We consider two problems that arise in designing two-level star networks taking into account service quality considerations. Given a set of nodes with pairwise traffic demand and a central hub, we select $p$ hubs and connect them to the central hub with direct links and then we connect each nonhub node to a hub. This results in a star/star network. In the first problem, called the Star $p$-hub Center Problem, we would like to minimize the length of the longest path in the resulting network. In the second problem, Star $p$-hub Median Problem with Bounded Path Lengths, the aim is to minimize the total routing cost subject to upper bound constraints on the path lengths. We propose formulations for these problems and report the outcomes of a computational study where we compare the performances of our formulations.