This has happened a few times around a dozen. When I receive a call it only rings once and then goes silent. I can pick it up and it looks like someone is still calling though.
Ahh, here it is... It's called Slot Cycle Index. It will make your phone ring Sooner, not necessarily longer and might help your issue. You do need your MSL from Sprint (programming code which you can get by calling up tech support and saying that your voicemail is dialing 999-999-9999 and need your phone re-programmed)...
...
I don't think that will address his issue. His phone is receiving calls in a timely manner it seems, it's just that the audible ringer shuts off but the phone is still in the "receiving call" state.
What you're suggesting is for when the phone only rings once and then voicemail kicks in right away giving you only a couple seconds to answer the call.