In short, how would one sample an Exponential Distribution with an exponentially-increasing rate parameter, e.g., $\lambda=e^t$? What distribution does such a random variable follow?
Background: The Exponential Distribution models "time until failure" of, say, lightbulbs. It is parametrized by a constant parameter $\lambda$ called the failure rate that is the average rate of lightbulb burnouts. If $\lambda$ is time-varying, say, proportionally to a power of time, then it turns out that the time until failure is distributed by the more general Weibull distribution. This question involves this relationship, although the time-variance of $\lambda$ is not made explicit.
My question is: What if $\lambda$ varies exponentially in time? Is there a distribution for this case? More simply, how would I sample a random variable that is exponentially distributed with time-varying rate parameter $\lambda = e^t$?
I've searched Google and Wikipedia without success. I'd really appreciate any advice or pointers. Thanks in advance.