"While spreadsheets and other programs can solve queuing models, the purpose of doing this PK$17$ exercise is to become familiar with equations to become somewhat familiar with solving this queuing models. The problem should be solved using a legible pen using equations and then scanned, not solved by using Excel. This problem relates to the collection of grapes which occurs in a short period of time. During this period most farmers deliver their truckloads of grapes to a giant central storage bin within a two$-$week span. Because of this, trucks waiting to unload and return to the fields have been known to back up in a queue at the receiving bin. The central bin is owned cooperatively, and it is to every farmer$$s benefit to make the unloading$/$storage process as efficient as possible. The cost of deterioration caused by unloading delays and the cost of truck rental and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying grape damage, it is easy to assign a waiting and unloading cost for truck and driver of $18$ per hour. The storage bin is open and operated $16$ hours per day, seven days per week during the collection of grapes and is capable of unloading $35$ trucks per hour, according to an exponential distribution. Full trucks arrive all day long $($during the hours the bin is open$),$ at a rate of about $30$ per hour, following a Poisson pattern. To help the cooperative get a handle on the problem of lost time while trucks are waiting in line or unloading at the bin, find the:

$($i$)$ Average number of trucks in the unloading system; $($ii$)$ The average time per truck in the system; $1($iii$)$ Utilization rate for the bin area; $($iv$)$ Probability that there are more than three trucks in the system at any given time; $($v$)$ Total daily cost to the farmers of having their trucks tied up in the unloading process; $($vi$)$ The cooperative, as mentioned, uses the storage bin only two weeks per year. Farmers estimate that enlarging the bin would cut unloading costs by $50\%$ next year. It will cost $9,000$ to do so during the off$-$season. Would it be worth the cooperative$$s while to enlarge the storage area?"