I want to understand the appropriate way to regress a duration variable on a count variable. For instance: I have data on turn around time of a process in a hospital ward. The ward admits patients with diverse clinical specialties. I have the daily average turnaround time of the process as the dependent variable. The independent variable is the count of medical specialties on a given day. The count variable has non-zero values ranging from 1 to 12.
What is good depends on the relationship in practice.
There is no reason I can imagine to rule out using the count as it comes if the relationship is linear.
As I understand it the intercept is unattainable, as zero specialties won't ever be observed, but that's not a novelty. If people's weights are predictable from their heights, we don't object that zero height cannot occur.
We would need to see the data to say more; I wouldn't be surprised if say log of time versus square root of count worked a bit better. Nor would I be surprised by enormous scatter and a weak relationship.