Distribution of a random sum of random non-iid variables

#1
I'm trying to find the probability distribution of a sum of a random number of variables which aren't identically distributed. Here's an example:

John works at a customer service call centre. He receives calls with problems and tries to solve them. The ones he can't solve, he forwards them to his superior. Let's assume that the number of calls he gets in a day follows a Poisson distribution with mean \(\mu\). The difficulty of each problem varies from pretty simple stuff (which he can definietly deal with) to very specialized questions which he won't know how to solve. Assume that the probability \(p_i\) he'll be able to solve the i-th problem follows a Beta distribution with parameters \(\alpha\) and \(\beta\) and is independent of the previous problems. What is the distribution of the number of calls he solves in a day?

More formally, I have:

\(Y\) = I(\(N\) > 0)\(\sum\)\(X_i\) for \(i\) = 0, 1, 2, ..., \(N\)

where \(N\) \(\sim\) Poisson(\(\mu\)) , (\(X_i\) | \(p_i\)) \(\sim\) Bernoulli(\(p_i\)) and \(p_i\) \(\sim\) Beta(\(\alpha\), \(\beta\))

Note that, for now, I'm happy to assume that the \(X_i\)'s are independent. I'd also accept that the parameters \(\mu\), \(\alpha\) and \(\beta\) do not affect each other although in a real-life exampl of this when \(\mu\) is large, the parameters \(\alpha\) and \(\beta\) are such so that the Beta distribution has more mass on low success rates \(p\). But let's ignore that for now.

I can calculate \(P\)(\(Y\) = 0) but that's about it. I can also simulate values to get an idea of what the distribution of \(Y\) looks like (it looks like Poisson but I don't know if that's down to the numbers of \(\mu\), \(\alpha\) and \(\beta\) I tried or whether it generalises, and how it might change for different parameter values). Any idea of what this distribution is or how I could go about deriving it?