The Abelian sandpile model was first introduced by Bak, Tang and Wiesenfeld in 1987. Since then, a lot of researchers have studied this model and similar models, all related by the concept of self-organized criticality. In this thesis, we study a variant on the classical model where dissipative and anti-dissipative vertices are incorporated in the model. These have an influence on the critical behaviour of the model. We first introduce a definition of criticality in this model and investigate which levels of dissipation are required to guarantee non-critical behaviour. In studying this variant, we encounter random walks and Green's functions.