Periodic Markovian replacement chains

We consider a periodic absorbing Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The resulting process is a Markov replacement chain and conditions are provided on the initial distribution and the probabilities of absorption for this to be periodic. It is shown among others that control on the entire chain is exercised only through the first cyclic subclass of the state space, both for stationary and time-dependent versions of the chain. The periodicity results revealing a real-time property of the physical system, are applied to topics and processes of related interest such as the fundamental matrix, recurrent processes and distributions of phase-type and population processes with Markovian replacements.Compensation Control of Markov chains Quasi-period chain Phase-type recurrent process Population replacement chains