We study the problem of modeling temporal networks constrained by the size of a concurrent set, a characteristic of temporal networks shown to be important in many application areas, e.g., in transportation, social, process, and other networks. We propose a competition-driven model for the generation of such constrained networks. Our method carries out turns of competitions along the timeline where each node in a network is labeled with a probability to gain outgoing edges in competitions. We present a thorough theoretical analysis to investigate the cardinality and degree distributions of the generated networks. Our experimental results demonstrate that our model simulates real-world networks well and generates networks efficiently and at scale.