A system has two different failure types: type 1 and type 2. After a type i-failure the system is said to be in failure state i ; i = 1, 2. The time L i to a type i-failure has an exponential...

A system has two different failure types: type 1 and type 2. After a type i-failure the system is said to be in failure state i ; i = 1, 2. The time L_i
to a type i-failure has an exponential distribution with parameter

Thus, if at time t = 0 a new system starts working, the time to its first failure is

The random variables L₁
and L₂
are assumed to be independent. After a type 1-fai-lure, the system is switched from failure state 1 into failure state 2. The respective mean sojourn times of the system in states 1 and 2 are μ₁
and μ₂. When in state 2, the system is being renewed. Thus, μ₁
is the mean switching time and μ₂
the mean renewal time. A renewed system immediately starts working, i.e. the system makes a transition from state 2 to state 0 with probability 1. This process continues to infinity. (For motivation, see example 5.7).

(1) Describe the system behaviour by a semi-Markov chain and draw the transition graph of the embedded discrete-time Markov chain.

(2) Determine the stationary probabilities of the system in the states 0, 1, and 2.