It seems to happen within a few seconds. Two maybe three. I can understand if that is a nojoy for you. Honestly for me, if I'm not using streaming, if the phone is off, I don't mind if messages show up on a periodic poll. I'm not really in the need to know more often that that if someone wants to send me a message via Whatsapp or email.
The thing I'm trying to determine, is VZN halting my connection when I'm streaming music, or is JD? Songs will play fine, then mysteriously, instead of transitioning to next song, I'll wake the phone to see "Waiting for Data Connection" or something like that.
Not a big issue for me to Disable JD if I want to stream music... Just not sure who is culprit.