4.15 Unloading at a warehouse A forklift transfers cargo from an intercity truck to a warehouse with the following operational characteristics (see Figure P4.15):

1 . The time spent on loading the forklift (at the truck location, A) is approxi- mately constant and equal to 15 seconds.

2. All cargo from each intercity truck is transferred to the same designated storage location (B) in the warehouse.

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3. For a given truck, any storage location in the warehouse is equally likely.

4. The average unloading time at B, including turning and reversing, is 10 seconds (negligible variance).

5. The forklift travels at a constant speed of 5 feet per second, and, because of the warehouse layout, in the east-west and north-south directions only.

6. Exactly 10 forklift loads are required to empty each truck.

7. Trucks are served one at a time and leave the dock as soon as they are emptied; the docking/undocking times for each truck last considerably less than a roundtrip by a forklift, so that no forklift time is wasted because of truck maneuvers.

In the following we shall refer to the truck parking lot and the dock as "the system."

a. Given that one truck arrives at the system every 50 minutes on the average and that the arrivals follow a Poisson process, find the average number of trucks in the system at any given time and the average time each truck remains in the system.

b. If two forklifts are used instead of one, answer approximately the questions in part (a) with an appropriately defined M/G/1 model. Assume that the forklifts do not interfere with one another throughout the operation.

Hint: What happens to the mean and variance of the service time?

c. Use approximate bounds for M/G/2 systems (Section 5.8) to answer the questions in part (b) and compare these answers with your answers in (b).