Question $1$: Hand$-$Simulation $(60$ points$)$

Consider a warehouse where trucks arrive to either pick$-$up or deliver items. There are two workers $($W$1,$ W$2)$ for transferring the items between the trucks and the warehouse, and one manager $($M$)$ keeps track of the incoming and outgoing item amounts and types.

Trucks arrive for one of two purposes: to pick up some items $($pick$-$up trucks$-$ PT$)$ or to deliver some items $($delivery trucks$-$ DT$).$ Pick$-$up trucks do not need the manager and are processed by either one of the two identical workers since the amount and types of items to be sent are already decided and approved by the manager on the previous day. Delivery trucks, however, need to get the manager$$s approval before leaving the warehouse. For delivery trucks, firstly, one of the workers takes whatever items are to be delivered to the warehouse, and then the manager inspects the items and gives approval. Upon this inspection, the truck can leave the warehouse.

In this system, since the trucks are loaded or unloaded from the rear $($behind$),$ a last$-$come$-$first$-$serve $($LIFO$)$ queue discipline is used. This is because only the rear of the last arriving truck is available. If$,$ however, a truck arrives while another is being processed, the arriving truck leaves some space between the truck that is in process and joins the queue. After a truck is done, the first idle worker moves on to the truck that has joined the queue most recently $($the last truck in the queue$).$

It takes $30$ minutes to load or unload the truck, so whenever a worker starts working on a truck, it will be done in $30$ minutes. The manager$$s approval $($which is only necessary for delivery trucks that are done with the unloading process$)$ takes $20$ minutes. The manager$$s queue follows a FIFO discipline. Each worker and the manager can work on a single truck at a time, and trucks arrive one by one in the system.

The system is empty, and all resources are idle at time $0.$ The manager is given the list of trucks that are planned to arrive tomorrow. She or he wants to calculate several performance measures on this system. Help the warehouse by creating a hand$-$simulation table and running the simulation until all entities leave the system.