$2.$

a$).$ Fuad & Sons Wholesale Fruit Distributors employs one worker whose job is to load fruit on outgoing company trucks. Trucks arrive at the loading gate at an average of $24$ per day, or $3$ per hour, according to Poisson distribution. The worker loads them at a rate of $4$ per hour, following approximately the exponential distribution in service times. Determine the operating characteristics of this loading gate problem. What is the probability that there will be more than three trucks either being loaded or waiting? Discuss the results of your queuing model computation.

b$)$ Fuad & Sons believes that adding a second fruit loader will substantially improve the firm$$s efficiency. He estimates that a two$-$person crew, still acting like a single$-$server system, at the loading gate will double the loading rate from $4$ trucks per hour to $8$ trucks per hour. Analyze the effect on the queue of such a change and compare the results with those found in Problem a$.$

c$).$ Truck drivers working for Fuad & Sons $($see Problem a and b$)$ are paid a salary of $$20$ per hour on average. Fruit loaders receive about $$12$ per hour. Truck drivers waiting in the queue or at the loading gate are drawing a salary but are productively idle and unable to generate revenue during that time. What would be the hourly cost savings to the firm associated with employing two loaders instead of one?

d$).$ Fuad & Sons $($of problem a$)$ is considering building a second platform or gate to speed up the process of loading the fruit trucks. This, he thinks, will be even more efficient than simply hiring another loader to help out the first platform $($as in Problem b$).$ Assume that each worker at each platform will be able to load $4$ trucks per hour and that trucks will continue to arrive at the rate of $3$ per hour. Find the waiting line$$s new operating conditions. Is this new approach indeed speedier than the other two considered?