Instead of building a brand new "complete" call center solution, one possible approach would be integration with existing CRM / support service. You would send REST API to CRM for events happening on the phone / calls.
- incoming call
- answered
- disconnect
- queued
- caller hang up without connection
- transfer
- three-way conf
Some of these events can be triggers using pivot or they can be also triggered by the phone (some phones do http port for events).
Live monitoring would be a little issue but I think there may be also some technical solutions online or saas.