Customers arrive at Paul Harrold¡¯s styling shop at a rate of 3 per hour, distributed in a Poisson fashion. Paul a perform haircuts at a rate of 5 per hour, distributed exponentially.
a) Find the average number of customers waiting for haircuts.
b) Find the average number of customers in the shop
c) Find the average time a customer waits until it is his or her turn.
d) Find the average time a customer spends in the shop
e) Find the percentage of time that Paul is busy.
To answer these questions, we can use the concepts of queuing theory and Little's Law. Little's Law states that the average number of customers in a queuing system is equal to the arrival rate multiplied by the average time spent in the system.
a) To find the average number of customers waiting for haircuts, we need to calculate the difference between the arrival rate and the service rate. In this case, the arrival rate is 3 customers per hour, and the service rate is 5 haircuts per hour. So, the average number of customers waiting for haircuts is 3 - 5 = -2. However, since the number of customers cannot be negative, we can consider it as zero.
b) The average number of customers in the shop is the sum of the average number of customers waiting and being served. In this case, since there are no customers waiting, the average number of customers in the shop is equal to the average number of customers being served, which is given by the service rate. So, the average number of customers in the shop is 5.
c) To find the average time a customer waits until it is their turn, we need to calculate the average time spent waiting in the queue. The average time spent waiting is given by Little's Law. In this case, the average number of customers waiting is 0, and the arrival rate is 3 customers per hour. So, the average time a customer waits is 0/3 = 0 hours.
d) To find the average time a customer spends in the shop, we can use Little's Law again. The average number of customers in the shop is 5, and the arrival rate is 3 customers per hour. So, the average time a customer spends in the shop is 5/3 = 1.67 hours.
e) The percentage of time that Paul is busy can be calculated by dividing the service rate by the arrival rate and multiplying by 100. In this case, the service rate is 5 haircuts per hour and the arrival rate is 3 customers per hour. So, the percentage of time that Paul is busy is (5/3) * 100 = 166.67%.