It could be an issue with your physical broadband connection but I'd eliminate Wifi issues first. If the modem is nearby, perhaps it may be worth trying to connect with an Ethernet cable and testing again to see if the issue persists? If there is a cordless phone or anything else near your modem, there may be some interference with your Wifi signal - be sure to position other electronic devices at least 1m away from your modem.
Looking at your snapshot I would say that your router dhcp is only leasing ip address for 15mins per session. Then everything connected to your router will have to ask to renew their assigned ip address.
You could set it so your router allocates to permanent ip address (static ip)
To get my tg589 working correctly with my xbox one, I had to turn on "game mode" on the router under firewall settings. (Not sure if urs will have game mode but worth a check)