Question 17.8: Consider a two-unit shared load parallel system where fh(t) ......
Consider a two-unit shared load parallel system where
f_h(t)=\lambda^{e-\lambda t} \quad t \geq 0 pdf for time to failure under half load
f_F(t)=5 \lambda e^{-5 \lambda t} \quad t \geq 0 pdf for time to failure under full load
(a) Find the system reliability function
(b) Find the MTTF.
(a) With \lambda_{h} = λ and \lambda_{f} = 5λ, using Equation (17.39), we have:
R_S(t)=e^{-2 \lambda t}+\frac{2 \lambda}{2 \lambda-5 \lambda}\left[e^{-5 \lambda t}-e^{-2 \lambda t}\right]=\frac{5}{3} e^{-2 \lambda t}-\frac{2}{3} e^{-5 \lambda t} .R_S(t)=e^{-2 \lambda_h t}+\frac{2 \lambda_h}{2 \lambda_h-\lambda_F}\left[e^{-\lambda_F t}-e^{-2 \lambda_h t}\right] (17.39)
(b) MTTF is given by:
\mathrm{MTTF}=\int_0^{\infty} R_S(t) d t \\ =\int_0^{\infty}\left(\frac{5}{3} e^{-2 \lambda t}-\frac{2}{3} e^{-5 \lambda t}\right) d t \\ =\frac{5}{3} \int_0^{\infty} e^{-2 \lambda t} d t-\frac{2}{3} \int_0^{\infty} e^{-5 \lambda t} d t \\ =\frac{5}{3} \frac{1}{2 \lambda}-\frac{2}{3} \frac{1}{5 \lambda}=\frac{7}{10 \lambda}.