How can I generate inter-arrival times from a Weibull distribution, p.d.f. $f(t)=\frac{\beta}{\alpha}\big(\frac{t}{\alpha}\big)^{\beta-1}\exp\big(-\frac{t}{\alpha}\big)^\beta$, where $\beta$ is the shape parameter and $\alpha$ is the scale parameter?
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The Weibull is a powered exponential distribution. Its survival function
$$q = 1 - F(x) = e^{(-x/\alpha)^\beta}$$
therefore has a particularly tractable form: it is readily inverted to give
$$x = \alpha(-\log(q))^{1/\beta}.$$
You therefore need only generate $q$ uniformly in the interval $(0,1)$ and compute $x$. Here's an R example:
alpha <- 2
beta <- 3
n <- 1e5 # Sample size
x <- alpha * (-log(runif(n))) ^ (1/beta) # Sample
hist(x, freq=FALSE, breaks=50)
curve(dweibull(x, beta, alpha), add=TRUE, col="Red", lwd=2)
The work is all done on a single line labeled "Sample". The output is a histogram on which the Weibull density function is plotted. The agreement in this simulation is excellent.
(R of course offers a built-in generator rweibull. The point of this answer is to show how it might work and to provide enough details to enable a generator to be implemented in any language.)
