{"id":"failure-rate","model_id":"Failure (or hazard) rate","name":"Failure (or hazard) rate","summary":"Failure rate, also called hazard rate, is the instantaneous conditional rate at which survivors in a non-repairable population fail at time t. It is written h(t) and defined as f(t)/(1-F(t)) or f(t)/R(t). The document also defines average failure rate over an interval using H(t) and R(t).","study_notes":["The failure rate is defined for non-repairable populations as the instantaneous rate of failure for the survivors to time t.","h(t) is the instantaneous conditional failure rate.","H(t) = ∫_0^t h(t) dt.","H(t) = -ln R(t).","F(t) = 1 - exp[-∫_0^t h(t) dt] = 1 - e^{-H(t)}.","h(t) = -d ln R(t) / dt.","Average failure rate: AFR(T1,T2) = (H(T2)-H(T1))/(T2-T1) = (ln R(T1)-ln R(T2))/(T2-T1).","AFR(0,T) = AFR(T) = H(T)/T = -ln R(T)/T."],"safety_notes":[],"sources":[{"url":"https://www.itl.nist.gov/div898/handbook/apr/section1/apr123.htm","title":"8.1.2.3. Failure (or hazard) rate","accessed":"2026-10-25"}],"verification":{"status":"source_checked","checked_at":"2026-10-25","values":6,"method":"llm_source_check"},"updated_at":"2026-10-25","type":"operational","spec":{"kind":"metric","formula":{"expression":"h(t) = f(t) / (1 - F(t)) = f(t) / R(t)","variables":[{"symbol":"h(t)","name":"failure rate or hazard rate"},{"symbol":"f(t)","name":"f(t)"},{"symbol":"F(t)","name":"F(t)"},{"symbol":"R(t)","name":"R(t)"}]},"related":[]},"url":"https://machineschema.com/operational/failure-rate","license":"https://creativecommons.org/licenses/by/4.0/"}