We are starting different places. The original post was if you show up to a line at 2:00 in the afternoon, you are explaining what happens when people show up at opening and they all rush to a line. This isn't giving you the wait time for everyone at any given time, it's the wait time for you if you show up and there are 1k in line. Maybe the original post should have stated that assumption that there were 1k already in line when you show up, but that was where he was going. I'll say again, it's a very simplistic model of what LL could be doing to wait times. Obviously there are a ton of variables that could go into this, but it gives an idea to the original question. The math works if you use the assumptions. Just like your math works if you assume as 17 people leave one a train, 17 more show up in that exact time frame.
Yes, indeed.
I'm saying it's unusual to start with 1K in line.
So, let's start the day with zero in line for a 1K-ride and 1K-guests show up over the course of that first hour. Theoretically, no wait.
So, what does LL do to that? Let's take a simple number, 20% of capacity for a 1K-ride.
So, let's run through the first hour... 1K guests show up for standby. But now, there are 200 guests that get to cut them in line. After one hour, the 200 LLs get through, but only 800 standby.
So, starting with hour-2 of the day, there are now 200 in standby. They all didn't wait in the queue a full hour. They showed up in the rate of 1K standby per hour. They had an average wait of 200/hour, which is 3-4 minutes.
In the next hour, another 1K standby show up and another 200 LL get to cut the line. Since we've gone past the tipping point of the line, guests in the queue will be accumulating hour over hour (until the wait is so long, that they chose not to ride). All 200 LL get to skip the line. And the end of the hour, there will be 400 in standby. There wait time will be, on average, 7 minutes.
And so it goes on. At this rate, the number in standby will eventually exceed even 1K in line.
Once the line gets to 1K in the queue, they will still let 200 LLs go on quickly. Those who now get on line will have more than an hour wait. It will be 72 minutes. In the next hour, it will be 86 minutes, etc...
The effect on line length depends on several factors, most notably, has the number of riders (LL and Standby) have gone past the 'tipping' point of the ride's capacity? If it does, standby time will continually inflate. But standby time continually inflates even if there is no LL, but the tipping point has been reached.
If standby and LL together doesn't hit the tipping point, no one will have to wait (e.g., picking Nemo as a LL).