There are a couple of problems with the above analysis. First, the calculations involving random event probabilities are wrong. For example, the probability of getting heads exactly once when you flip a coin twice is 50%, not 100%. Second, lists of possible wait times are averaged without weighting them by their probabilities. For example, the average of 10 minutes 90% of the time, and 4 minutes 10% of the time is 9.4 minutes, not 7 minutes.
A light time of say 1 sale / minute, then your time is 1 minute since you can see what cashier is open.
If 3 customers arrive over the course of a couple of minutes, there is a 25% chance of an issue with one of them, an 8% chance of an issue with 2 of them, and an issue with all 3 is extremely rare. In any of those cases, the smoothly flowing lines will start to back up until the issues are resolved.
A medium time of when you have say 4 ppl in a queue, which is 12 sales/minute. That means that there is 50% chance of hitting a line that is going to have an issue. The reason is that the queue is NOT dependent on which line you choose by the DEPTH of the queue. You have limited capabilities to decide just by looking at others if they will have issues. In addition, the time will take between 4-20 minutes to get to the cashier, with an average of over 10.
No, if issues are randomly distributed, you have a 59% chance of no problems in your line (0.875^4), a 28% chance of 1 problem (0.41 * 0.875^3), a 10% chance of 2 problems, a 3% chance of 3 problems, and a 0.4% chance of 4 problems. Since these cases each take 4, 8, 12, 16, and 20 minutes, respectively, the average wait is 6.3 minutes. If you simplify and say that on average there are 0.5 problems, you still get 4 minutes plus 50% of a 4 minute delay, which is 6 minutes.
Finally, when the queue hits 8, then it is 100% certain that you will have a slow down of some type. In addition, the time will take between
13 to 40 minutes to get to the cashier with an average of close to 20.
At length 8, you have a 34% chance of no problems, and a 66% chance of at least one problem. It will take 8-40 minutes, and the average is 13.8 minutes, not 20 minutes. In the simple case, you have on average one 4 minute issue, plus 8 minutes of normal wait, for a 12 minute typical wait.
Assume that it is the medium load, which is 12 sales. There will be 1.5 issues during that time, but at least 1 cashier will run full out.
As such, the time will be between 5-10 minutes, with an average of about 6. In addition, you will be moving through the line QUICKLY.
Yes, on average 1.5 issues divided across 3 cashiers is 0.5 delays per cashier, which gives the same 6 minutes as the 3 separate lines.
With the heavy load, that is a total of 24 sales. That means that there will be 3 issues.
That means that you have a time of between 8-12 minutes, with an average of 10 minutes.
No, your average is around 12 minutes just like the separate lines, but you are more likely to have a wait closer to 12 minutes, and less likely to have an 8 minute wait, or a 40 minute wait. Even though there are on average 3 issues, sometimes there are more, and sometimes there are less.
So yes, a single line makes you have wait time closer to the average more often, and reduces the likelihood of a very long or very short wait. But it does not reduce the average, nor change the best or worst case. You can't magically make the cashiers process more purchases per minute with a different line ordering.
But the point is, when one line is moving faster, more new customers get in that line. And therefore, you, as an individual customer, are more likely to have a shorter wait. Yet on average, you have an average wait.
Take the 8-person lines. About 34% of the time, one of them processes 8 people in 8 minutes (and 8 new people get in that line). 26% of the time, there is 1 issue and it only processes 4 people (so 4 new people get in that line). And 40% of the time, there are at least 2 issues, and only 2 people get through in 8 minutes. So in 8 minutes, 4.5 new people per cashier got in line, about 60% of them got in the fast line, 22% in the medium one, and 18% in the slow line. This is correct because we are talking about the probability of the last person in line having a certain experience throughout their wait, and combining that with how many people get in that line in each case.
Therefore in this example, you have a 60% chance of being in the fast line, and only an 18% chance of being in the slow line, just by getting in the line that just finished processing a customer.