Implications of mixed exponential occupancy distributions and patient flow models for health care planning

Abstract

There is considerable evidence that the distribution of the length of time that a patient occupies a bed in a hospital department is best described by a sum of two or three exponential terms, because of the presence of acute care, rehabilitation, and possibly long term care patients in the department. The patient flow models implied by these mixed exponential distributions are presented and fitting them to observed data when the admission rate fluctuates is discussed. Unlike single exponential distributions, mixed exponential distributions imply that the average length of stay of patients currently resident in the department is much longer than the average length of stay of a group of patients discharged over a period of time, so that the latter way of measuring will not correctly indicate what portion of the resources are being used by rehabilitation and long term care patients. Also, the expected additional length of stay increases dramatically with the time already spent in the department. Applications to predicting the effects of policy changes and to long term monitoring of hospital departments are presented. Two American hospitals are analyzed. The occupancy times in the government supported hospital follow a mixed exponential distribution similar to those found in the United Kingdom, but in the private hospital they fit a single exponential distribution, indicating markedly different management practices.