Problem
Nail spa is a small shop on campus that provides manicures and pedicures. Customer arrival follows a poisson process, at an average rate of 2 customers per hour. 60% of the customers want only a manicure, 40% of the customers want only a pedicure. No customer wants both a manicure and pedicure. Due to different requirements from different customers, the service time varies. The service time of a manicure follows an exponential distribution, and it takes 40 minutes, on average, to do manicure. The service time of a pedicure follows an exponential distribution, and it takes 40 minutes, on average, to do pedicure. The shop hires two employees. Manicure customers are served by employee A, following the first-come-first-served service rule. Pedicure customers are served by employee B, following the first-come-first-served rule. If a customer arrives and cannot be served immediately, the customer will wait in the waiting area.
The owner of the nail spa has decided to cross-train the two employees, so that both employees can perform manicures and pedicures. The service time of a manicure still follows an exponential distribution, and it takes 40 minutes, on average, to do manicure. The service time of a pedicure follows an exponential distribution, and it takes 40 minutes, on average, to do pedicure. After the employees are cross-trained, all customers are first-come-first served, regardless of the service they need. In other words, whenever there is a waiting line, the customer who is the first in the line will be served by the next available employee. Please answer questions for this setting.
a) What is the probability that a customer can be served as soon as she arrives in nail spa? Just use excel file for lambda and mu and tell me where the answer is found on the row
b) What is the probability that exactly 2 customers are waiting in the waiting area?