Instructions

$18-84.$ Eat & Gas convenience store operates a two$-$pump gas station. The lane leading to the pumps can house at most $3$ cars, excluding those being serviced. Arriving cars go elsewhere if the lane is full. The distribution of arriving cars is Poisson with mean $20$ per hour. The time to fill up and pay for the purchase is exponential with mean $6$ minutes. Determine the following:

$($a$)$ Percentage of cars that will seek business elsewhere.

$($b$)$ Percentage of time both pumps are in use.

$*($c$)$ Percent utilization of the two pumps.

$*($d$)$ Probability that an arriving car will not start service immediately, but will find an empty space in the lane.

$($e$)$ Capacity of the lane that will ensure that, on the average, no more than $10\%$ of the arriving cars are turned away.

$($f$)$ Capacity of the lane that will ensure that the probability that both pumps are idle is $\mathrm{.}1$ or less.