The elevator system of this present invention for multistory buildings includes at least one elevator shaft, each of which having a plurality of elevator units and at least one interlocking means, and an elevator control system. The elevator unit includes an elevator car and its guide means, a counterweight and its guide means, a drive means, and an elevator control system. The interlocking means includes a coupling mechanism and a bi-directional one-way clutch mechanism, and connecting means such as gears. The elevator system is operated by a plurality of computers including the schedule computer, shaft computers, and car computers. The acceleration (and deceleration) rate of the following elevator car is determined from the distance between the car and its leading car, the speeds of the two cars, and the acceleration rate of the leading car.