Failure rate of a system at time 't'

[francisg3]

I need to solve the following problem for a school assignment.

Let λ(t) denote the failuer rate of a system at time 't'. The failure rate is simple the number of failures in unit time. For example, if the unit time is one day, then λ is the average of failures per day. Let μ(t) denote the total number of failures from the first release (time t=0) until the current time, 't'. Then we have

(1) λ= dμ/dt

(2) μ = ∫λ(T) where the limits of integration are T=0 (lower) and T=t (upper)

Two models are used for estimating λ and μ. In the forumlae below, λ0 is the failure rate at time t=0, and α and β are constants

λ=λ0(1-μ/α)

λ=λ0e^- β μ



Use (1) or (2) to find λ and μ as functions of time for each model.



...I just need some direction. Thanks!

 

[issacnewton]

for the case 1

$$\lambda = \lambda_o (1-\frac{\mu}{\alpha})$$

$$\frac{d\mu}{dt}=\lambda_o(1-\frac{\mu}{\alpha})$$

$$\frac{d\mu}{\left(1-\frac{\mu}{\alpha}\right)}=\lambda_o \,\, dt$$

now integrate within the given limits

 

[francisg3]

so the resulting integration would be:

-α ln (μ -α) evaluated at 0 and 't' correct?

 

[issacnewton]

no...check the integration...remember to integrate both sides

and francis, i see that you have doubled up this thread...two threads are off
by half an hour. This is NOT a good practice. Somebody will report this to the mods.

Another thread going at

https://www.physicsforums.com/showthread.php?t=483138

 

[francisg3]

I know I double posted, realized that this was not homework/coursework section but I don't know how to delete a post. Sorry.

 

[kazo0]

I am also having problems with this question, I integrated:

$\int\frac{dμ}{(1-\frac{μ}{α})}$ = $\int$$\lambda$₀dt

And I got:

-αln(μ-α) = $\lambda$₀t

I'm not sure if this is going in the right direction and what would I have to do after this in order to find μ(t) and λ(t)?

Thanks

 

[issacnewton]

kazo, use properties of logarithm...

$$-\alpha \ln (\mu-\alpha)=\lambda_o t$$

$$\ln (\mu-\alpha)=-\frac{\lambda_o t}{\alpha}$$

$$\mu-\alpha= \mbox{exp}\left[-\frac{\lambda_o t}{\alpha}\right ]$$

$$\mu (t) =\alpha +\mbox{exp}\left[-\frac{\lambda_o t}{\alpha}\right ]$$

and plug this to get $\lambda$ as function of t

 

[Char. Limit]

Forgot a +C. That's important.