In UK, the following is the assistance procedure.
I had a fatal house fire in neighbours house in April. First attendance (FDNY 1st Alarm). brought three pumps. A make pumps 6 and an Ariel brought 3 extra pumps and a Combined Platform Ladder ( FDNY Tower Ladder).
Therefore the London fire, the first attendance which I think was 3 or 4 pumps and an Ariel , wanted the attendance to be 10 pumps , so a further number of pumps up to ten were dispatched. Once the fire was further assessed a Priority Message Make Pumps 20 a further 10 pumps were dispatched. Like wise 20 pumps to 40 pumps brought a further 20 pumps. With your revised system in FDNY, this would have gone to a 5th Alarm, then further "Special Requests" to arrived at 40 engines. Hope this explains UK system.