We demonstrate the ink-jet printed fabrication technique for TiO2-based memristor, followed by detailed analysis of electrical characteristics and development of a new model that considers observed phenomena of quantized conductance steps. The existence of pinched hysteretic current-voltage characteristics is evidence of memristive behavior, provided by the reversible atomic rearrangement taking place in the functional layer. For the first time, performed electrical measurement on the micrometer thickness devices based on TiO2 active layer has captured the plateaux steps of the conductance at integer multiples of elementary quantum conductance. This behavior is consistent with the assumption that transport from electrode to electrode emerges through confined paths of conductive filaments with radius in the nanometer size range. Moreover, we introduce a novel model, based on the diffusion equation for ballistic transport in memristive devices, which considers the conductance plateaux steps.