Final answer:
The initial scenario of a vending machine with a service time of 30 seconds and customer arrival rate of 80 per hour results in an average interarrival time of 45 seconds per customer. Any additional scenarios mentioning different arrival rates seem to be separate and unrelated to the question at hand.
Step-by-step explanation:
Understanding Arrival Rates and Service Times in Queuing Theory
Given the information that a vending machine dispenses drinks with a constant service time of 30 seconds per cup and that customers arrive at a mean rate of 80 per hour, we can analyze this system using the principles of queuing theory. To begin, we calculate the time between arrivals as follows:
- There are 60 minutes in an hour, and with 80 customers arriving per hour, we can divide 60 minutes by 80 customers to find the average time between arrivals.
- This calculation gives us 0.75 minutes or 45 seconds as the average time between arrivals of each customer.
With a service time of 30 seconds per cup and an arrival rate that exceeds this (one arrival every 45 seconds), it is apparent that the demand may exceed the service capacity during peak times, potentially leading to wait times and queuing.
Regarding part a of the given information, which talks about a scenario of 30 customers per hour, leading to a different result of one customer every two minutes, it seems incorrect based on the initial customer arrival rate of 80 per hour provided in the question. This is likely a typo or a separate scenario not linked to the given question.
For part b, calculations related to temperature, heat transfer, pace of evaporation, and excess time are part of physics and thermodynamics, which is not relevant to the initial scenario presented about average arrival times and service rates for hot drinks in a vending machine scenario.