And now, just for fun, a little quick & dirty animation (obviously nothing is to scale)...
This illustrates how the two-cab terminus model makes sure a cab is available (almost) at all times. The first engineer takes a cab and two astrophysicists arrive shortly thereafter (getting there a bit before the replacement cab), but they don't have to wait since the former Standby cab is there to serve them immediately. Then after a second replacement restores the terminus to its two-cab waiting mode, a third cab arrives bringing Scotty and the Jumpsuit Twins for a warp coil inspection. Thanks to the Overflow space, they are not delayed one bit.
I did create a very tiny delay for the Bald Botanist, and then contrived a bit of a longer wait for our heroes by throwing three
successive departures at the terminus. Even then, Kirk's wait would be just a few seconds at actual cab speeds, and he would probably hear the cab coming rather than wonder what's going on. (Here I decided that only Spock is paying attention with his Vulcan ears ... Kirk just keeps grousing. Must be on his way to Sickbay for his routine physical.)
If you watch for a few cycles, you'll see how the routing system tries to keep the terminus at its equilibrium of two empty cabs at all times, shuttling them in and out as quickly as it can without getting in the way of actual passengers. After Scotty's jumpsuit detail arrives, the router immediately tries to get rid of the extra cab — but after it senses the arrival of Chekov, it realizes that "extra" cab is now needed to fill the new deficit, so it parks it just out of the way (based on the known destinations of the two departing cabs) and brings it back in as soon as possible. And, off camera, it has also located a second nearby replacement cab, which is idling right behind since the terminus is now at a deficit of -2.
Unfortunately, in this rare case of three departures stacked atop each other, the replacements are not quite quick enough to keep the Captain's bad day from getting a little bit worse.
In most browsers you can stop the animation by pressing ESC. To start it again, you have to refresh the page (and it will start at the beginning).
Please don't anyone tell my wife how much time I wasted doing this.