Final answer:
Using the Poisson arrival process and the constant service rate, we calculated the average time between customer arrivals and the average time for three customers to arrive. We also determined that the exponential distribution is an appropriate model for this scenario.
Step-by-step explanation:
When analyzing the vending machine system, we have to calculate the traffic density, which is the balance between arrival rates and service rates. Since customers arrive at a mean rate of 76 per hour and service takes 30 seconds per cup, we first establish the average time between customer arrivals and the service rate.
The average interval between arrivals (A) is 60 minutes divided by the arrival rate, so A = 60/76 which is approximately 0.789 minutes between two successive arrivals. Since every two minutes on average one customer arrives, it will take approximately 2 minutes * 3 = 6 minutes on average for three customers to arrive.
To find the number of customers in the system (L), we use the formula L = λ / (u - λ), where λ is the arrival rate and u is the service rate. In our case,λ = 76/60 customers per minute and u is 2 cups per minute (since one cup is served every 30 seconds). We can now calculate L = (76/60) / (2 - 76/60) which gives us the average number of customers in the system.
For the time spent in the system (W), we use the formula W = 1 / (u - λ). By using the values we have, we can calculate the average time a customer spends in the system.
The probability of an event occurring in a given interval under the Poisson distribution can be calculated using the exponential distribution. With an average arrival rate of 30 customers per hour, the time between arrivals is exponentially distributed, which makes the exponential distribution a reasonable model for this scenario.