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 multitier 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 considering both the computation and communication performance using the queueing theory and stochastic geometry. We then analyze the effects of network parameters and bias factors in MEC server association on the successful edge computing probability. We provide how the optimal bias factors in terms of successful edge computing probability can be changed according to the user type and MEC tier, and how they are different to the conventional ones that did not consider the computing capabilities and task sizes. It is also shown how the optimal bias factors can be changed when minimizing the mean latency instead of successful edge computing probability. This study provides the design insights for the optimal configuration of MEC-enabled HetNets.