Suppose I have the random variables $Z_k=X_k/Y_k$ with a PDF $f_{Z_k}(z_k)$ for $k=1,\,2\,\ldots, K$, where $\{X_k, Y_k\}$ are i.i.d. random variables. I can find
$$\text{Pr}\left[\sum_{k=1}^3Z_k\leq \eta\right]$$
as
$$\int_{z_3=0}^{\eta}\int_{z_2=0}^{\eta-z_3}\int_{z_1=0}^{\eta-z_3-z_2}f_{Z_1}(z_1)f_{Z_2}(z_2)f_{Z_3}(z_3)\,dz_1dz_2dz_3$$
Now suppose I arrange the random variables $\{X_k\}_{k=1}^K$ as
$$X_{(1)}\leq X_{(2)}\leq \cdots \leq X_{(K)}$$
and define $G_k=X_{(k)}/Y_{i_k}$, where $i_k$ is the index of the $k^{\text{th}}$ order statistic.
How can I find
$$\text{Pr}\left[\sum_{k=1}^3G_k\leq \eta\right]$$