Call center operation model as a $MAP/PH/N/R-N$ system with impatient customers

We analyze a multiserver queueing system with a finite buffer and impatient customers. The arrival customer flow is assumed to be Markovian. Service times of each server are phase-type distributed. If all servers are busy and a new arrival occurs, it enters the buffer with a probability depending on the total number of customers in the system and waits for service, or leaves the system with the complementary probability. A waiting customer may become impatient and abandon the system. We give an algorithm for finding the stationary distribution of system states and derive formulas for basic performance characteristics. We find Laplace–Stieltjes transforms for sojourn and waiting times. Numeric examples are given.