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We consider a simple model of overhead in batch computer systems and message switching systems: an exponential, single-server queuing system with finite storage capacity, constant arrival rate, and queue-length-dependent service time. We consider cases in which the expected service time consists of a constant plus a term that grows linearly or logarithmically with the queue length. We show that the performance of this system -- as characterized by the expected number of customers in the system, the expected time in the system, and the rate of missed customers -- can undergo a sudden collapse as the result of small changes in the arrival rate, the overhead rate, or the queue capacity. The system has the interesting property that increasing the queue capacity can decrease performance. In addition to equlibrium results, we consider the dynamic behavior of the model. We show that the system tends to operate in either of two quasi-stable modes of operation -- one with low queue lengths and one with high queue lengths. System behavior is characterized by long periods of operation in both modes with abrupt transitions between them. We point out that the performance of a saturated system may be improved by dynamic operating procedures that return the system to the low mode. (Author)