This paper addresses the computational offloading of Deep Neural Networks (DNNs) to nearby devices with similar processing capabilities, to avoid the larger communication delays incurred for cloud offloading. We present a preemption aware scheduling approach for priority and deadline constrained task offloading in homogeneous edge networks. Our scheduling approach consists of two distinct scheduling algorithms, designed to accommodate the differing requirements of high and low priority tasks. To satisfy a task's deadline, our scheduling approach considers the availability of both communication and computational resources in the network when making placements in both the current time-slot and future time-slots. The scheduler implements a deadline-aware preemption mechanism to guarantee resource access to high priority tasks. When low-priority tasks are selected for preemption, the scheduler will attempt to reallocate them if possible before their deadline. We implement this scheduling approach into a task offloading system which we evaluate empirically in the real-world on a network of edge devices composed of four Raspberry Pi 2 Model B's. We evaluate this system under against a version without a task preemption mechanism as well as workstealing approaches to compare the impact on high priority task completion and the ability to complete overall frames. These solutions are evaluated under a workload of 1296 frames. Our findings show that our scheduling approach allows for 99\% of high-priority tasks to complete while also providing a 3 - 8\% increase in the number of frames fully classified end-to-end over both workstealing approaches and systems without a preemption mechanism.