Each time a machine is repaired it remains up for an exponentially distributed time with mean 20 days. It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to repair the machine is exponential with mean time 1/2 day; if it is a type 2 failure, then the repair time is exponential with mean 2 days. Each failure is, independently of the time it took the machine to fail, a type 1 failure with probability 0.8 and a type 2 failure with probability 0.2. What proportion of time is the machine down due to a type 1 failure? What proportion of time is it down due to a type 2 failure? What proportion of time is it up?

Question

We can model the state of the machine according to a continuous time markov chain X ( t ) , with state space { 0 , 1 , 2 } where
state 0 =machine is up and working
state 1 = type 1 failure
state 2= type 2 failure
Q1;
Find the transition rates:
q 0 = per day
q 10 = per day
q 20 = per day
Recall that q i j = q i p i j for i ≠ j .
q 01 = q 0 p 01 where p 01 =
q 02 = q 0 p 02 where p 02 =
q 1 =
q 2 =
Q2;
Find the steady-state probabilities of the machine being down due to type 1 failure ( π 1 ), machine being down due to type 2 failure ( π 2 ) and machine being up ( π 0 ). Either write your answers as decimals up to 4 decimal places or write them as fractions.
π 0 =
π 1 =
π 2 =

Answer 1

please solve this problem ASAP.

Answer 2

q0=1/20 per day
q10=2 per day
q20=1/2 per day
p01=.8
p02=.2
q1=2
q2=1/2
π0=25/26
π1=1/52
π2=1/52