If you recorded when someone enters the que, then when they exit. In the station for example. You would get an exact time each person has spent in the que, then calculate an average que time from that.
If you walked into a full que and was recorded by a system, maybe even something as simple as a pressure pad, the software would already know exactly how many people are in the que as it would have recorded them entering, when the 100th person to enter the que gets to the end, the system would know it is that exact person as 99 other people would have been recorded exiting the que. Thus giving a pretty exact time that specific person has been in the que for. You could then work out the average time spent from all the people in the que to give an approximate que time. T
The beauty and elegance of an algorithm like this is that along as you are just trying to give a que time for the main que. The merge point does not need to be monitored at all. As a very acurate que time can be given without monitoring it. You just need to monitor people entering the merge que.
Something like this would probably never be implemented but it is a good thought experiment, especially when you are a software engineer.