Question

Customers arrive to a hair cut salon according to a poisson process with a mean arrival rate of 5/hr. because of the reputation of the salon, customers were always willing to wait. customer processing time was exponentially distributed with an average of 10 min.

average no. of customers in the shop and average no. of customers waiting for hair cut.

Answer 1

The average time between two successive arrivals is 12 minutes. The average number of customers in the shop is 0.833 customers. There are no customers waiting for a haircut on average.

Step-by-step explanation:

The average time between two successive arrivals in a Poisson process can be found by taking the reciprocal of the arrival rate. In this case, the average arrival rate is 5 customers per hour, so the average time between two successive arrivals is 1/5th of an hour or 12 minutes.

For the average number of customers in the shop, we need to consider both the customers being served and the customers waiting. The average number of customers in the shop can be calculated using Little's Law, which states that the average number of customers in a steady-state system is equal to the arrival rate multiplied by the average time spent in the system. In this case, the arrival rate is 5 customers per hour, and the average processing time is 10 minutes. Therefore, the average number of customers in the shop is 5 * 10/60 = 0.833 customers.

The average number of customers waiting for a haircut can be found by subtracting the average number of customers being served from the average number of customers in the shop. In this case, the average number of customers being served is 1 (since there is only one hair stylist), so the average number of customers waiting is 0.833 - 1 = -0.167 customers. Since we cannot have a negative number of customers, it means that, on average, there are no customers waiting for a haircut.