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## Discrete-time Markov chain

A discrete-time Markov chain system is performed in multiple time-steps. A system change can only happen at a single discrete-time value. Let take an example of a board game such as ladders and chutes. In this game, the pieces move around the board when the dice are rolled. If you look at the board when the players are already halfway, you cannot tell how the players arrived at their current position, because the history of the game doesn't matter (previous history of the system).

## Continuous-time Markov chain

A continuous-time Markov chain system can experience changes at any period within a continuous interval. A good example would be the number of cars visiting a car wash on a given day. Arrivals are technically independent and cars can arrive at any time. If you know how many cars visited the car wash at say, 1100 hours, what happened before that won't give you any useful information on estimating how many cars will have driven in by, say, 1300 hours. This is under the assumption that the arrival of the cars follows a continuous-time Markov chain.

## The Markov decision process

The Markov decision process includes an agent that aids in making decisions. These decisions affect the evolvement of a system’s state over time. The agent selects actions and the system responds to these actions by presenting new situations to the agent. For example, in a supermarket’s inventory, the Markov decision process aspect comes to play if the manager decides to order more bread so that it arrives at certain times. Hence, the inventory level at 1100 hours will depend not only on customers arriving randomly and picking bread but also on the decision of the manager.