Consider a continuous operation in which raw green beans arrive to a

processing unit. The green beans arrive at an average steady rate of $80$ tons per hour from

noon to $5$ p$.$m$.$ every day. In other words, the incoming rate is $80$ tons per hour for a span

of $5$ hours. The processing begins as soon as the beans start arriving at noon. However,

the processing unit can only process at a steady rate of $25$ tons every hour; that is$,$

outgoing rate is $25$ tons per hour. If need be$,$ the processing unit can process for $24$ hours

a day. If beans arrive when the processing unit is busy, a queue of inventory will form.

a$.[1$ point$]$ How many hours a day does the processing unit operate?

b$.[1]$ What is the maximum inventory of unprocessed green beans?

c$.[1]$ What is the capacity utilization of the processing unit?

$($Hint: This is equivalent to the fraction of a day the plant operates$)$

d$.[2]$ How long on average does a bean stay in the queue?

$($Hint: First calculate average inventory$($INV$=$AREA$/$T$)$ and average throughput $($

l$)$

for the entire time the processing unit is operating. Then use Little$$s law to calculate

CT$=$INV$/$

l$.$