Retrial queuing system with Markovian arrival flow and phase-type service time distribution

We consider a multi-server queuing system with retrial customers to model a call center. The flow of customers is described by a Markovian arrival process (MAP). The servers are identical and independent of each other. A customer's service time has a phase-type distribution (PH). If all servers are busy during the customer arrival epoch, the customer moves to the buffer with a probability that depends on the number of customers in the system, leaves the system forever, or goes into an orbit of infinite size. A customer in the orbit tries his (her) luck in an exponentially distributed arbitrary time. During a waiting period in the buffer, customers can be impatient and may leave the system forever or go into orbit. A special method for reducing the dimension of the system state space is used. The ergodicity condition is derived in an analytically tractable form. The stationary distribution of the system states and the main performance measures are calculated. The problem of optimal design is solved numerically. The numerical results show the importance of considering the MAP arrival process and PH service process in the performance evaluation and capacity planning of call centers.