The wheat harvesting session in Saskatchewan is short, and farmers deliver their truckloads of

wheat to a giant central storage bin within a $2-$week span. Because of this, what filled trucks

waiting to unload and return to the fields have been known to back up for a block 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 grain 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 crop damage, it is easy to assign

a waiting and unloading cost for truck and driver of $$18\mathrm{.}5$ per hour. During the $2-$week harvest

season, the storage bin is open and operated $15$ hours per day, $7$ days per week, and can unload $35$

trucks per hours according to an exponential distribution. Full trucks arrive all day long $($during the

hours the bin is open$)$ at an average interval of $2$ minutes $($assume Poisson arrivals$).$

a$)$ What is the average number of trucks and the average waiting time per truck in the system?

b$)$ What fraction of time do you expect that there are three or less trucks in the system?

c$)$ As mentioned, the cooperative uses the storage bin heavily only $2-$weeks per year. Farmers

estimate that enlarging the bin would cut unloading cost by $50\%$ next year. It would cost $$9,000$ to

do so during the off season. Would it be worth the expense to enlarge the storage area? Explain.