Question 17.1: An electronic system consists of two parts that operate in s......
An electronic system consists of two parts that operate in series. Assuming that failures are governed by a constant failure rate \lambda_{i} for the ith part, determine (1) the system failure rate, (2) the system reliability for a 1000-hour mission, and (3) the system mean time to failure (MTTF).
The failure rates of the parts for this problem are given by:
\lambda_1=6.5 \text { failures } / 10^6 \text { hours } \\ \lambda_2=26.0 \text { failures } / 10^6 \text { hours. }For a constant failure rate, the reliability R_{i} for the ith part has the form:
R_i(t)=e^{-\int_0^t \lambda_{i} d \tau}=e^{-\lambda_{i} t}.
The reliability, R_{s}, of the series system is
R_s=e^{-\sum_{i=1}^n \lambda_{i} t}=e^{-\lambda_s t} \\ \lambda_s=\sum_{i=1}^n \lambda_i,
for a series system with parts assumed to have a constant failure rate. Substituting the given values:
\lambda_S=32.5 / 10^6 \text { hours. }The reliability for a 1000-hour mission is thus:
R_S(1000)=e^{-\left(32.5 \times 10^6\right) \times 1000}=0.968.
The MTTF for the system is:
\mathrm{MTTF}=\int_0^{\infty} R_s(t) d t=\int_0^{\infty} e^{-\lambda_s t} d t=1 / \lambda_s=30,770 hours.