This paper deals with the mathematical modeling of two-tier heterogeneous networks (HetNets) through measure theory, and the optimization of the related key performance indicators (KPIs). Thus, any HetNet can be described as α —algebra, on which measures can be defined. As a result, we make a connection between the mathematical and the physical features of the KPIs of HetNets, such as the capacity or the energy efficiency (EE), which are extensive and intensive performance metrics, respectively. Then, it is shown that the extensive property remains in presence of inter-cell interferences. Through examples, we highlight that the optimization of extensive variables is more simple than the optimization of intensive variables, since extensive variables are often concave/convex functions of their parameters. Finally, the mathematical modeling of HetNets is illustrated through applications and examples, such as the maximization of the capacity in a long term evolution (LTE) macrocell/femtocell HetNet, as well as the minimization of the power consumption in a broadcast/unicast hybrid network.

Keywords:
Heterogeneous network, Key performance indicator, Measure, Optimization, LTE, Broadcast, Capacity, Energy efficiency