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Failure rate of a system at time 't' 
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#1
Mar2111, 06:59 PM

P: 32

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(1μ/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! 


#2
Mar2111, 09:13 PM

P: 610

for the case 1
[tex] \lambda = \lambda_o (1\frac{\mu}{\alpha})[/tex] [tex] \frac{d\mu}{dt}=\lambda_o(1\frac{\mu}{\alpha})[/tex] [tex] \frac{d\mu}{\left(1\frac{\mu}{\alpha}\right)}=\lambda_o \,\, dt [/tex] now integrate within the given limits 


#3
Mar2111, 09:46 PM

P: 32

so the resulting integration would be:
α ln (μ α) evaluated at 0 and 't' correct? 


#4
Mar2111, 10:15 PM

P: 610

Failure rate of a system at time 't'
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 http://www.physicsforums.com/showthread.php?t=483138 


#5
Mar2111, 10:26 PM

P: 32

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



#6
Feb1912, 02:57 PM

P: 2

I am also having problems with this question, I integrated:
[itex]\int\frac{dμ}{(1\frac{μ}{α})}[/itex] = [itex]\int[/itex][itex]\lambda[/itex]_{0}dt And I got: αln(μα) = [itex]\lambda[/itex]_{0}t 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 


#7
Feb1912, 03:25 PM

P: 610

kazo, use properties of logarithm...
[tex]\alpha \ln (\mu\alpha)=\lambda_o t [/tex] [tex] \ln (\mu\alpha)=\frac{\lambda_o t}{\alpha}[/tex] [tex]\mu\alpha= \mbox{exp}\left[\frac{\lambda_o t}{\alpha}\right ] [/tex] [tex]\mu (t) =\alpha +\mbox{exp}\left[\frac{\lambda_o t}{\alpha}\right ] [/tex] and plug this to get [itex]\lambda[/itex] as function of t 


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