In The literature, it was said that Semi Markov processes are a continuous-time extension of Markov Renewal Process.
We know that a sequence of bi-variate random variables {(Yn, Tn)} is called a Markov Renewal Process if {Tn} is a sequence of non-negative iid random variables and Markov Property must to be met. In Semi Markov Process, same thing was happened. What is the difference between SMP and RMP? What make an SMP an SMP?
After a lot searching, finally I found this link helpful.
the main difference between an MRP and a semi-Markov process is that the former is defined as a two-tuple of states and times, whereas the latter is the actual random process that evolves over time and any realisation of the process has a defined state for any given time. The entire process is not Markovian, i.e., memoryless, as happens in a CTMC. Instead the process is Markovian only at the specified jump instants. This is the rationale behind the name, Semi-Markov.