Why do my arms fall asleep when im on my back?

Position. It may have something to do with the position of your head and neck or the position of your arms. In either case this may be causing some decreased blood flow to the nerves in the neck or the nerves that cross the shoulder which would then cause your arms to fall asleep. A physiatrist would be a good specialist to sort this out.
Paresthesias. Could be due to kinking your neck while reading or watching tv in bed. If there is cervical neck nerve compression due to arthritis or trauma (recent or previous), the nerves radiating from the neck can be compressed by the kinking. Some who have injury/arthritis in their elbow crease are sensitive to lying flat on their backs via pressure to ulnar nerve emerging under elbow. Carpal tunnel ?