looking for a continuous distribution analogue of ZI Poisson?

[#1]

I have product complaint data which are counts, but the areas of opportunity are different because batch sizes vary.  Complicating matters is the fact that many of our batches have zero complaints.  These data are generally analyzed on a PPM basis - the number of complaints are divided by the batch size and then multiplied by a million, making it a continuous positive distribution with a disproportionately large number of zeroes.  Is there a continuous distribution I can use that produces similar results to a zero-inflated Poisson on the count data?  I have attached a modified version of the Fishing.jmp sample data file to illustrate the issue.

[#2]

I am wondering if the zero-inflated beta binomial might be part of the solution, although it will not allow you to use the data in parts per million.

I have attached a version of the table with a script for a model using this distribution in Gen Reg in JMP Pro.

Hopefully this is helpful.

[#3]

There is a class of ZI distributions supported in the Life Distribution platform. It includes: ZI Lognormal, ZI Loglogistic, ZI Weibull, and ZI Frechet.

I attach the updated data table with a new script. This platform, however, does not take covariates.

Here is a screenshot of fitted ZI Loglogistic.