Computational offloading is a promising approach for overcoming resource constraints on client devices by moving some or all of an application's computations to remote servers. With the advent of specialized hardware accelerators, client devices can now perform fast local processing of specific tasks, such as machine learning inference, reducing the need for offloading computations. However, edge servers with accelerators also offer faster processing for offloaded tasks than was previously possible. In this paper, we present an analytic and experimental comparison of on-device processing and edge offloading for a range of accelerator, network, multi-tenant, and application workload scenarios, with the goal of understanding when to use local on-device processing and when to offload computations. We present models that leverage analytical queuing results to derive explainable closed-form equations for the expected end-to-end latencies of both strategies, which yield precise, quantitative performance crossover predictions that guide adaptive offloading. We experimentally validate our models across a range of scenarios and show that they achieve a mean absolute percentage error of 2.2% compared to observed latencies. We further use our models to develop a resource manager for adaptive offloading and show its effectiveness under variable network conditions and dynamic multi-tenant edge settings.