Continuous-Time Markov Chains

Continuous-Time Markov Chains (CTMCs) are mathematical models used to describe systems that transition randomly between different states over continuous time. In these models, the likelihood of moving from one state to another depends only on the current state, not the past history, making them "memoryless." The system jumps between states at random times, with transition rates determining how often changes occur. CTMCs are widely used in fields like queuing theory, biology, and finance to analyze processes such as customer arrivals, disease progression, or stock price shifts, providing insight into their long-term behavior and dynamics.