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The number of vehicles that a road network can carry is limited. When the limit is exceeded, the network system is not able to function effectively. In this paper, an updated cellular automaton model for grid networks with all-way stop-controlled intersections is proposed to simulate the network level critical density and carrying capacity under different conditions, which essentially indicates the limit number of vehicles that the network can handle before going into gridlock. In the proposed model, two update rules, including lane-changing and the longitudinal location update, are adopted to describe the vehicle's movements on road segments according to the driving condition on the road and the vehicle's direction in the downstream intersection. The vehicle's movements in intersection areas are prioritized based on vehicle's position, so as to prevent collisions within the intersection area. The simulation results show that an increase in network size is able to expand the carrying capacity of a road network, whereas the expansion rate is lower than the change rate in the network size. The carrying capacity is also associated with the structure of road network. The carrying capacity is inversely proportional to the number of intersections, and proportional to the length of the road in the network. Also, optimizing the origin-destination (O-D) distribution can increase the carrying capacity of an urban road network.