Robert$$ Olney's Barbershop is a popular haircutting and styling salon near the campus of the University of New Haven. Four barbers work$$ full$-$time and spend an average of $15$ minutes on each customer. Customers arrive all day long at an average rate of $12$ per hour. Arrivals tend to follow the Poisson$$ distribution, and service times are negative exponentially distributed. The software described in this quantitative module may be used to answer the following questions.

Part $2$

$$a$)$ The probability that the shop is empty is

$$

enter your response here $($round your response to four decimal$$ places$).$

Part $3$

$$b$)$ The average number of customers in the barber shop is

$$

enter your response here customers $($round your response to two decimal$$ places$).$

Part $4$

$$c$)$ The average time spent in the shop is

$$

enter your response here hours $($round your response to three decimal$$ places$).$

Part $5$

$$d$)$ The average time that a customer waits to be called to the barber chair is

$$

enter your response here hours $($round your response to three decimal$$ places$).$

Part $6$

$$e$)$ The average number waiting to be served is

$$

enter your response here customers $($round your response to two decimal$$ places$).$

Part $7$

$$f$)$ Robert is thinking of adding a fifth barber. The utilization rate of his shop will be

$$

enter your response here $($round your response to four decimal$$ places$).$