This paper proposes a closed-form eye-diagram model for on-chip distortionless transmission lines with intentionally inserted shunt conductance. We derive expressions of eye-opening both in voltage and time, by assuming a piece-wise linear waveform model. The model is experimentally verified with various length, shunt conductance and resistive termination. We also apply the proposed model to design space exploration, and demonstrate that the proposed model helps estimate the optimal shunt conductance and resistive termination according to required signaling length and throughput.