Q$1(10$ points$)$ Consider a bar that operates between $5$pm and $3$am$.$ The demand rate is

variable, and can be described as follows:

Between $5$pm and $9$pm$,$ the demand rate is $100$ customers$/$hr

Between $9$pm and $11$pm$,$ the demand rate is $200$ customers$/$hr

Between $11$pm and $1$am$,$ the demand is constant and equal to $50$ customers$/$hr

After $1$am$,$ no new customers arrive

To serve this demand, the bar has bartenders that work on different shifts. The restaurant

knows that each bartender has a capacity of $50$ customers$/$hour$,$ and suppose that the

bartenders can serve customers in parallel. In addition, we know that:

Between $5$pm and $8$pm$,$ there is only one bartender working at the restaurant

After $8$pm$,$ and until the bar closes, there are two bartenders working in parallel

$($a$)$ Plot the evolution of the demand rate and the service rate $($or capacity$)$ of the

restaurant. $(5$ points$)$

$($b$)$ Draw the inventory build$-$up diagram. $(5$ points$)$