How can we prevent waiting on a messaging ticket when a customer has disconnected?

Our agents handle multiple channels of incoming ticket, with our primary use being phones and messaging. In order to attend to as many inquiries as possible, agents alternate between messaging and phones, having only one live messaging chat or phone call at any given time. 

We are looking to avoid wasting time on a messaging ticket where the end user has abandoned, disconnected, or does not reply, but there is no indication to the agent when a guest has disconnected or has gone inactive. Our end users use the messaging widget on our desktop site to reach out to us, and they do not have to log in to our support site to do so.

At times, an end user can even abandon a messaging ticket completely (while still appearing as Active to an agent) and begin a new messaging ticket which is routed to a different agent. 

Is there a way we can implement a manual or automated "End Chat" for messaging, other than simply waiting an arbitrary amount of time with no response and setting an abandoned ticket to Solved?