# failure rate of a system at time 't'

by francisg3
Tags: failure, rate, time
 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(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!
 P: 602 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
 P: 32 so the resulting integration would be: -α ln (μ -α) evaluated at 0 and 't' correct?
P: 602

## 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

 P: 2 I am also having problems with this question, I integrated: $\int\frac{dμ}{(1-\frac{μ}{α})}$ = $\int$$\lambda$0dt And I got: -αln(μ-α) = $\lambda$0t I'm not sure if this is going in the right direction and what would I have to do after this in order to find μ(t) and λ(t)? Thanks
 P: 602 kazo, use properties of logarithm... $$-\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