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The vaccination process in Tukey for preventing COVID-19 disease has just started. Consider a family health center in Güngören, Istanbul where citizens wait in a single line to be vaccinated and a single doctor is responsible for the vaccination. Based on his expert knowledge, the doctor believes that the time that a person spends in tht family health center from her arrival to the completion of vaccination process is 30 minutes and the average amount of time that passes between the arrivals of two people is exponentially distributed with a mean value of 5 minutes. Let us also assume that the time that is necessary for the doctor to vaccinate a person is also exponentially distributed. a) Define the states and draw the birth-death model for this process. b) Now assume that the number of doctors that are responsible for vaccination in this family health center is two. Assuming that the service rates for each doctor are the same as in part (a) and independent from each other, find the expected amount of time that a person spends waits in the queue before she gets vaccinated. c) Now suppose that regulations require a policy that does not allow more than 3 people to be present at the health center at the same time this does not include the two doctors). Write the corresponding balance equations for this new system (do not solve the equations!).
 
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