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1IMM - DTU02407 Stochastic Processes2012-12-7BFN,DAME/bfnDTU MASTER OF SCIENCE EXAMINATION Page 1 of 6Course: Stochastic Processes Course no.: 02407<strong>Take</strong> <strong>home</strong> <strong>exam</strong>fall 2012Over the last decades stochastic models have been used to some extent in health care modelling,and their usage has increased over the recent years. In the literature we can find different modelsthat optimise the use of hospital resources in order to improve patient care. In this exercise we willinvestigate such stochastic models related to health care.A proper solution will mention the assumptions necessary and state further assumptions wheneverrequired. For some of the questions access to some kind of computer facilities is advantageous; MAT-LAB should be a sufficient tool here. However, in some questions even MAPLE or MATHEMATICAcan be used. It is perfectly acceptable to apply such tools. However, the use should be documentedand described.Part 1: Equipment replacementThe Al Ahli Hospital in Doha, Qatar has to install new equipment in the area of intensive care.This equipment is very expensive and must be handled carefully so it is necessary that the area isempty at the time of installation. Currently there are M patients in the area. In order to proceed withthe installation the intensive care area will not receive more patients before the new equipment hasbeen installed.Every morning a doctor evaluates the condition of the patients to see if they can be discharged.Each patient has a probability p to leave the area and a probability 1 − p to stay, regardless of whathappens with the other patients. No one can enter the area until after the equipment is installed.Question 1Show that the system described can be modelled as a discrete time Markov chain, identify andclassify the states.Problem set continues ...


2Question 2If the system has M patients initially, what is the probability that after t days the system hasM − 1 patients? What is the probability that at day t the equipment will be installed?Suppose now that M = 10 and p = 0.3.Question 3Calculate the probability that there are two patients left in the area, at the beginning of dayt = 5, i.e. before the evaluation by the doctor, and that these two patients are discharged thatday.Now suppose that the installation of the equipment already took place, so patients can reach thearea of pediatrics from the intensive care unit. It is known that the probability that k patients arrivein any given day is q k = λkk! e−λ . The area has 20 beds so if someone comes and there is no bedavailable, the patient is referred to another hospital. Consider that a person who enters into the areais hospitalised for at least one day, and that the probability of being discharged any given day is p.Question 4Make a model for this scenario, then discuss and classify it.Assume now that p = 1 4 .Question 5Show graphically how the utilisation of the pediatrics beds vary with λ.Assume now that λ = 9 2 .Question 6What is the mean time (in days) between two days, where all pediatric beds are used.Part 2: Ambulance dispatchThe hospital serves two districts of the city. The hospital has two ambulances. If the hospitalreceives an emergency call from district i, i ∈ {1, 2} then, they send ambulance i when the twoambulances are free, while if it receives a call when only one ambulance is available, that ambulanceis dispatched. If both ambulances are busy, the call will be served by another hospital. Supposethat emergency calls from districts 1 and 2 arrive randomly with time independent rates λ 1 and λ 2respectively. The service times of calls are independent random variables where the time to handlethe request from district j for ambulance i is exponentially distributed with rate µ ij .Problem set continues ...


3Question 7Formulate a model describing the ambulance dispatch system.Question 8Calculate the probability that at any given time a patient can be served by the ambulanceallocated to the district of the patient. Then calculate the probability that a patient can beserved by an ambulance of the hospital.Suppose now that the parameters of the model are known in a specific time unit, that is λ 1 = 2, λ 2 =4, µ 11 = 4, µ 12 = 2, µ 21 = 2, µ 22 = 6.Question 9What is the fraction of calls by patients that will be served by another hospital?The hospital wants to investigate another strategy where there is no districts and all ambulancesserve all patients. Patients can not be served by another hospital and will have to wait for an availableambulance. Suppose that the calls by patients now occur with an average of 6 calls per time unit. Theaverage rate with which an ambulance can serve a patient is 3 per time unit.Question 10How many ambulances are needed in order to serve all patients?Question 11What is the probability that a patient will wait more than 10 time units to be served providedthat the number of ambulances is the minimal number needed.The hospital is exploring yet another model for ambulance service. This is for a new area, whichis to be served by a single ambulance. Here, the patients will also have to wait until the ambulancebecomes available. The patients will be served on a first come first serve basis. Previous experiencehas shown that the patients in this area calls randomly with no mutual dependence with a rate of 12patients per hour. The time required for the ambulance to serve them can be modelled as a constanttime interval of 2 minutes + a variable time interval which can be reasonably approximated by anexponential distribution with mean 2.Question 12What is the expected value and the variance of the number of patients awaiting the arrival ofthe ambulance.Problem set continues ...


