Building and analysis of the contact center model as a queuing system with multi-level IVR and impatient applications

A mathematical model for the receipt and servicing of applications in a modern call-center in the form of a queuing system with impatient requests has been developed and investigated. Call-centers include companies working in the field of transport services, communications, sales, etc. The input of the system in question receives a stationary stream of requests. The service time for applications by each device consists of four phases. If the necessary information is not received using two-level voice service (IVR - Interactive Voice Response), the client’s application is sent with a certain probability to the buffer with an unlimited number of places (third phase) and awaits service by the operator (fourth phase), and leaves the system with an additional probability. During waiting times, impatient requests may leave the system. The article presents an algorithm for finding the probability distribution of the state of a queuing system. Using the method of moments, we found solutions of the four differential equations of the system that determine the average probabilistic characteristics of the number of serviced customer applications in each phase. The proposed mathematical model of the call processing process in the form of a four-phase queuing system allows you to optimize the activities of the contact center, speed up the customer service process and optimize costs.