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We model the probabilistic behavior of a system comprising a failure detector and a monitored crash-recovery target. We extend failure detectors to take account of failure recovery in the target system. This involves extending QoS measures to include the recovery detection speed and proportion of failures detected. We also extend estimating the parameters of the failure detector to achieve a required QoS to configuring the crash-recovery failure detector. We investigate the impact of the dependability of the monitored process on the QoS of our failure detector. Our analysis indicates that variation in the MTTF and MTTR of the monitored process can have a significant impact on the QoS of our failure detector. Our analysis is supported by simulations that validate our theoretical results.