We consider cluster systems with multiple nodes where each server is prone to run tasks at a degraded level of service due to some software or hardware fault. The cluster serves tasks generated by remote clients, which are potentially queued at a dispatcher. We present an analytic queueing model of such systems, represented as an M/MMPP/1 queue, and derive and analyze exact numerical solutions for the mean and tail-probabilities of the queue-length distribution. The analysis shows that the distribution of the repair time is critical for these performability metrics. Additionally, in the case of high-variance repair times, the model reveals so-called blow-up points, at which the performance characteristics change dramatically. Since this blowup behavior is sensitive to a change in model parameters, it is critical for system designers to be aware of the conditions under which it occurs. Finally, we present simulation results that demonstrate the robustness of this qualitative blow-up behavior towards several model variations.