The problem I have is, given $n$ independent normal distributions describing the times that $n$ random events occur at, what is the most likely order that they will occur in?
This questions follows on somewhat from this question on calculating the probability of a given ordering of normal random variables, and is related to this question on the probability that one event occurs before all other events. In the case of the 2nd post, the answers all used Monte Carlo simulation to find the answer. Ideally I'd like to find an analytical method of finding the most likely order.
My initial thought was that you could simply order the events by their mean. However, a simple example shows that this is not the case. For 3 events, $A$, $B$, and $C$, with times $N(17.836,2.968^2)$, $N(18.067,4.638^2)$, and $N(18.209,2.982^2)$ respectively, the most likely order of events is $BAC$ with a probability of $19.7\%$. The order $ABC$ only has a probability of $13.9\%$.
Unfortunately, it is looking more likely to me now that an exhaustive search over all possible orders is required. Edit: As joriki has pointed out, branch and bound can be used to reduce the search space in most cases. However, that can still be quite slow (as the method for calculating the probability of an order is slow).