The 'auto' system descends, IIRC, at about 500fpm. Therefore if you are at FL400, say, (cabin at about 8000ft), you need about 16 mins to get the cabin 'down'. Make that descent in 15 minutes and you 'catch the cabin'. Perhaps the other a/c you mention have either a lower cabin altitude or higher cabin descent rate? The programmed descent does not begin until around ?2000ft? (1500ft on the 'classic' 737) below cruise altitude so it can be a good idea in the circumstances you describe to make that first 2000ft 'drop' early.
Putting the system in 'standby' (where fitted) will also allow an early cabin descent but does produce a 'bump' in pressure when selected.