In this paper, we study additive cyclic codes over the direct product of two finite commutative chain rings. Let $ S= \mathbb{F}_{q}[u]/ \langle u^2\rangle$ = $ \mathbb{F}_{q}+u\mathbb{F}_{q}$ and $R=\mathbb{F}_{q}[u]/ \langle u^3\rangle$ = $\mathbb{F}_{q}+u\mathbb{F}_{q} + u^{2}\mathbb{F}_{q}$ are two finite chain rings, where $ u^{2}=0=u^{3} $ and $ q $ is a power of a prime number. We construct a class of $ SR $-additive cyclic codes generated by pairs of polynomials, where $ S $ is a $ R $-algebra and $ SR $-additive cyclic code is a $ R $-submodule of $ S^{\alpha} \times R^{\beta} $ . Based on probabilistic arguments, we study the asymptotic behavior of the rates and relative minimum distances of a certain class of the codes. We show that there exists an asymptotically good infinite sequence of $ SR $-additive cyclic codes with the relative minimum distance of the code is convergent to $ \delta $, and the rat is convergent to $ \frac{2}{q+q^{2}} $ for $ 0 < \delta < \frac{1}{1+q} $, and $ q $ is a power of a prime number.
