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Failure rate of a system at time 't'

  1. Mar 21, 2011 #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=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. Mar 21, 2011 #2

    Char. Limit

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    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. Mar 21, 2011 #3
    so i just differentiate with respect to μ?
  5. Mar 21, 2011 #4
    so the resulting integration would be:

    -α ln (μ -α) evaluated at 0 and 't' correct?
  6. Mar 21, 2011 #5

    Char. Limit

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    Gold Member

    Well, don't EVALUATE it at those two points. Instead, set that equal to t+C.
  7. Mar 21, 2011 #6
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