A space ship warp engine has enough energy to perform two warp$-$jumps going between star systemsA machine has two engines and a control module. It cannot operate with a malfunctioning control

module, but only needs one engine, i$.$e$.,$ one engine is used for operation and one serves as a backup.

At the beginning of a day, if machine is operational $($i$.$e$.,$ at least one working engine and a working

control module$),1$ engine and the control unit will be used. During a day of operation one of $3$ cases

may happen:

i$.$ engine will malfunction with probability $0\mathrm{.}0$ ;

ii$.$ the control module will malfunction with probability $0\mathrm{.}15$ ;

iii. no malfunctions will be observed with probability $0\mathrm{.}8.$

If at the beginning of the day the machine cannot be operated $($either both engines or the control

module are malfunctioning$),$ then repairs will be initiated, as follows:

i$.$ repair of each of the two engines will be attempted independently, succeeding with probability

$0\mathrm{.}9$ ;

ii$.$ the control unit will be replaced with a new one $($with probability $1).$

Model the the system as a Markov chain with states representing the status of each component $(2$

engines and control module$)$ and transitions ones a day. Draw transition diagram and transition

matrix.Hint: each state should correspond to the status of the components, e$.$g$.,2$ engines and $1$

module working, $1$ engine and module working, etc.

$($i$.$e$.,$ the warp battery has $2$ units of charge$).$ The ship can operate in two modes: survey the current

system or charge battery. On a given day the ship will complete the survey of the current system

with probability $0\mathrm{.}24.$ Then, if battery has capacity, it will move to a new system, consuming $1$ unit

of charge. If the battery is empty, the ship will switch to the charging mode. In the charging mode,

the ship will recover $1$ unit of charge on a given day with probability $0\mathrm{.}95.$ The ship will stay in the

charging mode until the battery is full. Model the behavior of the ship as a Markov chain.