Queueing system with heterogeneous customers as a model of a call center with a call-back for lost customers

A multiserver queueing system with infinite and finite buffers, two types of customers, and two types of servers as a model of a call center with a call-back for lost customers is investigated. Type 1 customers arrive to the system according to a Markovian arrival process. All rejected type 1 customers become type 2 customers. Type r, r=1,2, servers serve type r customers if there are any in the system and serve type r′, r′=1,2, r′≠r, customers if there are no type r customers in the system. The service times of different types of customers have an exponential distribution with different parameters. The steady-state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution of type 2 customers is derived. The problem of optimal choice of the number of each type servers is solved numerically.