A recent analysis at a FroyoIYkYk...by the store manager indicates that, during operating

hours, there are $45$ customers arriving every hour, with a standard deviation of inter$-$arrival

times of $1\mathrm{.}5$ minutes. This arrival pattern is consistent throughout the day. The checkout is

currently operated by one employee, who needs on average $1\mathrm{.}25$ minutes to check out a

customer. The standard deviation of this checkout time is $3\mathrm{.}25$ minutes, primarily because

of customers buying different numbers of toppings and customizations.

$1.$ If you assume that every customer buys at least one item $($i$.$e$.$ has to go to check out$),$ what is the

average time a customer has to wait in line before being served by the checkout employee?

$2.$ What if the processing time decreased by $\mathrm{.}25$ seconds?

$3.$ What if the standard deviation of processing time decreased to $1\mathrm{.}5$ minutes?

$4.$ What is the total time a customer spends in line and at the checkout counter being served?

$5.$ What is the average number of customers who are in the system $($i$.$e$.,$ either waiting or being

served$)?$

$6.$ The manager decides that the waiting time is way too long and hires $4$ more checkout clerks, for a

total of $4.$ What is the waiting time $($in the line, before being served$)$ in this new situation?