Markov Chain

A Markov chain is a mathematical model for a sequence of states with the Markov property: the probability of the next state depends on the current state rather than the full path taken to reach it. Transition probabilities describe how the process moves, and repeated transitions can reveal long-run behavior or the likelihood of future outcomes. Markov chains appear in queueing, language modeling history, simulation, economics, and reinforcement-learning foundations. They are useful when the current state captures all relevant information, but that assumption may be unrealistic. Designers must define states carefully, estimate transitions from appropriate data, and check whether time-varying or hidden influences invalidate the model.