In this study, we consider an edge cloud server in which a lightweight server is placed near a user device for the rapid processing and storage of large amounts of data. For the edge cloud server, we propose a latency classification algorithm based on deadlines and urgency levels (i.e., latency-sensitive and latency-tolerant). Furthermore, we design a task offloading algorithm to reduce the execution time of latency-sensitive tasks without violating deadlines. Unlike prior studies on task offloading or scheduling that have applied no deadlines or task-based deadlines, we focus on a comprehensive deadline-aware task scheduling scheme that performs task offloading by considering the real-time properties of latency-sensitive tasks. Specifically, when a task is offloaded to the edge cloud server due to a lack of resources on the user device, services could be provided without delay by offloading latency-tolerant tasks first, which are presumed to perform relatively important functions. When offloading a task, the type of the task, weight of the task, task size, estimated execution time, and offloading time are considered. By distributing and offloading latency-sensitive tasks as much as possible, the performance degradation of the system can be minimized. Based on experimental performance evaluations, we prove that our latency-based task offloading algorithm achieves a significant execution time reduction compared to previous solutions without incurring deadline violations. Unlike existing research, we applied delays with various network types in the MEC (mobile edge computing) environment for verification, and the experimental result was measured not only by the total response time but also by the cause of the task failure rate.