Hi @sverzijl. I’ve taken the liberty to move this question to a new topic (please feel free to edit the title).
If I’m understanding you correctly, one observation consists of a count (total number of cartons of any type, call this y), and an indicator of what types of cartons are involved (say x = (0, 0, 1, 0, 1, 0) if line was generating the 3rd and 5th cartons during that 8 hour period).
I would treat this as having a random variable \vec r, where r_i \sim \mathrm{Beta}(a,b) is the rate of production of the i-th type, a collection of random variables \phi \sim \mathrm{Dir}_0(\alpha) where \phi_i gives the proportion of the 8-hour period spent making the i-th type. By \mathrm{Dir}_0(\alpha) I mean that, for a given binary vector x^{(j)} and proportion vector \phi^{(j)}, \phi^{(j)}=0 wherever x^{(j)} = 0$ and the non-zero components of \phi^{(j)} are Dirichlet-distributed with the appropriate dimensionality.
One way to model this tractably is to pretend that the cartons being produced are interleaved rather than sequentially preduced (all type-1 followed by all type-2 followed by …). Under this assumption, the total count looks like
\mathrm{Count} \sim \mathrm{Pois}(r_1\phi_1) + \mathrm{Pois}(r_2\phi_2) + \dots + \mathrm{Pois}(r_k\phi_k)
=\mathrm{Pois}(r_1\phi_1 + r_2\phi_2 + \dots + r_k\phi_k)
=\mathrm{Pois}(\langle r, \phi \rangle)
=\mathrm{Pois}(\Phi r, \mathrm{shape}=n_\mathrm{obs})
So I think a Poisson likelihood is a defensible choice to use.
If the variance of the Poisson (equal to the mean) bothers you, you could use a Generalized Poisson Distribution which allows for over-dispersion or under-dispersion:
https://journals.sagepub.com/doi/pdf/10.1177/1536867X1201200412
Or use an adjusted normal approximation to the poisson (N(\lambda, \delta \lambda)).