In a recent paper, Khojasteh emph{et al.} [F. Khojasteh, S. Shukla, S. Radenovi'c, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), 1189-–1194] presented a new class of simulation functions, say $mathcal{Z}$-contractions, with unifying power over known contractive conditions in the literature. Following this line of research, we extend and generalize their results on a $b$-metric context, by giving a new notion of  $b$-simulation function. Then, we prove and discuss some fixed point results in relation with existing ones.