16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Reliability over lifetime tr R(1)=P(T>1) dt Suppose a system contains n components all of which must operate if the system is to operate(no redundancy), and component failures are considered independent. This is called a simplex system For the system: R,(D)=P(T>1) =P(T1>1,2>1,…,Tn>D =P(T1>1)P(72>1).P(Tn>1) This is the same form for the system as for the individual components. Thus the system also has an exponential distribution of time to failure, with the indicated mean time to failure λ=∑ If all the components have the same mean time to failure In=In,I= Note the importance of part count n to system reliability Example: Lifetime systemm reliability Suppose we wish to achieve a 99% probability of successful system operation over a mission lifetime tl. Page 3 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Reliability over lifetime tl: ∞ − τ R tl () = PT ( > tl) = ∫ 1 e tm dτ t t m ∞ ⎛ ⎞ ⎛ ⎞ −⎜ ⎟ τ tl −⎜ ⎟ tm t ⎝ ⎠ m = − e = e ⎝ ⎠ τ =tl Suppose a system contains n components all of which must operate if the system is to operate (no redundancy), and component failures are considered independent. This is called a simplex system. For the system: Rt () = PT ( > t) s s ( t T2 = P T , 1 > > t,..., T > t) n = P T( ) ( ( 1 > t P T > t)...PT > t) 2 n ⎛ ⎞⎛ ⎞ ⎛ ⎞ −⎜ ⎟⎜ ⎟ −⎜ ⎟ t t t − ⎜ ⎟⎜ ⎟ ⎜ ⎟ mn = e ⎝ ⎠⎝ ⎠ t t m1 e tm2 ...e ⎝ ⎠ ⎛ ⎞ −⎜ ⎟ t ⎜ ⎟ tms = e ⎝ ⎠ This is the same form for the system as for the individual components. Thus the system also has an exponential distribution of time to failure, with the indicated mean time to failure. n 1 = ∑ 1 t i=1 t ms mi n λs = ∑λi i=1 If all the components have the same mean time to failure, t = t , 1, 2,..., i = n mi mc 1 n = t t ms mc 1 t = t ms mc n Note the importance of part count n to system reliability. Example: Lifetime system reliability Suppose we wish to achieve a 99% probability of successful system operation over a mission lifetime tl. Page 3 of 6