Determining ratio of mean time in the queue

Question

Assignment:

At Martha's café, Martha herself operates the espresso machine. Customers arrive and demand cups of espresso according to a Poisson process with an average rate of 30 cups/h. The time it takes for Martha to make a cup of espresso is exponential with a mean of 75 s.

a. Determine L, L_q, W, and W_q for this queuing system. Show that the results are the same if you use the expressions available for M/M/1 and M/G/1 systems, respectively.

b. Assume that Martha buys a new fully automated espresso machine that makes a cup of espresso in exactly 75 seconds always. Determine L, L_q, W, and W_q under these new conditions.

c. Determine the ratio of the mean time in the queue, L_q, for the systems in (a) and (b). What are your conclusions?

Provide complete and step by step solution for the question and show calculations and use formulas.