The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues

The aim of this paper is to treat an analytical explicit solution of the moments of a system of an overflow queue (M/M/1/k, M/M/∞), with the balking, reneging and an additional server for longer queues. An explicit solution for M₍₁₎, M₍₂₎ and M_(l) is given. Finally, we deduce some particular cases for the two systems (M/M/1/k, M/M/∞) and (M/M/2/k, M/M/∞).