@gyoung 

It sounds like whatever is giving the router internet does not recognize it unless the device giving internet is rebooted. Usually if a modem does not recognize a router, the router needs to remain on and the modem only rebooted. In this case, it could be the modem turns on first, then the router after. 

If the router is not on a surge protector consider placing the router on one. There have been cases of a router WAN port fried from it.