Lecture9_Process Reliability

Lecture9_Process Reliability - Lecture 9 Reliability...

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Lecture 9 Reliability measures and prediction System reliability

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Solve this Problem: Your company has developed a new microcontroller. A lot of 1000 units are tested for 2000 hours, after which 500 good units remain. What is the is the probability that one unit will have failed after in the first 100 hours of operation?
Probability of Failure Let F(t) be the define the probability of failure probability fa f(t) as: ilure den ( ) sity ( ) d f t F t dt = 0 It follows that: ( ) ( ) t F t f d τ τ =

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Define R(t) as: R(t) = 1 - F(t), or R(t) reliabilit + F(t y ) = 1 ( ) ( ) 0 ( ) ( ) Differentiating gives d d R t F t dt dt d d R t F t dt dt + = = - Reliability, R(t)
O F S F O S number of failures per unit time t Total failed survived number of units at time t=0 Number of units at time t Number of units at time t Define h per unit failure ra (t) a te s : ( ) = N N N N N + N h t = = = = otal units remaining Per Unit Failure Rate, h(t)

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number of failures per unit time ( ) total units remaining [ ] [ ( / )] / [ ] 1 ( ) ( ) ( ) O S F S S S O S O h t d dt N N d dt N d dt N N N N d dt N N R R h t h t h t = - - = = = = - Per Unit Failure Rate, h(t)
[ ] ( ) ( ) ( ) ln d dt h t dt R R h t h t dt R A R Ae - - = - = + = ( ) 1, since 1 at 0 h t dt A R t R e - = = = =

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Bath Tub Model for h(t) (per unit failure rate) (Hyman)
( ) Suppose failure rate ( ) = λ = constant / then, system reliability = 1/ mean time to failure h t dt h t t t R e e e λ τ τ λ - = - - = = = = = O O O / good units unit will Thus, if have N 0 then N remain N = N N at time where: at t t t t e e λ τ - - = = System Reliability when Failure Rate is Constant

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