Telecommunications network is modeled as a continuous time Markov chain with two states, denoted as O and 1. State O indicates that the network is available, and state 1 is interpreted as a failure. Instant transition probabilities are defined for ( 0) as follows:

P(X(t+h) = 1|X(t) = 0) = 1.h+o(h) and P(X(t+h) =0|X(t) = 0) = x+h+o(h)
Set daily rates as follows: X = 4 and u=1.

1. Consider Time To Fail defined as T = min (t > 0: X(t) = 1] and derive its conditional distribution, given that X(0) = 0.

2. Consider Time to Repair defined as S = min (t > 0X(t) = 0] and find its conditional distribution, given that X(0) = 1.

3. Derive the stationary distribution of the process X = (X(t): t > 0) and express it in terms of parameters and u.

4. Find network availability defined as stationary probability of the state 0.