While mobile edge computing (MEC) alleviates the computation and power limitations of mobile devices, additional latency is incurred when offloading tasks to remote MEC servers. In this work, the power-delay tradeoff in the context of task offloading is studied in a multi-user MEC scenario. In contrast with current system designs relying on average metrics (e.g., the average queue length and average latency), a novel network design is proposed in which latency and reliability constraints are taken into account. This is done by imposing a probabilistic constraint on users' task queue lengths and invoking results from extreme value theory to characterize the occurrence of low- probability events in terms of queue length (or queuing delay) violation. The problem is formulated as a computation and transmit power minimization subject to latency and reliability constraints, and solved using tools from Lyapunov stochastic optimization. Simulation results demonstrate the effectiveness of the proposed approach, while examining the power-delay tradeoff and required computational resources for various computation intensities.