A finite capacity single-server queue with customer loss

We present a model for a finite capacity, single-server M/G/1 queue in which customers, who would cause the system capacity of unfinished work to be exceeded, are lost. Problems are formulated for: (i) the stationary density of the unfinished work; (ii) the mean time to empty the queue; (iii) the mean length of a busy period; (iv) the mean total unfinished work during a busy period; and (v) the stationary rate at which customers are lost. We construct exact solutions for these quantities for an M/M/l queue with constant arrival and service rates. When the arrival and service rates are state dependent, we construct approximate expressions for these quantities using singular perturbation techniques. When specialized to state-independent queues, these approximations are shown to agree with the asymptotic expansions of the exact solutions.