Ion induced beam instability is one critical issue for the electron damping ring of the International Linear Collider (ILC) due to its ultra small emittance of 2 pm. The beam ion instability for the latest DTC lattice has been studied both analytically and numerically. Different from previous studies, multi-gas species and exact beam filling patterns have been modeled simultaneously in this study. This feature makes the studies more realistic and accurate. The numerical code has been benchmarked with SPEAR3 experimental data and there is a good agreement between the simulation and observations. It uses the optics from MAD and can handle arbitrary beam filling pattern and vacuum. Our analyses agree well with expensive simulations.