AllWings restaurant offers half$-$price wings every Wednesday. The restaurant opens at $7$ pm$.$ However, customers start lining up at $6$:$30$ pm$.$ Customers arrive at a rate of $45/$hr until $8$:$30$ pm and then at a rate of $30/$hr until $10$ pm$.$ The restaurant can serve at a rate of $35/$hr$,$ and the counter works till all customers served.

a$)$ Draw the Inventory Build$-$Up diagram showing the number of customers waiting in line in $30-$ minute time slots. Feel free to use a table representing arrivals$/$capacity$/$excess demand$/$excess capacity to ease up preparing the diagram. Start the graph at $6$:$30$pm and show the number of waiting customers until the line becomes empty.

b$)$ Using the inventory build$-$up diagram, calculate the average number of customers in line.