A single crew is provided for unloading and$/$or loading each truck that arrives at the loading dock of a warehouse. These trucks arrive according to a Poisson input process at a mean rate of one per hour. The time required by a crew to unload and$/$or load a truck has an exponential distribution $($regardless of the crew size$).$ The expected time required by a one$-$person crew is $1$ hour. The cost of providing each additional member is $$20$ per hour. The cost that is attributable to having a truck not in use $($i$.$e$.,$ a truck standing at the loading dock$)$ is estimated to be $$30$ per hour.

$($a$)$ Assume that the mean service rate of the crew is proportional to its size. What should be the size to minimize the expected total cost per hour?

$($b$)$ Assume that the mean service rate of the crew is proportional to the square root of its size. What should be the size to minimize the expected total cost per hour?