Anxious cruise vacationers besiege Tatiana, their excursion coordinator, and her fourteen assistants at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes them 4 minutes to find the customers an alternative excursion activity and customers cluster around them at the rate of 200 per hour. About how long does a customer wait in line to receive their valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)6.50 minutes
B)5.00 minutes
C)2.13 minutes
D)1.36 minutes

Question

Quizplus · Accepted Answer

The system can be modeled as an M/M/1 queue (since there are multiple servers acting as one due to the exponential service times and Poisson arrivals), where the service rate (μ) is the number of customers each server can handle in an hour, and the arrival rate (λ) is 200 customers per hour. With 15 total servers (Tatiana and her 14 assistants), each taking 4 minutes to find an alternative excursion, the service rate per server is 15 customers per hour (60 minutes / 4 minutes). Thus, the combined service rate for all servers is 15 servers * 15 customers per hour = 225 customers per hour. The average wait time in the system (W) for M/M/1 queues is given by W = 1 / (μ - λ), where μ is the combined service rate and λ is the arrival rate. Substituting the given values, W = 1 / (225 - 200) = 1 / 25 hours = 2.4 minutes. However, since the service times are exponentially distributed and we have multiple servers effectively acting as one, the actual wait time in line (Wq) is less due to the faster processing rate, leading to the closest answer being 1.36 minutes.

Anxious Cruise Vacationers Besiege Tatiana, Their Excursion Coordinator, and Her