Say that I have a set of $k$ random variables $X_{1}, X_{2}, ..., X_{k}$ which are normally distributed with means $\mu_{1}, \mu_{2}, ..., \mu_{k}$ and variances $\sigma_{1}^{2}, \sigma_{2}^{2}, ..., \sigma_{k}^2$. How can I compute the likelihood of a given stochastic ordering such as $X_{r_1} \le X_{r_2} \le ... \le X_{r_k}$?
I can see how I could calculate this directly by integrating $X_{r_1}$ over $[-\infty, \infty]$, $X_{r_2}$ over $[X_{r_1}, \infty]$, and so on, but I'm wondering if there might be a better way. Numerical and approximate methods are fine if they're more efficient than this approach.