A fast$-$food franchise is considering drive$-$up window food$-$service operation at a busy intersection in Chicago. It is expected that cars arrival follows a Poisson process with an average inter$-$arrival time of $2\mathrm{.}5$ minutes. Customers place the order for food$/$drinks at an intercom station at the back of the parking lot and then drive to the service window to pay and receive their order. Following three alternatives are being considered.

Alternative $1$: A single channel operation where one employee fills the order and takes the money from the customer. The service time is exponentially distributed with a mean of $2$ minutes. The employee is paid $$6\mathrm{.}50$ per hour.

Alternative $2$: A single channel operation where one employee fills the order and a second employee the takes the money from the customer. The service time is exponentially distributed with a mean of $1\mathrm{.}25$ minutes. Each employee is paid $$6\mathrm{.}50$ per hour.

Alternative $3$: A two channel operation with two service windows, where each window is served by one employee. The service time is exponentially distributed with a mean of $2$ minutes. The employee at each window is paid $$6\mathrm{.}50$ per hour.

Given that waiting is very costly for a $$fast food$$ business, the franchise attaches a value of $$25/$hour for customer waiting time. To account for equipment and space an additional cost of $$20/$hour is attributable to each channel.

$.$

Show the answers to one decimal place accuracy

Alt $1$ Alt $2$ Alt $3$

$1)$ Probability that there are no cars in the system $-$ in $\%20\%50\%43\%$

$2)$ Probability that a car must wait for service $-$ in $\%80\%50\%50\%$

$3)$ Average number of cars waiting for service $3\mathrm{.}2\mathrm{.}5\mathrm{.}152$

$4)$ Average number of cars in the system $41\mathrm{.}952$

$5)$ Average time a car waits for service $-$ in minutes $7\mathrm{.}981\mathrm{.}248\mathrm{.}36$

$6)$ Average time spent in the system $-$ in minutes $10\mathrm{.}0022\mathrm{.}5022\mathrm{.}4$

Determine the lowest cost design.