Final answer:
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.