Compliance mismatch is one of the reasons of the coronary artery bypass graft (CABG) failure. The purpose of this study is to investigate the effect of compliance mismatch on the End to Side bypass graft. In order to model non Newtonian behavior of the blood flow, the Carreau–Yasuda model was employed and the graft and artery wall was assumed to be isotropic and modeled as a linearly elastic. In this study also the effects of blood rheology and wall distensibility on the wall shear stress distribution and velocity profile were investigated. The results of the simulation show that the maximum deformation occurs in the critical position of graft-artery junction and compliance mismatch cause smaller wall deformation in comparison to the cases in which the materials of the graft and artery are the same which leads to a higher intramural shear stress in graft-artery junction. The anastomotic wall deforms in a way that always tends to separate the graft and artery. Wall shear stress distribution on the bed centerline and the toe of the bypass graft indicates that the differences between the homologous and non-homologous material case are visible only when the internal pressure is lower than the external one. In the distal location of the artery after the toe of the anastomotic, the values of wall shear stress in the homologous material case are lower than the non-homologous material one.