In this work, we propose a cross-layer framework for optimizing user association, packet offloading rates, and bandwidth allocation for Mission-Critical Internet-of-Things (MC-IoT) services with short packets in Mobile Edge Computing (MEC) systems, where enhanced Mobile BroadBand (eMBB) services with long packets are considered as background services. To reduce communication delay, the 5th generation new radio is adopted in radio access networks. To avoid long queueing delay for short packets from MC-IoT, Processor-Sharing (PS) servers are deployed at MEC systems, where the service rate of the server is equally allocated to all the packets in the buffer. We derive the distribution of latency experienced by short packets in closed-form, and minimize the overall packet loss probability subject to the end-to-end delay requirement. To solve the non-convex optimization problem, we propose an algorithm that converges to a near optimal solution when the throughput of eMBB services is much higher than MC-IoT services, and extend it into more general scenarios. Furthermore, we derive the optimal solutions in two asymptotic cases: communication or computing is the bottleneck of reliability. Simulation and numerical results validate our analysis and show that the PS server outperforms first-come-first-serve servers.