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I am working on an operations issue. We process patients in care and pay close attention to their discharge rate, which is more or less calculated as an incidence rate (discharges per 100 patients). We also pay close attention to the length of stay of patients in their facilities.

I don't have a clear understanding of the relationship between survival time and discharge rate. That is, if we have a target discharge rate, what is the ideal median or average length of stay to achieve that target? It is easier at an operational level for the facilities to reduce length of stay than focus on how many patients are discharged. I know this depends on how many individuals their are in the facility at any given time.

Any thoughts would be appreciated.

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If all patients have the same constant risk of discharge, then you have exponential distribution of length of stay and average time of stay = 1/incidence rate. Or, to put it differently, if discharges occur with a Poisson process with intensity $\lambda$, then the average duration is $1/\lambda$. This may be a good starting point for your model.

In reality

1) there are patients with different discharge probabilities. At any point of time, the patients with low discharge probability are overrepresented in the hospital. 2) The individual probability changes over time, in particular patients recover.

These two effects probably make the exponential model above only a crude approximation. You can figure it out if you have the data (I would look at Weibull models) and do the corresponding calculations.

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