This paper deals with a merging section in a personal rapid transit system, and a new control strategy that allows a waiting time limit is proposed as a method to reduce the waiting queue. Vehicles arrive at a merging section from main and sub lines. Since vehicles on the main line have priority, a queue is formed on the sub line in order to avoid the collision of vehicles. Vehicles on the main line need to stop when the waiting time of the last vehicle on the sub line queue exceeds the limit. The relation between the waiting time limit and the behavior of traffic on the main and sub lines (characterized by the mean queue length and queuing delay) is derived explicitly by using a Markov chain model. Moreover, the control strategy for the queuing delay on each line is developed on the basis of the obtained relation. We compare the results obtained by using the present strategy with those obtained by using another strategy. The comparison shows the validity of the present strategy in cases where the probability of vehicle arrival on the sub line is low.