Motivated by the proliferation of real-time applications in multimedia communication systems, tactile Internet, networked controlled systems, and cyber-physical systems, supporting delay-constrained traffic become critical for the communication system. In delay-constrained traffic, each packet has a hard deadline and if it cannot be delivered before its deadline, it becomes useless and will be removed from the system. In this work, we consider a slotted ALOHA system where multiple stations need to deliver delay-constrained traffic to a common receiver by accessing a shared channel. We prove that, under the frame-synchronized traffic pattern, the maximum system timely throughput converges to 1/e = 36.8% as the number of stations goes to infinity, which is the same as the asymptotic maximum system throughput for delay-unconstrained slotted ALOHA system with saturate traffic. While this is not completely surprising, we further investigate the speed of such a maximum system throughput approaching 1/e under borderline traffic.