An important class of physical systems is those that are driven and dissipative. One such system is the Olami-Feder-Christensen (OFC) model which was proposed as a model for earthquake faults. Previous investigations have shown that the OFC model can be described by equilibrium methods under certain circumstances. This observation has explained several aspects of the OFC model and earthquake faults. However, these studies were done at a relatively high level of noise and for systems with a high degree of homogeneity. But real earthquake faults have a very low level of noise and a large degree of inhomogeneity. In this work, a careful study of the noise and its relation to the stress transfer range, dissipation parameter, and system size is performed. In addition, a modified form of the OFC model is studied, where we added asperities. We carefully examined the effect of the asperities on various statistical properties of the model.