What is continuous time Markov chain? Explain why the birth-and-death process is a special type of continuous time Markov chain.

A Continuous Time Markov Chain (CTMC) is a type of stochastic process that undergoes transitions from state to state in a continuous time domain. The defining characteristic of a CTMC is the Markov property, which states that the probability of transitioning to any particular state depends solely on the current state and time spent in the current state, and not on the sequence of states that preceded it.

The birth-and-death process is a special type of CTMC. In a birth-and-death process, the state changes are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state by one. The system can only transition between adjacent states, and the rates of births and deaths may depend only on the current state.

The reason why the birth-and-death process is a special type of CTMC is because of its specific transition rules. While a general CTMC can transition from one state to any other state, a birth-and-death process can only transition to adjacent states. This restriction to adjacent states is what makes the birth-and-death process a special case of CTMC.