Anxious cruise vacationers besiege Tatiana, their excursion coordinator, at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes her 2.5 minutes to find the customers an alternative excursion activity and customers cluster around her at the rate of 20 per hour. About how long does a customer wait in line to receive her valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)12.5 minutes
B)5 minutes
C)21 minutes
D)15 minutes

Question

Quizplus · Accepted Answer

The system described follows the characteristics of an M/M/1 queue, where the arrival rate (λ) is 20 customers per hour and the service rate (μ) is 24 customers per hour (since it takes 2.5 minutes to serve one customer, and there are 60 minutes in an hour, 60/2.5 = 24). The average waiting time in the system (W) for an M/M/1 queue is given by W = 1 / (μ - λ). Substituting the given values, W = 1 / (24 - 20) = 1 / 4 hours = 15 minutes. However, this includes the service time, so the average waiting time in line (Wq) is W - (1/μ), which is 15 minutes - 2.5 minutes = 12.5 minutes.

Anxious Cruise Vacationers Besiege Tatiana, Their Excursion Coordinator, at the Port