We propose two new simple lepton flavor models in the framework of the $S_4$ flavor symmetry. The neutrino mass matrices, which are given by two complex parameters, lead to the inverted mass hierarchy. The charged lepton mass matrix has the 1-2 lepton flavor mixing, which gives the non-vanishing reactor angle $theta_{13}$. These models predict the Dirac phase and the Majorana phases, which are testable in the future experiments. The predicted magnitudes of the effective neutrino mass for the neutrino-less double beta decay are in the regions as $32~text{meV}lesssim |m_{ee}|lesssim 49~text{meV}$ and $34~text{meV}lesssim |m_{ee}|lesssim 59~text{meV}$, respectively. These values are close to the expected reaches of the coming experiments. The total sum of the neutrino masses are predicted in both models as $0.0952~text{eV}lesssim sum m_ilesssim 0.101~text{eV}$ and $0.150~text{eV}lesssim sum m_ilesssim 0.160~text{eV}$, respectively.
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