Template:Ipl ex rel function

IPL-Exponential Reliability Function

The IPL-exponential reliability function is given by:

[math]\displaystyle{ R(T,V)={{e}^{-TK{{V}^{n}}}} }[/math]

This function is the complement of the IPL-exponential cumulative distribution function:

[math]\displaystyle{ R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT }[/math]

or:

[math]\displaystyle{ R(T,V)=1-\mathop{}_{0}^{T}K{{V}^{n}}{{e}^{-K{{V}^{n}}T}}dT={{e}^{-K{{V}^{n}}T}} }[/math]

Conditional Reliability

The conditional reliability function for the IPL-exponential model is given by:

[math]\displaystyle{ R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-K{{V}^{n}}t}} }[/math]

Reliable Life

For the IPL-exponential model, the reliable life or the mission duration for a desired reliability goal, [math]\displaystyle{ {{t}_{R}}, }[/math] is given by:

[math]\displaystyle{ R({{t}_{R}},V)={{e}^{-K{{V}^{n}}{{t}_{R}}}} }[/math]

[math]\displaystyle{ \ln [R({{t}_{R}},V)]=-K{{V}^{n}}{{t}_{R}} }[/math]

or:

[math]\displaystyle{ {{t}_{R}}=-\frac{1}{K{{V}^{n}}}\ln [R({{t}_{R}},V)] }[/math]