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Multi-access Edge Computing (MEC) facilitates the deployment of critical applications with stringent QoS requirements, latency in particular. This paper considers the problem of jointly planning the availability of computational resources at the edge, the slicing of mobile network and edge computation resources, and the routing of heterogeneous traffic types to the various slices. These aspects are intertwined and must be addressed together to provide the desired QoS to all mobile users and traffic types still keeping costs under control. We formulate our problem as a mixed-integer nonlinear program (MINLP) and we define a heuristic, named Neighbor Exploration and Sequential Fixing (NESF), to facilitate the solution of the problem. The approach allows network operators to fine tune the network operation cost and the total latency experienced by users. We evaluate the performance of the proposed model and heuristic against two natural greedy approaches. We show the impact of the variation of all the considered parameters (viz., different types of traffic, tolerable latency, network topology and bandwidth, computation and link capacity) on the defined model. Numerical results demonstrate that NESF is very effective, achieving near-optimal planning and resource allocation solutions in a very short computing time even for large-scale network scenarios.