[Q] Slow push notifications when 4G radio is sleeping?

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This could just be an abnormality happening tonight and not the norm (I just got this phone recently), but I'm noticing my notifications are a bit delayed when my 4G radio goes to sleep (the icon has "zzz") vs when I'm actually using data. I tested this informally for a few minutes sending myself Google Voice texts, and it seems to take a couple minutes to receive a notification if the radio is asleep. Needless to say, that's a pain.

Is this normal?