Understanding Reliability Theory in Engineering and Mathematics
Reliability theory, presented by S. Ithaya Ezhil Manna, explains the concept of reliability as the probability of a component functioning properly over time. The theory defines reliability in terms of the random variable X representing component life or time to failure. Key points include the definition of reliability function R(t), failure rate or hazard rate, and the assumptions related to component reliability.
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RELIABILITY THEORY Presented By S.ITHAYA EZHIL MANNA Assistant Professor In Mathematics St.Joseph s College Trichy-2.
Let the random variable X be the life time or the time to failure of the component. The reliability of the component is defined by R(t) = P(X> t).
Let the random variable X be the life time or the time to failure of the component. The reliability of the component is defined by R(t) = P(X> t). Remarks: 1. The component is assumed to be working properly at time t = 0 (i.e) R(0) = 1.
Let the random variable X be the life time or the time to failure of the component. The reliability of the component is defined by R(t) = P(X> t). Remarks: 1.The component is assumed to be working properly at time t = 0 (i.e)R(0) = 1. 2.No component can work forever without failure limn R(t) = 0.
Let the random variable X be the life time or the time to failure of the component. The reliability of the component is defined by R(t) = P(X> t). Remarks: 1.The component is assumed to be working properly at time t = 0 (i.e)R(0) = 1. 2.No component can work forever without failure limn R(t) = 0 3.R(t) is a monotone non-increasing function of t.
Let the random variable X be the life time or the time to failure of the component. The reliability of the component is defined by R(t) = P(X> t). Remarks: 1.The component is assumed to be working properly at time t = 0 (i.e)R(0) = 1. 2.No component can work forever without failure limn R(t) = 0 3.R(t) is a monotone non-increasing function of t. 4. For t < 0, reliability has no meaning, but we let R(t) = 0, for t < 0.
FAILURE RATE OR HAZARD RATE: The failure rate h(t) at time t is defined to be ( ) f t Density Function = = ( ) h t ( ) R t Survival Function
FAILURE RATE OR HAZARD RATE: The failure rate h(t) at time t is defined to be ( ) f t Density Function = = ( ) h t ( ) R t Survival Function t ( ) h x dx 0 = ( ) . R t e
FAILURE RATE OR HAZARD RATE: The failure rate h(t) at time t is defined to be Density t R ) ( ( ) f t Function = = ( ) h t Survival Function t ( ) h x dx 0 = ( ) . R t e Cumulative Failure Rate: t = ( ) ( ) . H t h x dx 0
FAILURE RATE OR HAZARD RATE: The failure rate h(t) at time t is defined to be Density t R ) ( ( ) f t Function = = ( ) h t Survival Function t ( ) h x dx 0 = ( ) . R t e Cumulative Failure Rate: t (t ) H = = ( ) ( ) . ( ) . H t h x dx R t e 0
PARAMETERS OF RELIABILITY 1.MTTF: Mean Time To Failure = ( ) . MTTF R t dt 0
PARAMETERS OF RELIABILITY 1.MTTF: Mean Time To Failure = ( ) . MTTF R t dt 0 2.MTBF: Mean Time Between Failures 1 FR FR + + = , where FR is the failure MTBF + ... FR 1 2 n rate of component each of system the up to ' components n' .
PARAMETERS OF RELIABILITY 1.MTTF: Mean Time To Failure = ( ) . MTTF R t dt 0 2.MTBF: Mean Time Between Failures 1 FR FR + + = , where FR is the failure MTBF + ... FR 1 2 n rate of component each of system the up to ' components n' . 3.MTTR: Mean Time To Repair MTBF MTTR = . MTTF
EXPONENTIAL DISTRIBUTION Suppose the random variable X (be the lifetime of a component) is said to follow the exponential distribution,then
EXPONENTIAL DISTRIBUTION Suppose the random variable X (be the lifetime of a component) is said to follow the exponential distribution,then 0 , . . ) ( = = e f d p T f T . 0 T and T T = = = = ( ) . . ( ) ( ) 1 . F T c d f P X T f T dT e 0 T = = = = ( ) ( > T ) 1 T ( ) 1 ( ) . R T P X P X T F T e ( ) f T e = = = ( ) . h T T ( ) R T e
EXPONENTIAL DISTRIBUTION Suppose the random variable X (be the lifetime of a component) is said to follow the exponential distribution,then 0 , . . ) ( = = e f d p T f T . 0 T and T T = = = = ( ) . . ( ) ( ) 1 . F T c d f P X T f T dT e 0 T = = = = ( ) ( > T ) 1 T ( ) 1 ( ) . R T P X P X T F T e ( ) f T e = = = ( ) . h T T ( ) R T e Hence,the exponential lifetime distribution is a constant failure rate [ It s the only distribution with a constant failure rate].
WEIBULL DISTRIBUTION: The density function is given by, t t 1 = ( ) for , all t 0, 0 > and > 0. f t e t t 1 = = = = ( ) . . ( ) ( ) 1 . F t c d f P X t f t dt e 0 t 1 = = = = ( ) ( > ) t 1 ( ) 1 ( ) . R t P X P X t F t e t 1 t 1 ( ) f t e t = = = ( ) . h t t 1 ( ) R t e
Failure rate of the weibull distribution with various values of and = 1 1 . If <1,the function is DFR. 2. If = 1,the function is CFR. 3. If >1,the function is IFR.
APPLICATIONS 1. Reliability engineering - Reliability engineering that emphasizes dependability in the lifecycle management of a product. Dependability, or reliability reliability, describes the ability of a system component to function under stated conditions for a specified period of time. 2. 2. System Reliability System Reliability - - The probability that a system including all hardware, firmware, and software, will satisfactorily perform the task for which it was designed or intended, for a specified time and in a specified environment. 3. Structural reliability. Reliability engineering is system or system,
REFERENCES: 1 . J.Medhi,Stochastic Processes,New Age International Publishers,Second Edition,New Delhi,1994. 2 . U.Narayan Bhat,Elements of Applied stochastic Processes,second Edition,John Wiley & Sons,New York,1972. 3 . N.V.Prabhu,stochastic processes,Macmillan,NewYork,1970.
