Question

Frenzied 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 many customers are clustered around them, either receiving service or waiting in line to receive their valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?

A. 17.86
B.. 14.17
C. 13.25
D. 19.66

Answer 1

Using queuing theory and the given arrival and service rates, we calculate the average number of customers in the system. Since the arrival rate is lower than the service rate, the system should not be overwhelmed. The exact number will depend on the Poisson and exponential distributions.

Step-by-step explanation:

To answer the question about how many customers are clustered around the excursion coordinators and their assistants, either receiving service or waiting in line, we need to use the queuing theory principles and calculations that involve the Poisson distribution for arrivals and the exponential distribution for service times.

Given the rate of arrivals is 200 per hour, on average, one customer arrives every 0.3 minutes (60 minutes/200 customers). With fifteen service agents (Tatiana and her fourteen assistants), they can handle a combined total of 15 customers every 4 minutes. This equates to 3.75 customers per minute (15 customers / 4 minutes). Since the arrival rate is less than the service rate (0.3 customers per minute < 3.75 customers per minute), the customer service system should not be overwhelmed in the long term.

However, the exact number of people clustered at any given moment will depend on the stochastic nature of the Poisson arrivals and the exponential service times. As the service times are exponentially distributed, the average number of customers in the system can be given by the formula L = λ / (μ - λ), where λ is the arrival rate and μ is the service rate. Plugging in the given rates (λ = 200/60 customers per minute and μ = 15/4 customers per minute), we can calculate the average number of customers in the system.