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In many applications, we observe a system transitioning through various possible states and the probability of observing the system in any given state is conditioned by states occupied at previous times. A key example is a queue, or waiting line, where jobs arrive for service and the state of the system is the number of jobs in the system. This number is random and, at any point in time, depends on arrivals to the system and service to jobs in the system prior to that point. Since arrivals and service times are random, so is the state of the system. In the next three sections, we study systems for which conditional state probabilities depend only on the most recent conditioning event. The first two sections treat mainly discrete-time processes; the third treats continuous-time processes.
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