We study the information-theoretic limit of reliable information processing by a server with queue-length dependent quality of service. We define the capacity for such a system as the number of bits reliably processed per unit time, and characterize it in terms of queuing system parameters. We also characterize the distributions of the arrival and service processes that maximize and minimize the capacity of such systems in a discrete-time setting. For arrival processes with at most one arrival per time slot, we observed a minimum around the memoryless distribution. We also studied the case of multiple arrivals per time slot, and observed that burstiness in arrival has adverse effects on the system. The problem is theoretically motivated by an effort to incorporate the notion of reliability in queuing systems, and is applicable in the contexts of crowdsourcing, multimedia communication, and stream computing. © 2017 IEEE.