This is just a hint; actually yours is a question I planned to ask myself sooner or later: What's the model for the gamma distribution with non-integer parameter $\lambda$?

Since possibly the most relevant aspect is the semigroup law w.r.to convolution, my feeling is that it should describe properly waiting times for reaching given amount $\lambda$ of (continuous) mass, arriving randomly under independence assumptions, like e.g. rain, falls of powder on the soil. I'm very interested in further more precise answers.

This is just a hint (actually yours is a question I planned to ask here myself sooner or later): What's the model for the gamma distribution with non-integer parameter $\lambda$?

Since possibly the most relevant aspect is the semigroup law w.r.to convolution, my feeling is that it should describe properly waiting times for reaching given amount $\lambda$ of mass, arriving continuously and randomly under independence assumptions, like e.g. rain, falls of powder on the soil. I'm very interested in further more precise answers.