Birth and Death processes are a special type of Markov processes in which there are only two state transitions. One of them increases the state variable by one and is called birth and the second one decreases the state variable by one and is called death. In this paper we are expressing some birth and death processes as stationary Markov Chain and they are reconstructed as a probabilistic metric space. In this case a distance distribution function can be assigned to any pair of states of the processes. More properties of the processes can be explored using the theoretical background of probabilistic metric spaces.