A facility operates in two phases: from $4$am till $4$pm is the manufacturing phase, and from $4$pm till

$4$am is the packaging and shipping phase. Output of the manufacturing phase is uncertain $$ during

one day of production $2$ units will be produced $15\%$ of the time, $1$ unit $25\%$ of the time and no units

$60\%$ of the time. At the end of the manufacturing phase, all produced units are moved to storage.

Storage has capacity for $3$ units, and any units that do not fit $($at $4$pm at the end of manufacturing

phase$)$ have to be scraped. $70\%$ of the time there is demand for the product, in which case $1$ unit of

product will be packaged and shipped during the second phase. Otherwise, nothing happens during

the second phase $($facility does not have capacity to ship more than $1$ unit per day$).$

Model the system as a Markov Chain. Define the states as the status of the storage at $4$am$,$ i$.$e$.,$ the

number of units stored before mafacturing starts.

i$.$ List possible states.

ii$.$ Find the probability that there will be $1$ unit in storage tomorrow, given that there is noting in

storage today $($this is transition probability from state $0$ to state $1).$

iii. Fill in the rest of the transition matrix.

iv$.$ Given the Markov model, calculate the probability that the storage will be full the day after

tomorrow, if it is empty today