Module 7: Reliability July 16, 2023 Module 7 July 16, 2023 1 / 18 Introduction Reliability: The probability that a failure may not occur in a given time interval. Definition: The measure of ability of a product or part to perform its intended function under a prescribed set of conditions. Mathematically, it is defined in terms of probability. Module 7 July 16, 2023 2 / 18 Introduction Reliability: The probability that a failure may not occur in a given time interval. Definition: The measure of ability of a product or part to perform its intended function under a prescribed set of conditions. Mathematically, it is defined in terms of probability. Let T be a random variable representing the life time of a system. Then reliability can be defined as: Module 7 July 16, 2023 2 / 18 Introduction Reliability: The probability that a failure may not occur in a given time interval. Definition: The measure of ability of a product or part to perform its intended function under a prescribed set of conditions. Mathematically, it is defined in terms of probability. Let T be a random variable representing the life time of a system. Then reliability can be defined as: R(t) = P(T > t), where 0 < t < ∞. Module 7 July 16, 2023 2 / 18 Introduction Reliability: The probability that a failure may not occur in a given time interval. Definition: The measure of ability of a product or part to perform its intended function under a prescribed set of conditions. Mathematically, it is defined in terms of probability. Let T be a random variable representing the life time of a system. Then reliability can be defined as: R(t) = P(T > t), where 0 < t < ∞. R∞ If T has p.d.f. f (t), then R(t) = P(T > t) = t f (s), ds. Module 7 July 16, 2023 2 / 18 Introduction Reliability: The probability that a failure may not occur in a given time interval. Definition: The measure of ability of a product or part to perform its intended function under a prescribed set of conditions. Mathematically, it is defined in terms of probability. Let T be a random variable representing the life time of a system. Then reliability can be defined as: R(t) = P(T > t), where 0 < t < ∞. R∞ If T has p.d.f. f (t), then R(t) = P(T > t) = t f (s), ds. From this can you tell between what range does R(t) lie and its relationship with f (t)? Module 7 July 16, 2023 2 / 18 Introduction Reliability: The probability that a failure may not occur in a given time interval. Definition: The measure of ability of a product or part to perform its intended function under a prescribed set of conditions. Mathematically, it is defined in terms of probability. Let T be a random variable representing the life time of a system. Then reliability can be defined as: R(t) = P(T > t), where 0 < t < ∞. R∞ If T has p.d.f. f (t), then R(t) = P(T > t) = t f (s), ds. From this can you tell between what range does R(t) lie and its relationship with f (t)? f (t) = −R ′ (t). Module 7 July 16, 2023 2 / 18 Hazard Function or Hazard Rate Now given that a system has survived up to time t, the probability that it will fail in the time interval (t, t + ∆t), for infinitesimally small ∆t is called hazard rate or hazard function. (Notation: λ(t)) Module 7 July 16, 2023 3 / 18 Hazard Function or Hazard Rate Now given that a system has survived up to time t, the probability that it will fail in the time interval (t, t + ∆t), for infinitesimally small ∆t is called hazard rate or hazard function. (Notation: λ(t)) >t) f (t) λ(t) = lim P(t<T <t+∆t/T = R(t) . ∆t ∆t→0 Module 7 July 16, 2023 3 / 18 Hazard Function or Hazard Rate Now given that a system has survived up to time t, the probability that it will fail in the time interval (t, t + ∆t), for infinitesimally small ∆t is called hazard rate or hazard function. (Notation: λ(t)) >t) f (t) λ(t) = lim P(t<T <t+∆t/T = R(t) . ∆t ∆t→0 From the fact that f (t) = −R ′ (t), it could be derived that, Rt Rt R(t) = e − 0 λ(t) dt =⇒ f (t) = λ(t)e − 0 λ(s) ds . Module 7 July 16, 2023 3 / 18 Hazard Function or Hazard Rate Now given that a system has survived up to time t, the probability that it will fail in the time interval (t, t + ∆t), for infinitesimally small ∆t is called hazard rate or hazard function. (Notation: λ(t)) >t) f (t) λ(t) = lim P(t<T <t+∆t/T = R(t) . ∆t ∆t→0 From the fact that f (t) = −R ′ (t), it could be derived that, Rt Rt R(t) = e − 0 λ(t) dt =⇒ f (t) = λ(t)e − 0 λ(s) ds . Given a value of reliability say, r, the value of t such that R(t) = r is called design life. Module 7 July 16, 2023 3 / 18 Conditional Reliability Given that a system has survived up to the warranty period T0 , the probability that it will live for t more years