We study reliability and sensitivity analysis of a retrial machine repair problem with working breakdowns operating under the F-policy. The F-policy studies the most common issue of controlling arrival to a queueing problem and it requires startup time before allowing failed machines to enter the orbit. The server is subject to working breakdowns only when there is at least one failed machine in the system. When the server is busy it works at a fast rate, but when it is subject to working breakdowns, it works at a slow rate. Failure and repair times of the server are exponentially distributed. The Laplace transform technique is utilized to develop two system performance measures such as system reliability and mean time to system failure (MTTF). Extensive numerical results are carried out to reveal how performance measures are affected by changing various values of each system parameter.