ABSTRACT
This chapter presents the notation and preliminaries of hidden Markov renewal chains. Hidden Markov models are stochastic models that are widely used in numerous applications in reliability and DNA analysis, seismology, speech recognition, and so forth. In the literature of semi-Markov models (SMM), many reliability indicators have been introduced, including availability, maintainability, mean times to failure, hazard rates, and so on. SMMs might be used to describe loading data and conditional failure occurrence rates could be determined to allow the monitoring of mechanical systems. In a continuous-time semi-Markov context, concentrate on the failure occurrence rate (ROCOF) was studied in for finite state space processes and in for general state space processes, respectively. ROCOF was studied for discrete hidden Markov renewal chains in for the case where failures are associated with specific observations rather than hidden states. Counting processes were used to define an empirical estimator whose asymptotic behavior was studied.
