A machine produces items one at a time, the production times being iid exponential with

mean $\frac{1}{}.$ The items produced are stored in a warehouse of capacity $K.$ When the warehouse

is full, the machine is turned off, and it is turned on again when the warehouse has space

for at least one item. Demand for the items arises according to a $PP().$ Any demand that

cannot be satisfied is lost. Suppose the mean manufacturing time is $1$ hour and the demand

rate is $20$ per day. The warehouse's capacity is $10.$

$($a$)(3$ pts$)$ You need to compute the fraction of the time the machine is turned off in the

long run. Briefly explain how you would solve the problem using concepts in the CTMC

chapter only. $($you just need to explain it without much formulas. Two sentences would

be enough.$)$

$($b$)(7$ pts$)$ Now use the skills from queuing systems, compute the fraction of the time the

machine is turned off in the long run.