This is related to my previous question: How to update Poisson conjugate prior with observations of arrival time instead of counts?
Using the same notation, suppose $N \sim \operatorname{Pois}(\mu)$ is a Poisson distributed random variable, with interarrival times between observations distributed accordingly as $A \sim \operatorname{Exp}(\mu)$. Let $\mu$ be distributed according to the conjugate prior $\mu \sim \operatorname{Gamma}(\alpha, \beta)$.
Suppose I start waiting for the next observation of my Poisson process, and I have waited a long time $t$ without observing any, that is, in time $t$, there are $n = 0$ observations. Is it possible to derive a posterior distribution for $\mu|n = 0, t$? Would it be:
$$\mu|n = 0, t \sim \operatorname{Gamma}(\alpha, \beta + t)$$
I am not sure how this is consistent with Wikipedia article on updating the prior of an exponential distribution, as that explicitly relies on at least one observation of an interarrival time.