Question $2(20$ points$)$

In an effort to become the most advanced airport in the world, Hampton International decided to automate its luggage handling facilities. After lengthy delays, the system was put into operation but was immediately unable to provide the anticipated throughput. Passengers still had to wait for a long time for their luggage, and some aircraft were kept waiting. The problem appears to be in the transportation of luggage from the aircraft to the distribution network.

Under the new system, luggage is brought from the aircraft to a loading station where a series of transporters convey the luggage to the distribution point. Before reaching the distribution station, each transporter is inspected by Homeland Security. Once at the distribution point, the luggage is sent to its destination $($PICK UP area$)$ while the transporter is returned to the loading station.

The basic operating parameters for the system are as follows:

On the average, luggage arrives at the transporter loading area from the aircraft about every $5$ seconds $($distributed exponentially$).$

The loading area has space enough for $100$ bags to wait to be loaded onto transporters.

Eight pieces of luggage are put on a transporter. It takes about $20$ seconds to fill the transporter as soon as all the pieces of luggage are present.

The transporter brings the luggage to the inspection station.

The unloading process at the distribution point takes $45$ seconds, after which the transporter returns to the loading station.

Inspecting a transporter takes an average of $120$ seconds.

The system has $15$ transporters.