The department chair is out of town at a conference and you, as the assistant department chair, find yourself besieged by dozens of students needing permission to get into a very popular (but closed)class. Student arrivals are governed by the Poisson distribution with a mean of 5 minutes. Telling the students that they can't get into the closed class takes about 3 minutes, and this average is exponentially distributed. About how long does it take a student to wait in line and receive your valuable service?
A)0.125 minutes
B)0.075 hours
C)4.5 minutes
D)7.5 minutes

Question

Quizplus · Accepted Answer

The average arrival rate of students is governed by a Poisson distribution with a mean of 5 minutes, which means on average, a student arrives every 5 minutes. The service time, which is the time it takes to tell a student they can't get into the class, is exponentially distributed with a mean of 3 minutes. To find the average time a student spends waiting in line and receiving service, we can use the formula for the average time in the system in a M/M/1 queue, which is 1/(μ - λ), where μ is the service rate (1/service time) and λ is the arrival rate (1/arrival time). Here, μ = 1/3 minutes^-1 and λ = 1/5 minutes^-1. Plugging these values into the formula gives 1/(1/3 - 1/5) = 1/(2/15) = 15/2 = 7.5 minutes.

The Department Chair Is Out of Town at a Conference