Failure rate of a system at time 't'

  1. 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!
     
  2. jcsd
  3. Char. Limit

    Char. Limit 1,986
    Gold Member

    Well, Assuming that your first equation reads as such:

    [tex]\lambda = \lambda_0 \left(1-\frac{\mu}{\alpha}\right)[/tex]

    You should be able to substitute λ=dμ/dt and get a seperable differential equation in μ. Then you differentiate that equation to get λ.

    So you just need to solve:

    [tex]\frac{d\mu}{dt} = \lambda_0 \left(1-\frac{\mu}{\alpha}\right)[/tex]
     
  4. so i just differentiate with respect to μ?
     
  5. so the resulting integration would be:

    -α ln (μ -α) evaluated at 0 and 't' correct?
     
  6. Char. Limit

    Char. Limit 1,986
    Gold Member

    Well, don't EVALUATE it at those two points. Instead, set that equal to t+C.
     
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