Example 1. Let positive numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy $a_{1}+a_{2}+\cdots+a_{n}=1$. Find
$$\begin{array}{l}
\frac{a_{1}}{1+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}} \\
+\cdots+\frac{a_{n}}{1+a_{1}+\cdots+a_{n-1}}
\end{array}$$

the minimum value. (82 West Germany Competition Problem)