Janet Wilson writes:
> queen quality (which should be defined)

The question of queen quality has been studied for decades. For example, Dave Tarpy has authored more than 40 papers on the subject over a span of more than 20 years:

Hatch, S., Tarpy, D. R., & Fletcher, D. J. C. (1999). Worker regulation of emergency queen rearing in honey bee colonies and the resultant variation in queen quality. Insectes Sociaux, 46(4), 372-377.
... 
McAfee, A., Milone, J. P., Chapman, A. A., Foster, L. J., Tarpy, D. R., & Pettis, J. (2020). Candidate stress biomarkers for queen failure diagnostics. bioRxiv.

Most of what beekeepers "believe" about queen rearing and queen quality is not supported by research:

Lee, K. V., Goblirsch, M., McDermott, E., Tarpy, D. R., & Spivak, M. (2019). Is the brood pattern within a honey bee colony a reliable indicator of queen quality?. Insects, 10(1), 12.

I would suggest reading some of these and considering there may not be a simple mathematical answer to a complex biological question.

Peter Borst
Ithaca NY  USA

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