That is just how I work on WP sites, I may also be working on multiple WP sites at one time,which are hosted on the same server. I like to have multiple tabs open for each area so I don't have to keep reloading pages or re-navigate to them for the needed information. Efficiency Reasons.
You are right just have one or two open is the easiest and most obvious solution along with closing down the admin section whenever not in use. That is a Process/Systems Solution I am going to implement. I don't see that as a fix to the issue, because if someone accidentally leave some open over night or during the day, I don't want all my sites to go down again.
My ultimate goal is to talk to a WP developers to see if this is a known or expected experience. If not maybe add something to stop the heartbeat after a certain sleep setting or combine all the heartbeats into POST or something.
Thanks for the help, hopefully someone else has some other ideas.