Live Chat WebSocket our best live chat solution hosted on your server. Please find here the minimum requirements for Live Chat WebSocket.
Additonal for ssl access via websocket you would need a stunnel configuration on your server which will forward the port to handle secure connections.
That's about it, Live Chat WebSocket is the fastest and resource friendliest chat available on the market.
If you are on a VPS or Dedicated Server you certainly will have access to SSH and therefore you can run a PHP Daemon from the console.