My course notes (3rd-year module in Bayesian Statistics, unpublished) contain the following section.
Assume we have data on the number of people queuing at an ATM at a specific hour for several randomly chosen days, and that we are interested in the mean time between two consecutive arrivals to the queue at some arbitrary point within this time.
First, let $\textbf{x} = \{x_1,\dots,x_n\}$ represent the observed data and assume these are a random sample from $\text{Po}(x_i\;|\;\lambda)$. We will use the conjugate prior $\text{Ga}(\lambda\;|\;a,b)$ to represent our prior knowledge, thus $$\pi(\lambda\;|\;\textbf{x})\propto\left[\lambda^se^{-n\lambda}\right]\times\left[\lambda^{a-1}e^{-b\lambda}\right]=\text{Ga}(\lambda\;|\;a^\ast,b^\ast),$$ where $a^\ast=a+s$ and $b^\ast=b+n$, with $s=\sum_{i=1}^nx_i$ a sufficient statistic for $x_i$.
Using standard queuing theory, the time within two consecutive arrivals, $y$, is distributed $\text{Ex}(y\;|\;\lambda)$. Thus, the predictive distribution is $$f(y\;|\;\textbf{x})=\int_0^\infty\lambda e^{-y\lambda}\frac{(b^\ast)^{a^\ast}}{\Gamma(a^\ast)}\lambda^{a^\ast-1}e^{-\lambda b^\ast}\;\text{d}\lambda=\frac{a^\ast}{b^\ast}\left(1+\frac{y}{b^\ast}\right)^{-(a^\ast+1)}.$$ Finally, the expected predictive time between consecutive arrivals is $$\mathbb{E}[y\;|\;\textbf{x}]=\int_0^\infty y\frac{a^\ast}{b^\ast}\left(1+\frac{y}{b^\ast}\right)^{-(a^\ast+1)}\;\text{d}y=\frac{b^\ast}{a^\ast-1}.$$
But, by my reasoning, \begin{align} & \begin{aligned} f(y\;|\;\textbf{x}) & = \frac{a^\ast}{b^\ast}\left(1+\frac{y}{b^\ast}\right)^{-(a^\ast+1)} \\ & = \frac{a^\ast b^{\ast a^\ast}}{(y+b^\ast)^{a^\ast+1}} \\ & = \text{Pa}(y+b^\ast\;|\;a^\ast,b^\ast) \end{aligned} \\[1em] \therefore \hspace{1em} & \begin{aligned}[t] \mathbb{E}[y\;|\;\textbf{x}] & = \mathbb{E}[y+b^\ast\;|\;y\sim\text{Pa}(a^\ast,b^\ast)] \\ & = \mathbb{E}[y\;|\;y\sim\text{Pa}(a^\ast,b^\ast)] + b^\ast \\ & = \frac{a^\ast b^\ast}{a^\ast-1} + b^\ast \\ & = \frac{(2a^\ast-1)b^\ast}{a^\ast-1} \\ & \neq \frac{b^\ast}{a^\ast-1} \text{ (unless $a^\ast=1$)}. \end{aligned} \end{align}
What's gone wrong?