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In this paper, we study the category $\mathbb{C}(\text{Rep}(\mathcal{Q}, \mathcal{G}))$ of complexes of representations of quiver $\mathcal{Q}$ with values in a Grothendieck category $\mathcal{G}$. We develop a method for constructing some model structures on $\mathbb{C}(\text{Rep}(\mathcal{Q}, \mathcal{G}))$ based on componentwise notion. Moreover we also show that these model structures are monoidal. As an application of these model structure we introduce some descriptions of the derived category of complexes of representations of $\mathcal{Q}$ in $\text{Mod-} R$. In particular we set $\mathcal{Q}=A_2$ and consider the morphism category $\mathbf{H}(R)$ and its two full subcategories, monomorphism category $\mathbf{S}(R)$ and epimorphism category $\mathbf{F}(R)$. We show that the well know equivalence between $\mathbf{S}(R)$ and $\mathbf{F}(R)$ can be extended to an auto-equivalence of derived category of $\mathbf{H}(R)$.

Model Structures on the Category of Complexes of Quiver Representations