A machine con-

sists of 4 independent components in series, i.e.,

all

the components must be in working

condition for the machine to be functional. When the machine is functional at the be-

ginning of the

n

th day, each component has a probability

p

of failing, independent of

other components. Failure probability 0

< p <

1 is same for each component. More

than one component can fail on the same day. When the machine fails (i.e., there

exists at least one failed component), a single repair person repairs the failed compo-

nent(s) one by one. It takes exactly one day to repair one failed component, and the

repairman starts working at the beginning of the day. When all the failed components

are repaired, the machine is functional again and behaves as before. When the ma-

chine is in repair, the working components do not fail. Let

X

n

be the number of failed

components at the beginning of the

n

th day, after all the failure and repair events at

that time are accounted for.

Show that {Xn , n 0} is a DTMC.