  In this paper, we study the category $C(Rep(\mathcal{Q}, \mathcal{A}))$ of
complexes of representations of quiver $\mathcal{Q}$ with values in an abelian
category $\mathcal{A}$. We develop a method for constructing some model
structures on $C(Rep(\mathcal{Q}, \mathcal{A}))$ based on componentwise notion.
Moreover, we also show that these model structures are monoidal. As an
application of these model structures, we introduce some descriptions of the
derived category of complexes of representations of $\mathcal{Q}$ in $\Mod R$.
