{"id":"mtbf","model_id":"MTBF","name":"Mean Time Between Failures (MTBF)","summary":"MTBF is the mean time between failures of a repairable system, defined in the Homogeneous Poisson Process (HPP) model as the reciprocal of the repair rate λ. In the HPP model, failures occur at a constant rate λ, so the expected number of failures by time T is λT and the waiting time to the next failure follows an exponential distribution. It is used for repairable equipment and systems operating in the flat, constant repair rate portion of the bathtub curve.","study_notes":["In the HPP model the repair rate (ROCOF) m(t) = λ is constant, and M(t) = λT is the expected number of failures by time T.","The cumulative distribution of the waiting time to the next failure (interarrival time) is F(t) = 1 - e^(-λt).","The probability of exactly k failures by time T is P{N(T)=k} = (λT)^k e^(-λT) / k!.","HPP fits the flat portion of the empirical bathtub curve; most systems spend most of their lifetimes in that constant repair rate region.","HPP is the only model that applies to the flat portion of the bathtub curve, which is the stated reason for its wide use in system reliability evaluation and reliability test planning.","Planning reliability assessment tests under the HPP assumption, estimating MTBF from system failure data, and calculating upper and lower confidence limits are covered in later sections of the document."],"safety_notes":[],"sources":[{"url":"https://www.itl.nist.gov/div898/handbook/apr/section1/apr171.htm","title":"8.1.7.1. Homogeneous Poisson Process (HPP)","accessed":"2026-10-25"}],"verification":{"status":"source_checked","checked_at":"2026-10-25","values":4,"method":"llm_source_check"},"updated_at":"2026-10-25","type":"operational","spec":{"kind":"methodology","formula":{"expression":"MTBF = 1 / λ","variables":[{"symbol":"λ","name":"repair rate (ROCOF)"}],"output_unit":"time"},"related":[]},"url":"https://machineschema.com/operational/mtbf","license":"https://creativecommons.org/licenses/by/4.0/"}