Why do the kolmogorov equations for time inhomogeneous markov chains take into consideration the time spent (waiting time) in a state whereas this is not so in case of time homogeneous markov chains ???
Because, in time homogeneous Markovian, rate is independent of time...' if it depends, it'll be time-inhomogeneous...
Ok ! So we can say that only in time inhomogeneous Markov process the kolmogorov equations take into account the probability of waiting in a state. Well, can u tell me then why is this so ? I mean why is the probability of transition affected by the probability of waiting in a particular state even though we are taking into account the time inhomogenity in its transition rate?
Yes, affect of waiting time to transitions can be present in time-inhomogeneous case only. suppose, a persons in sick for a year and (s)he has transition rate of \(\mu \) to death... but after 2-3 more years of sickness, transition rate to the death would be more... in other words, here transition rate increase(time inhomogenous) with time spent in Sick State
