Your inquiry really encompasses two separate questions:
1. What percentage of discharges occur by the 11:00 desired time? This is a binary variable, illustrated with a p-chart.
2. What is the distribution of discharges by hour throughout the day (the Poisson distribution you referred to previously)?
The second will give you some idea of how much movement will be necessary to achieve whatever goal is set for the first measure. For instance, moving a discharge up from noon to 11:00 would be more feasible than moving one from 22:00 up by 11 hours.
After that it is probably a slog to examine those “feasible” post-11:00 discharges by service, by physician, etc. to look for patterns for possible interventions. It could be (to choose one real-life example from our system), for instance, that the patient is ready to go home at 10:00, but it takes the pharmacy more than one hour to fill discharge prescriptions because of the crunch of morning discharge orders.
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