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.