Queueing system MAP|PH|N|N+RMAP|PH|N|N+R with impatient heterogeneous customers as a model of call center

A multi-server queueing system with a Markovian arrival process (MAP), a finite buffer and impatient heterogeneous customers useful in modeling a call center is investigated. The servers are identical and independent of each other. The service time of a customer has a Phase type distribution (PH). If all servers are busy at a customer arrival epoch, the customer moves to the buffer with probability, which depends on the number of customers in the system, or, alternatively, leaves the system forever. During a waiting period the customers can be impatient and leave the system. An efficient algorithm for calculating the stationary probabilities of system states is proposed. A special approach for reducing the system dimension is used. Some key performance measures are calculated. The Laplace–Stieltjes transforms of the sojourn and waiting time distribution are derived. Some illustrative numerical results are presented.