RESTART - Estimating Residual Time

Residual time is: given that a task has run for time tau, it is the estimate of the time remaining until task completion from time tau.

$E[X|\tau ]=\int \limits _{\tau }^{\infty }t{\frac {f(t)}{R(\tau )}}\,dt$

Since we are looking for the estimated remaining time under RESTART, replace t with the restart function

$E[X_{R}|\tau ,\beta ]=\int \limits _{\tau }^{\infty }{\frac {e^{\beta t}-1}{\beta }}{\frac {f(t)}{R(\tau )}}\,dt$

$=\int \limits _{\tau }^{\infty }\left({\frac {e^{\beta t}-1}{\beta }}\right){\frac {\lambda e^{-\lambda t}}{e^{-\lambda \tau }}}\,dt$

$={\frac {e^{\lambda \tau }}{\beta }}\int \limits _{\tau }^{\infty }\left(\lambda e^{-\left(\lambda -\beta \right)t}-\lambda e^{-\lambda t}\right)\,dt$

$={\frac {e^{\lambda \tau }}{\beta }}\left({\frac {-\lambda e^{-\left(\lambda -\beta \right)t}}{\left(\lambda -\beta \right)}}+e^{-\lambda t}\right){\Bigg |}_{\tau }^{\infty }$

$={\frac {e^{\lambda \tau }}{\beta }}\left({\frac {\lambda e^{\left(\lambda -\beta \right)\tau }}{\left(\lambda -\beta \right)}}-e^{-\lambda \tau }\right)$

$E[X_{R}|\tau ,\beta ]={\frac {1}{\beta }}\left({\frac {\lambda e^{\beta \tau }}{\left(\lambda -\beta \right)}}-1\right)$

If $\tau =0$

${\frac {\lambda }{\beta \left(\lambda -\beta \right)}}-{\frac {1}{\beta }}$

$={\frac {\lambda }{\beta \left(\lambda -\beta \right)}}-{\frac {\left(\lambda -\beta \right)}{\beta \left(\lambda -\beta \right)}}$

$={\frac {1}{\left(\lambda -\beta \right)}}$

Taking the limit as $\beta \to 0$

$\lim _{\beta \to 0}{\frac {\lambda e^{\beta \tau }-\lambda +\beta }{-\beta ^{2}+\lambda \beta }}$

$=\lim _{\beta \to 0}{\frac {d\left({\frac {\lambda e^{\beta \tau }-\lambda +\beta }{-\beta ^{2}+\lambda \beta }}\right)}{d\beta }}$

$=\lim _{\beta \to 0}{\frac {\lambda \tau e^{\beta \tau }+1}{-2\beta +\lambda }}$

$=\tau +{\frac {1}{\lambda }}$