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Problem 4_ [This problem is based on Example 6.11.]device works for an exponentially distributed amount of time with rate 2 days before breaking down; the repair ti...

Question

Problem 4_ [This problem is based on Example 6.11.]device works for an exponentially distributed amount of time with rate 2 days before breaking down; the repair time is exponentially distributed with rate day, and after repair the device is returned to service. Assume that the device is in working condition at time 0. Explain how the state of the device can be modeled using birth-and-death process where "state means that the device is working and "state 1" means that the device i

Problem 4_ [This problem is based on Example 6.11.] device works for an exponentially distributed amount of time with rate 2 days before breaking down; the repair time is exponentially distributed with rate day, and after repair the device is returned to service. Assume that the device is in working condition at time 0. Explain how the state of the device can be modeled using birth-and-death process where "state means that the device is working and "state 1" means that the device is under repair_ Be sure to state the birth-rates A; and the death-rates /;- (b) Use the Kolmogorov Backward Equations to show that the transition probability func- tions satisfy the equations: "oo (t) 2[Pio(t) Poo (t)]. Pio(t) Poo(t) Pio(t) By solving the equations in part (b) show that the probability that the device is working at time =4is Poo(4) = {(2e 12 + 1)_



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The article "Stochastic Modeling for Pavement Warranty cost Estimation" $(J .$ of Constr. Engr.
and $M$ gmt. $2009 : 352-359$ ) proposes the following model for the distribution of $Y=$ time to
pavement failure. Let $X_{1}$ be the time to failure due to rutting, and $X_{2}$ be the time to failure due to transverse cracking; these two rvs are assumed independent. Then $Y=\min \left(X_{1}, X_{2}\right) .$ The probability of failure due to either one of these distress modes is assumed to be an increasing
function of time $t .$ After making certain distributional assumptions, the following form of the cdf for each mode is obtained:
$\Phi\left(\frac{a+b t}{\sqrt{c+d t+e t^{2}}}\right)$
where $\Phi$ is the standard normal cdf. Values of the five parameters $a, b, c, d,$ and $e$ are $-25.49$ ,
$1.15,4.45,-1.78,$ and. 171 for cracking and $-21.27, .0325,972,-.00028,$ and .00022 for rutting.
Determine the probability of pavement failure within $t=5$ years and also $t=10$ years.


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