Multi-Access Edge Computing (MEC) paradigm has been widely studied as a potential solution to cope with the challenges emerging from new generations of mobile networks. By processing applications' data closer to the users, service providers are able to offload origin servers and their underlying network infrastructure, which consequently reduces users' experienced latency. In this paper, we consider internet-based applications with strict latency tolerance which are primarily enabled by the MEC architecture. Moreover, nodes at the edge may host application-related tasks as well as assist in their provision. We formalize the Task Distribution Problem (TDP), where the objective is to maximize the overall Quality of Service (QoS) while ensuring that tasks' latency requirements are satisfied. The TD problem is modeled as an Integer Programming problem, taking into account three components: (i) tasks' priority assignment, (ii) placement and (iii) routing through the MEC network. We propose to approach the TDP through an approximation scheme, called Shortest-Flow Approximation (SFA), which considers predetermined network flows selected according to a Betweennessbased criterion. In our experiments, we first extract useful insights on the TDP that help characterize its solution. Then, we evaluate SFA's performance empirically across different experimental settings, showing numerical results confirming its proximity to the optimal solution.