I had experience of this when I first took over my current team.
It's a nightmare to manage, because if you don't have the capacity to answer all the chats, we ended up with people going on the forum and kicking off.
Our agents handle email tickets, calls and when they had 2-3 chats at a time to handle, it honestly started to come apart at the seams, and we ended up having to knock it on the head.
One of the biggest issues is customers literally sitting on the chats for hours. No joke. If I remember correctly the record was 6 hours.
I suppose it depends on what you're providing support for, but when it's a build it yourself ecommerce solution and you give customers the ability to tie up an agent for as long as they want, with no charge, they'll take full advantage!