A professor holds office hours during final exam week and discovers much to his dismay that his sudden popularity affords him little time to engage in his favorite activity, watching cat videos.Fifteen students per hour arrive at his office and he patiently answers questions, averaging three minutes per student.On average, how many students are waiting to ask questions?
A)2.00
B)2.25
C)2.50
D)2.75

Question

Quizplus · Accepted Answer

Since the professor spends an average of three minutes per student, he can see 20 students in an hour (60 minutes / 3 minutes per student). However, 15 students arrive per hour. This means there is a constant flow of students, and at any given time, there are students waiting. To calculate the average number of students waiting, we consider the rate at which students are served and the rate at which they arrive. The difference in these rates (20 students served per hour - 15 students arriving per hour = 5 students per hour) indicates how quickly the queue decreases. However, since the question asks for how many are waiting on average, we look at the accumulation over time. Given the constant arrival rate and service rate, the queue will build up. Since he can serve 20 students an hour but only 15 arrive, the queue builds at the rate they arrive because he is serving faster than they are arriving, which means no buildup actually occurs, leading to a misunderstanding in the initial explanation. The correct approach to find the average number of students waiting involves calculating the average wait time or the average queue length directly from the given information, considering the process as a queueing system. In a system where the service rate is higher than the arrival rate, the direct calculation should consider the arrival rate and service time to find the average number of students waiting. Given the mistake in the initial explanation about the buildup, the correct calculation involves the arrival rate (15 students per hour) and the service rate (20 students per hour), focusing on how the queue operates within these constraints. The average number of students waiting is calculated based on the rate at which students are served and the time it takes to serve them, leading to the formula for the average number of people waiting in the system (L = λW), where λ is the arrival rate and W is the average time a student spends in the system. Given the error in the buildup explanation, the focus should be on the correct queueing theory formula and its application to the given scenario. The correct answer, taking into account the proper calculation method for the average number of students waiting, involves re-evaluating the given rates and the system's operation without the incorrect assumption of a buildup due to the service rate being higher than the arrival rate. The calculation should accurately reflect the dynamics of the queueing system, considering both the arrival and service rates to determine the average number of students waiting at any given time.

A Professor Holds Office Hours During Final Exam Week and Discovers