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Failure rate of a system at time 't'

by francisg3
Tags: failure, rate, time
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francisg3
#1
Mar21-11, 07:36 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(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!
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Char. Limit
#2
Mar21-11, 07:56 PM
PF Gold
Char. Limit's Avatar
P: 1,951
Quote Quote by francisg3 View Post
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!
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]
francisg3
#3
Mar21-11, 08:06 PM
P: 32
so i just differentiate with respect to μ?

francisg3
#4
Mar21-11, 09:46 PM
P: 32
Failure rate of a system at time 't'

so the resulting integration would be:

-α ln (μ -α) evaluated at 0 and 't' correct?
Char. Limit
#5
Mar21-11, 10:05 PM
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P: 1,951
Well, don't EVALUATE it at those two points. Instead, set that equal to t+C.
issacnewton
#6
Mar21-11, 10:22 PM
P: 610
francis has started two threads for the same question. another thread at

http://www.physicsforums.com/showthread.php?t=483125

I am answering the same question at this thread. Beware francis


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