4Part 3: Power supplyThe hospital has an integrated system that includes backup power generators. Its main objectiveis to provide reliable emergency power for critical needs during power outages. The reserve systemsupplies the intensive care unit, operating room and other treatment rooms, as well as lighting,elevators, computer equipment and a portion of the air conditioning system.Suppose that the hospital has only one general generator taking care of everything, but when itbreaks down it has to be repaired immediately. Assuming that the times between each failure leadingto repair are exponentially distributed with parameter mean 10 days and assume approximately thatthe time needed for repair is quite short and can be ignored.Question 13Determine the probability that the generator has 5 or more repairs during the time span of 50days.Now suppose that the time between failures that need repair is given by a distribution with densityf(t)f(t) = t20 e− t 9t 3t4 + e− 5012500and that a failure leading to repair has just occurred.Question 14What is the expected number of failures leading to repair in the first 30 days after that failure.If you cannot calculate the value analytically or numerically then state which formula to useand the ingredients entering the formula.Suppose now that a new repair person is hired at a random point in time.Question 15What is the probability that the new repair person will have to make the first repair within 10days, and what is the expected number of repairs (s)he is expected to make in the first 20 days.Question 16What is the covariance between the time since the last repair and the time to the next repair.Problem set continues ...


6Part 5: Share valueThe Al Ahli Hospital is owned by a public company. Some transformation is made of the share pricein order to make the transformed values be well-described by some adequate mathematical model.After transformation one can interpret the transformed values at time t as the result of a number ofsmall cumulated independent changes. The initial value of the transformed values is set to 0 and thevariance of the transformed value at time t is proportional to t with proportionality constant 1 (one).Question 23Calculate the probability that the transformed share price reaches a value of 4 within 3 timeunits.Question 24What is the mean time for the transformed share price to reach a value of 4? If the mean timeexists then state the value, if not, then show that it does not exist.After a change in the macro economic conditions the assumptions for the model are no longer valid.The underlying mechanism is unchanged but now also the mean of the transformed value of the shareprice changes over time. Like the variance the mean can be considered to be proportional to t withproportionality constant 1 2, while the proportionality constant of the variance is now 4.Question 25Calculate the probability that the transformed share price ever reaches a value of −1.It turns out that the transformation needed to get the transformed share price was to take the naturallogarithm of the share price.Question 26Calculate the probability that the share price reaches a value of 1 2before it reaches a value of 2.For the last question we still consider the share price of the Al Ahli Hospital. This time we don’t knowthe proportionality constant for the mean of the transformed values. The variance parameter of thetransformed values will be 4 as in the previous question. We assume that the proportionality constantof the mean of the transformed values is a save indicator for the general economy of the country. As asimplification we assume that if the proportionality constant of the mean in the transformed processis 5 2then the economy is in an upward drift while the economy is on a down slop if the proportionalityconstant of the mean is 1. A share holder is observing the share price and will only buy further sharesif certain that the general economy is growing. On the contrary the share holder sells shares whencertain the economy is on a down slope. The share holder wants to limit the probability of buyingshares when the economy is on a down slope to 5% and to limit the probability of selling shares whenthe economy is in an upward drift to 10%.Question 27Determine the share holders decision criteria for buying and selling.Problem set ends.

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