Expected ranked position of a set of normal-distributed random variables

Given several normal-distributed random variables X1 .. Xn with means m1,m2,m3 .. mn and variance s1,s2,s3 .. sn respectively, how can the ranking (when sorted from highest to lowest) of a particular sample can be predicted in general?

For example, if X1 ~ N(1,1) , X2 ~ N(2,1) and X3 ~ N(1,2), what will be the average "ranked position" (after sorting) of the X1 sample, the X2 sample and the X3 sample?

I tried to research the problem, but all I came up was a solution for the case with two variables.

Thanks in advance for help and hints!