(Solution) - A major New York bank commissioned a study to measure -(2025 Original AI-Free Solution)

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A major New York bank commissioned a study to measure the service levels of its over 500 ATMs in N.Y.. The ATMs were located in pairs inside vestibules attached to bank branches. The team used the finite queue version of multi server queuing model (i.e., Model 6) and suggested that the proportion of customers who balk (i.e., refuse to join the queue) should be used as the measure of service as opposed to the average wait time or average queue length, [t was suggested that if the probability of balking during busiest times is greater than 5%, then a vestibule should be expanded and an additional ATM installed in it. Consider a specific vestibule with 2 ATMs in it. There is space for 10 people in the vestibule (2 being served, 8 in line); average service time is 1.28 minutes per customer (approximately Exponential), and average arrival rate is 86 customers per hour (approximately Poisson) during the busiest times of the week (noon-1 pm or 1-2 pm weekdays). Determine the probability of balking at this vestibule. Should the vestibule be enlarged?