.