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A control algorithm and a probability-theoretical traffic analysis of a grade-crossed intersection of the computer-controlled vehicle system, a computer-controlled rapid transit system developed in Japan, are presented. The control algorithm is simple but realistic and optimal in the sense that delay is minimized under the following constraints: the first-come first-served rule, the maneuver length limitation, and the collision-freeness condition. The analysis is completely based on this detailed control algorithm; random arrival of vehicles is assumed, and the behaviour of the algorithm is formulated as a simple Markov chain known as a one-dimensional random walk. Exact and explicit expressions for some performance measures such as delay, abort rate, and throughput are obtained.