In the beginning, when the concept of customer support had begun, everything seemed to happen over a telephone call. Telephonic method did not allow an operator to deal with many customers at the same time. However, the Live chat feature, we have now, allows an operator to deal with many website visitors/customers at the same time, without any hindrance.
The Operator concurrent chat limit configuration allows the administrator to set a maximum chat limit, the operator can handle at a particular time. This in turn decides the operator's availability when a visitor sends a message.
To explain this further, consider an operator who is busy assisting the customers and has reached his/her maximum chat limit, then the next upcoming visitor's chat will be forwarded to another operator. Meanwhile as the operator finishes a chat session, he will be assigned another visitor chat.This improves customer management support and delivers solutions to a maximum number of visitors at a time.
So go to your Portal Settings tab under Settings and scroll down to Operator availability configuration and set the operator concurrent chat limit. This limit setting can be overridden at an individual operator level.