# Failure rate of a system at time 't'

1. Mar 21, 2011

### francisg3

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. Mar 21, 2011

### Char. Limit

$$\lambda = \lambda_0 \left(1-\frac{\mu}{\alpha}\right)$$

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:

$$\frac{d\mu}{dt} = \lambda_0 \left(1-\frac{\mu}{\alpha}\right)$$

3. Mar 21, 2011

### francisg3

so i just differentiate with respect to μ?

4. Mar 21, 2011

### francisg3

so the resulting integration would be:

-α ln (μ -α) evaluated at 0 and 't' correct?

5. Mar 21, 2011

### Char. Limit

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

6. Mar 21, 2011