Queueing Models Based on The Birth$-$and$-$Death Process

The Friendly Neighbor Grocery Store has a single checkout stand with a full$-$time cashier. Customers arrive randomly at the stand at a mean rate of $30$ per hour. The service$-$time distribution is exponential, with a mean of $1\mathrm{.}5$ minutes. This situation has resulted in occasional long lines and complaints from customers. Therefore, because there is no room for a second checkout stand, the manager is considering the alternative of hiring another person to help the cashier by bagging the groceries. This help would reduce the expected time required to process a customer to $1$ minute, but the distribution still would be exponential. The manager would like to have the percentage of time that there are more than two customers at the checkout stand down below $25$ percent. She also would like to have no more than $5$ percent of the customers needing to wait at least $5$ minutes before beginning service, or at least $7$ minutes before finishing service.

A$.$ Use the formulas for the M$/$M$/1$ model to calculate L $,$ W $,$ Wq $,$ Lq $,$ P$0,$ P$1,$ and P$2$ for the current mode of operation. What is the probability of having more than two customers at the checkout stand? $($On hand calculations$).$

B$.$ Use the Excel template for this model to check your answers in part $($A$).$

C$.$ Repeat part $($A$)$ for the alternative being considered by the manager. $($On hand calculations$).$

D$.$ Use the Excel template and repeat part $($C$)$ for this alternative.

E$.$ Which approach should the manager use to satisfy her criteria as closely as possible?