A fast food drive$-$through restaurant has a single serving booth staffed by a single operative. There is space for ten vehicles to wait if the service booth is occupied. If all ten

spaces are occupied when a car arrives, it has to drive past and the customer is lost. Cars arrive at random at a rate of $55$ per hour. Service times are exponential with a

mean of one minute.

Employees are paid $10\mathrm{.}30$ per hour. Time waiting before receiving service has a high value in the fast food industry. Here it is valued at $95$ per hour. The average revenue

per customer is $15.$

For this queueing situation, use the queueina template spreadsheet $($and enter answers to the accuracy displayed by the spreadsheet$)$ to calculate the average number of cars

at the drive$-$through $($the whole system$)$

and the average time a car is waiting tot be able to drive up to the booth and make their order

and the probability a customer will fail to achieve entry to the drive$-$through

Note: Fill the first three blanks with $4$

decimal places.

Calculate the hourly cost of the system under present circumstances

Note: Enter this and all further costs in this question to $2$ decimal places.

The manager of the drive$-$through restaurant could choose to build and staff extra serving booths if it resulted in an overall cost saving. However, the additional space required

would mean a reduction in the space for cars to wait for service $-$ for each additional booth, two waiting spaces are lost.

To minimise costs,

additional booths should be built. The waiting space is reduced to

This achieves a minimum hourly

cost of

$($In the calculations above you may ignore the capital costs of performing infrastructure work$)$