System crash in a finite capacity M/G/1 queue

We analyze a finite capacity M/G/1 queue in which an arrival that causes the system capacity of unfinished work to be exceeded, results in the loss of all jobs in the system. We sbudy the stationary density of unfinished work, the system utilization, the mean length of the busy period, and the probability that the system crashes before completing a busy period. We obtain formulas for these quantities for an M/M/1 queue with constant arrival and service rates. Approximate expressions for these quantities are constructed for M/G/1 queues in which the arrival rates and service densities depend on the amount of unfinished work in the system at the instant a new customer enters. These approximations, obtained by using singular perturbation techniques, are shown to agree with the exact results for state-dependent M/M/1 queues.