Question $1.$ Queuing in Fast$-$food Restaurant Operations $(6$ marks$)$

The district manager of a fast$-$food restaurant chain has been receiving numerous complaints about long wait times at the counter for placing and receiving orders. During a peak hour, an audit team found that customers were arriving at an average rate of one customer per minute, following a Poisson distribution. The standard service time for each customer was $2\mathrm{.}5$ minutes, following an exponential distribution. The store had three cash registers $($service channels$)$ open during the audit, though a fourth cash register was available but not in use due to scheduling constraints.

Using the appropriate queuing model and the Queuing Calculator, answer the following questions. Show the output of queuing calculator in your answers.

a$)$ What is the probability that the system is empty? $(1$ mark$)$

b$)$ What is the probability that a customer must wait for service? $(1$ mark$)$

c$)$ What is the average number of customers waiting for service in the queue? $(1$ mark$)$

d$)$ What is the average wait time in the queue $($in minutes$)?(1$ mark$)$

e$)$ How would opening the fourth cash register impact system performance? $(2$ marks$)$