Monte Carlo Hospital

Transcription

1 Monte Carlo Hospital Let s take a break from springs and things. This model will be inspired by a paper by Schmitz and Kwak [1972]. When a hospital decides to increase the number of beds administrators must consider the increased demand on the other hospital services. More beds mean more patients in which will necessitate more OR s and recovery rooms, RR s. How many Extra OR s and RR s are needed to deal with the extra beds? Our hospital plans to add 144 beds next year. Our model should answer the questions : How many more surgical procedures will MC Hospital perform because of the 1

2 increased bed capacity. How much OR time and space will the surgical procedures require? How much RR time and space will the surgical procedures require? The first question can be answered by examining the past years hospital data. 42% of medical-surgical patients actually require surgery. So that s ' 60 beds will be used by those needing surgery. We can assume that the patient mix will remain the same year after year. That is, 4.5% of total surgical procedures are ophthalmology cases. so we need = 2.7 ophtha beds. Average length of stay in an ophtha bed is 7.4 days. How many optha patients can we cycle thru in one optha bed during a 365 day year? ' 49 So we have 2.7 optha beds and 49 per bed, 2

3 that s the 132 in the chart below. type new cases Ophthalmology 132 Gynecology 282 Uro log y 264 Orthopedic 202 ENT 1098 Dental 715 Other 683 Total 3376 So there will be 3376 more surgeries on top of the 6293 normally for a total of 9669 surgeries. That s ' 27 per day. First question answered. So,howmanyextraOR sdoweneedfor 27 procedures per day? The average length per procedure is 1 hour and MD s work 8 to 5. Is that reasonable? In a sample of 445 procedures, 7 of them took more than 4 hours. One of those patients will occupy an OR for half a day and blow the whole schedule. On any given day the 27 procedure will be 3

4 a random mix, you need some extra capacity to absorb overflow. 4

5 The curve seems to be a negative exponential distribution, the probability that an operation lasts at most t hours is given by P (t) =1 e µt What is the expected value, the variance of this distribution? Side Bar, we plan to simulate a day at MC Hospital with 27 randomly chosen patients. We need to determine the time needed for each operation. We need to generate random variables that are independent and identically distributed according to the above Cumulative Probability Distribution, CPF. All we can produce are iid U i [0, 1] with a PRNG. We have two ways to change these into a desired distribution. 0.1 Rejection Method The rejection method spends at least 2 U s to make one X i f(x) that is a random variable that is drawn from the desired distribution. 5

6 The first chooses an x [a, b] How? The next chooses a y [0, sup f(x)] by the same linear transform. Write an algorithm for the Rejection Method. Note that this method is good for discrete distributions or piecewise continuous, whatever. 0.2 Inversion Method The inversion method only spends one U to get one X so it is more than twice as fast. In a nut shell consider the target cpf f(x) =1 e µx find the inverse and f 1 (U i ) are iid with the proper target distribution. Prove the Inversion Method works: Homework: Generate and graph 300 X 0 i s random iid variables distributed by a Weibull density function with scale parameter b>0 6

7 and shape parameter a>0,thatis,the function f(u; a, b) =ab a u a 1 e ua b a for u 0,andis0 otherwise y u Weibull density functions (a, b) =(.5, 1), (1, 1), (3,.5), (3, 1) The Weibull cumulative distribution is defined by the integral Z x F (x; a, b) =ab a u a 1 e ua b a du =1 e xa b a 0 and the inverse Weibull distribution function µ 1 1 G (α; a, b) =b ln a 1 α Verify the above equations and their relations, let a =3,b =1for your random variables. 7

8 So in our simulated hospital day, we can generate 27 OR times from the cumulative probability distribution P (x) =1 e µx with µ =1.(why?) Let s simulate 27 patients by the following algorithm 1. Generate an OR time.t i 2. If T i < 0.5 then T i =0.5These 0.5 hour operations are of three types ENT RR time 1.5 hours Urology to RR (50%), 1.5 hours Urology, no RR (50%) Ophthalmology, no RR 3. If T i > 0.5 then RR time is 3 hours (major surgery) 4. Surgerystartsat7.5(7:30am) 5. We always leave 0.25 hours between procedures to clean and prep the OR 6. It takes 0.8 hours to transport a patient from OR to RR 8

9 7. An RR bed needs 0.25 hours to clean and prep between patients 8. The first OR vacant is the first filled when the need arrives 9. The first RR bed vacant is the first filled when the need arrives. 10. If there is no empty RR bed available when a new patient arrives, a new bed is created. 9