In this work we report on the role of ion transport for the dynamic behavior of a double barrier quantum mechanical Al/Al$_2$O$_3$/Nb$_{text{x}}$O$_{text{y}}$/Au memristive device based on numerical simulations in conjunction with experimental measurements. The device consists of an ultra-thin Nb$_{text{x}}$O$_{text{y}}$ solid state electrolyte between an Al$_2$O$_3$ tunnel barrier and a semiconductor metal interface at an Au electrode. It is shown that the device provides a number of interesting features for potential applications such as an intrinsic current compliance, a relatively long retention time, and no need for an initialization step. Therefore, it is particularly attractive for applications in highly dense random access memories or neuromorphic mixed signal circuits. However, the underlying physical mechanisms of the resistive switching are still not completely understood yet. To investigate the interplay between the current transport mechanisms and the inner atomistic device structure a lumped element circuit model is consistently coupled with 3D kinetic Monte Carlo model for the ion transport. The simulation results indicate that the drift of charged point defects within the Nb$_{text{x}}$O$_{text{y}}$ is the key factor for the resistive switching behavior. It is shown in detail that the diffusion of oxygen modifies the local electronic interface states resulting in a change of the interface properties of the double barrier device.