What is the main difference between Markov renewal process and Semi-Markov Process? 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?
 A: 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.
A: Markov-Renewal processes (MRP) encompass Semi-Markov Processes (SMP) as a special case. While SMP requires the memory of the process to be renewed each time it reaches a state, MRP relaxes this assumption. I.e., there can be states in an MRP where the Markovian assumption need not hold. Such states are classified as non-regenerative and their counterparts as regenerative. Consequently, a Markov Regenerative Process (MRRP), which is a part of MRP, is a more generalized version.
What are the applications of MRRP? Imagine a system performing a certain task at a particular rate. Two regenerative states are zero- and full-completion. But what if the system fails while attempting to complete? Well, the system's ability (to complete after it recovers from failure) is likely going to be impacted. So, the failure of the system is a non-regenerative state as the Markov assumption need not hold true.
Hope this helps!