past T0 is called conditional reliability of t given T0 and is given by: − R(t + T0 ) =e R(t/T0 ) = P(T > T0 + t/T > t0 ) = R(T0 ) TR 0 +t T0 λ(t) dt . The value of T0 is called as wear-in period or guarantee period or burnin period. Module 7 July 16, 2023 4 / 18 Mean Time to Failure (MTTF) MTTF MTTF is given by E (T ) = R∞ tf (t) dt. Using the fact that f (t) = −R ′ (t), 0 and by integration by parts we have, Z∞ MTTF = E (T ) = R(t) dt. 0 Module 7 July 16, 2023 5 / 18 Problem The density function of the time to failure in years of the gizmos (for use on widgets) manufactured by a certain company is given by f (t) = 200 , t ≥ 0. (t + 10)3 (i) Derive the reliability function and determine the reliability for the first year operation. (ii) Compute the MTTF. (iii)What is the design life for a reliability 0.95? Module 7 July 16, 2023 6 / 18 Problems √ Given that R(t) = e − 0.001t ; t ≥ 0, (a) Compute the reliability for a 50 hours mission (b) Given a 10 hour wear in period, compute the reliability for a 50 hour mission. (c) What is the design life for a reliability of 0.95? (d) What is the design life for a reliability of 0.95, given a 10 hour wear in period? Module 7 July 16, 2023 7 / 18 Problems The time to failure in operating hours of a critical solid state power unit has hazard rate function λ(t) = 0.003(t/500)1/2 ; t ≥ 0. (i) What is the reliability if the power unit must operate continuously for 50 hours? (ii) Determine the design life if reliability of 0.90 is desired. Module 7 July 16, 2023 8 / 18 Problems The time to failure in operating hours of a critical solid state power unit has hazard rate function λ(t) = 0.003(t/500)1/2 ; t ≥ 0. (i) What is the reliability if the power unit must operate continuously for 50 hours? (ii) Determine the design life if reliability of 0.90 is desired. (iii) If a burn-in period of 50 hours is given, then does the reliability in part (i) improve? If so, what is the new reliability? Module 7 July 16, 2023 8 / 18 Problems 2 The reliability of a turbine blade is given by R(t) = 1 − tt0 ; 0 ≤ t ≤ t0 where t0 is the maximum life of the blade. (a) Show that the blades are experiencing wear out. (b) Compute MTTF as a function of the maximum life. (c) If the maximum life is 2000 operating hours, what is the design life for a reliability of 0.90? Module 7 July 16, 2023 9 / 18 Problems (i) Compute the reliability and hazard function of Exponential distribution (ii) Compute the reliability and hazard function of Weibull distribution. Module 7 July 16, 2023 10 / 18 Problems A one year guarantee is given based on the assumption that no more than 10% of the items shall be returned. Assuming exponential distribution, what is the maximum failure rate that can be tolerated? Module 7 July 16, 2023 11 / 18 Problems A one year guarantee is given based on the assumption that no more than 10% of the items shall be returned. Assuming exponential distribution, what is the maximum failure rate that can be tolerated? A manufacturer determines that, on the average, a television set is used 1.8 hours per day. A one-year warranty is offered on the picture tube having a MTTF of 2000 hours. If the distribution is exponential, what percentage of the tubes will fail during the warranty period? Module 7 July 16, 2023 11 / 18 System Reliability Any system is an assembly of smaller components. A system’s reliability depends on its components’ reliability Based on how the components are arranged systems are basically of two types: (i) series and (ii) parallel. Module 7 July 16, 2023 12 / 18 Reliability of Series System Figure: Components connected in series Module 7 July 16, 2023 13 / 18 Reliability of Series System Figure: Components connected in series Reliability of this system is R(t) = RC1 (t) × RC2 (t). Module 7 July 16, 2023 13 / 18 Reliability of Parallel System Figure: Components connected in parallel Module 7 July 16, 2023 14 / 18 Reliability of Parallel System Figure: Components connected in parallel Reliability of this system is R(t) = 1 − [(1 − RC1 (t))(1 − RC2 (t))]. Module 7 July 16, 2023 14 / 18 Reliability of a System Figure: A Series-Parallel System Module 7 July 16, 2023 15 / 18 Reliability of a System Figure: A Series-Parallel System Reliability of this system is R(t) = [1 − (1 − RC1 (t))(1 − RC2 (t))] × [1 − (1 − RC3 (t))(1 − RC4 (t))]. Module 7 July 16, 2023 15 / 18 Problem Determine the overall reliability of the following system: Module 7 July 16, 2023 16 / 18 Problem Determine the overall reliability of the following network: Module 7 July 16, 2023 17 / 18 Problems Find the reliability of the following system: Module 7 July 16, 2023 18 / 18
