Abstract

Emergency medical service is an important part of the health care delivery system, and the optimal allocation of resources and their efficient utilization are essentially demanded. Since these conditions are the prerequisite to prompt treatment which, in turn, will be crucial for life saving and in reducing the undesirable sequelae of the event. This study, taking the hyperbaric chamber for carbon monoxide poisoning as an example, is to develop a stochastic approach for solving the problems of optimal allocation of such emergency medical facility in Korea. The hyperbaric chamber, in Korea, is used almost exclusively for the treatment of acute carbon monoxide poisoning, most of which occur at home, since the coal briquette is used as domestic fuel by 69.6 percent of the Korean population. The annual incidence rate of comatous and fatal carbon monoxide poisoning is estimated at 45.5 per 10,000 of coal briquette-using population. It offers a serious public health problem and occupies a large portion of the emergency outpatients, especially in the winter season. The requirement of hyperbaric chambers can be calculated by setting the level of the annual queueing rate, which is here defined as the proportion of the annual number of the queued patients among the annual number of the total patients. The rate is determined by the size of the coal briquette-using population which generate a certain number of carbon monoxide poisoning patients in terms of the annual incidence rate, and the number of hyperbaric chambers per hospital to which the patients are sent, assuming that there is no referral of the patients among hospitals. The queueing occurs due to the conflicting events of the "arrival" of the patients and the "service" of the hyperbaric chambers. Here, we can assume that the length of the service time of hyperbaric chambers is fixed at sixty minutes, and the service discipline is based on "first come, first served". The arrival pattern of the carbon monoxide poisoning is relatively unique, because it usually occurs while the people are in bed. Diurnal variation of the carbon monoxide poisoning can hardly be formulated mathematically, so empirical cumulative distribution of the probability of the hourly arrival of the patients was used for Monte Carlo simulation to calculate the probability of queueing by the number of the patients per day, for the cases of one, two or three hyperbaric chambers assumed to be available per hospital. Incidence of the carbon monoxide poisoning also has strong seasonal variation, because of the four distinctive seasons in Korea. So the number of the patients per day could not be assumed to be distributed according to the Poison distribution. Testing the fitness of various distributions of rare event, it turned out to be that the daily distribution of the carbon monoxide poisoning fits well to the Polya-Eggenberger distribution. With this model, we could forecast the number of the poisonings per day by the size of the coal-briquette using population. By combining the probability of queueing by the number of patients per day, and the probability of the number of patients per day in a year, we can estimate the number of the queued patients and the number of the patients in a year by the number of hyperbaric chambers per hospital and by the size of coal briquette-using population. Setting 5 percent as the annual queueing rate, the required number of hyperbaric chambers was calculated for each province and for the whole country, in the cases of 25, 50, 75 and 100 percent of the treatment rate which stand for the rate of the patients treated by hyperbaric chambers among the patients who are to be treated. Findings of the study were as follows. 1. Probability of the number of patients per day follows Polya-Eggenberger distribution. Hourly arrival pattern of the patients turned out to be bimodal, the large peak was observed in 7:00~8:00 a.m., and the small peak in 11:00~12:00 p.m. 2. In the cases of only one or two hyperbaric chambers installed per hospital, the annual queueing rate will be at the level of more than 5 percent. Only in case of three chambers, however, the rate will reach 5 percent when the average number of the patients per day is 0.481. 3. According to the results above, a hospital equipped with three hyperbaric chambers will be able to serve 166, 485, 83, 242, 55, 495 and 41, 620 of population, when the treatment rate are 25, 50, 75 and 100 percent. 4. The required number of hyperbaric chambers are estimated at 483, 962, 1,441 and 1,923 when the treatment rate are taken as 25, 50, 75 and 100 percent. Therefore, the shortage are respectively turned out to be 312, 791, 1,270 and 1,752. The author believes that the methodology developed in this study will also be applicable to the problems of resource allocation for the other kinds of the emergency medical facilities.