The mobile edge computing (MEC) has been introduced for providing computing capabilities at the edge of networks to improve the latency performance of wireless networks. In this paper, we provide the novel framework for MEC-enabled heterogeneous networks (HetNets), composed of the multi-tier networks with access points (APs) (i.e., MEC servers), which have different transmission power and different computing capabilities. In this framework, we also consider multiple-type mobile users with different sizes of computation tasks, and they offload the tasks to a MEC server, and receive the computation resulting data from the server. We derive the successful edge computing probability (SECP), defined as the probability that a user offloads and finishes its computation task at the MEC server within the target latency. We provide a closed-form expression of the approximated SECP for general case, and closed-form expressions of the exact SECP for special cases. This paper then provides the design insights for the optimal configuration of MEC-enabled HetNets by analyzing the effects of network parameters and bias factors, used in MEC server association, on the SECP. Specifically, it shows how the optimal bias factors in terms of SECP can be changed according to the numbers of user types and tiers of MEC servers, and how they are different to the conventional ones that did not consider the computing capabilities and task sizes.