"Before ordering a taxi...", and "After ordering a taxi..."; the noun-verb follows a preposition and is a gerund.
"Waiting for a taxi, patience is needed." Or "Waiting for a taxi, I dropped my wallet," are I think both participles, even though the noun being qualified is only explicit in the second.
In the case of...
"While ordering a taxi, patience is very important."
"When ordering a taxi, reliability is very important."
...are while and when currently treated as adverbs qualifying verbal adjectives (ptc) or are they categorised as prepositions (before, while, after) taking objects (ger)? Or is there another analysis?