Final answer:
To calculate the average number of customers at the coffee urn, the given arrival rate (4.0 per minute) and service rate (conversion of 11 seconds per customer) are used to determine the traffic intensity, resulting in an average of 11/15 customers at the urn.
Step-by-step explanation:
The question is about determining the average number of customers at a coffee urn with a known arrival rate and service time distribution. Arrivals at the urn follow a Poisson distribution at a rate of 4.0 per minute, and the service times are exponentially distributed with an average of 11 seconds per customer. To find the average number of customers at the coffee urn, we use the traffic intensity formula \(\rho = \lambda / \mu\), where \(\lambda\) is the arrival rate and \(\mu\) is the service rate.
The arrival rate is given as 4 customers per minute, and the service rate can be calculated as the reciprocal of the average service time. Given 11 seconds per customer, we convert this to minutes to find \(\mu\), which would be \(60 / 11\) customers per minute. The traffic intensity \(\rho\) is then \(4 / (60 / 11)\), which simplifies to \(11 / 15\). This value represents the average number of customers being served at the coffee urn at any given time.