This paper studies the dynamic application placement for edge computing with a variety of random task arrivals (or task offloading requests). Since the storage capacity of the edge server is inherently limited, for better serving mobile devices with heterogeneous computation demands, the edge server is required to update its application placement, leading to a potential latency-energy paradox (i.e., infrequent updates may result in growth of latency while frequent updates may introduce high energy consumption). To this end, we propose a novel application placement policy, consisting of a response threshold and a waiting duration for installing and uninstalling the application for each type of tasks, respectively. Furthermore, we leverage the vacation queue for analyzing the performance of such system, and formulate a joint optimization problem for deriving the optimal configurations of application placement, along with the computation resource allocations, to minimize the average service latency with a desired energy constraint. A branch-and-bound method integrating an inner convex approximation approach is proposed to solve the problem, and numerical simulations are conducted to evaluate our proposed policy, which demonstrate its superiority over counterparts.