In the fifth generation (5G) wireless communications, ultra-reliable and low-latency communications (URLLC) service that deliveries short packets will coexist with enhanced mobile broadband (eMBB) service with long packets. In this work, we study how to analyze delay and optimize offloading scheme of short packets in a mobile edge computing (MEC) system with both URLLC and eMBB services. To reduce computing delay for short packets, a processor-sharing (PS) server is deployed in the MEC system, where the total computing ability is equally allocated to all the packets in the server. As such, short packets can bypass long packets in the server. We derive a closed-form expression of the complimentary cumulative distribution function of latency experienced by short packets in the PS server, and minimize the end-to-end (E2E) delay of short packets by optimizing offloading probabilities subject to the reliability requirement. Simulation results validate our analysis and show that the PS server outperforms two kinds of first-come-first-serve servers in the scenarios with both short and long packets.