Gunicorn lacks native busy worker count metrics

Gunicorn does not expose built-in metrics for the number of workers currently processing requests (busy) versus idle workers. Without this visibility, it is difficult to determine if the worker pool is appropriately sized or to proactively detect saturation before users experience latency.

Implement custom busy worker tracking using Gunicorn's pre_request and post_request hooks with a multiprocessing.Semaphore to maintain an atomic counter across worker processes. Store the busy count in a custom request header (e.g., x-busy) and include it in the access_log_format using %({x-busy}i)s. Monitor this metric to right-size the worker pool.