<p>Mobile edge computing (MEC) has emerged as a promising technology to provide computational resources and data storage at the edge of networks, which is essential for the Internet of Things (IoT). However, the burst of tasks, the energy limitation of terminal devices, and the external disturbances can affect the time constraints of tasks and the reliability of systems, posing significant challenges for MEC. In this paper, we propose a primary backup redundancy scheme in which a task access threshold is set for the cache of the primary server and a retry mechanism is implemented for the backup server to offload tasks effectively in MEC-enabled wireless systems. To evaluate the performance of our proposed scheme, we build a five-dimensional continuous-time Markov chain (CTMC) and use the matrix-geometric method to derive key performance indexes consisting of blocking rate, task response latency, and mean time between failures (MTBF). Experimental results exhibit the change trends of the system performance and demonstrate the superiority of our proposed scheme over various baseline schemes. In addition, we consider an application scenario without centralized control over the network connectivity and introduce a set of variables referring to the probabilities of tasks being offloaded to the edge layer for terminal devices. Under the integration architecture of the terminal layer and edge layer, we formulate a multi-device burst task offloading game problem and prove the existence of a Nash equilibrium. We develop a distributed algorithm based on the alternating direction method of multipliers (ADMM) and the sine and cosine algorithm for computing equilibria.</p>