Final answer:
To keep the average number of customers in the service system to four, the average time spent in the system should be 12 minutes. The probability of more than four customers in line can be calculated using the Poisson distribution. The average waiting time in line for each customer is 48 minutes.
Step-by-step explanation:
To keep the average number of customers in the service system to four, we can use Little's Law which states that the average number of customers in a system is equal to the arrival rate multiplied by the average time spent in the system. In this case, the arrival rate is 20 customers per hour, and we want the average number of customers to be four. Therefore, the average time spent in the system should be 4/20 = 0.2 hours or 12 minutes.
b. To find the probability that more than four customers are in line and being served, we can use the Poisson distribution. The Poisson distribution allows us to calculate the probability of a certain number of events occurring in a fixed interval of time, given the average rate at which the events occur. In this case, the average rate is 20 customers per hour, so we can calculate the probability of more than four customers using the Poisson probability formula.
c. To find the average waiting time in line for each customer, we can use Little's Law again. The average waiting time is equal to the average number of customers in the system multiplied by the average time spent in the system. With an average of four customers in the system and an average time spent in the system of 12 minutes, the average waiting time in line for each customer is 4 * 12 = 48 minutes. This average waiting time may not be satisfactory for a fast-food business as it can lead to long wait times and potentially dissatisfied customers.