Problem 6: 20 points (= 5 + 5 + 5 + 5) Telecommunications engineers consider a continuous time Markov chain with three states, so the state space is S = {0,1,2} The process is at state 0 means that network is functioning, while states 1 and 2 indicate two different types of failure. Instant transition rates, 0 1 and 0 2 are denoted as 11 and 12, respectively. They are viewed as failure rates. Instant restoration (or repair) rates denoted as wi and M2, determine transitions 1 0 and 2 0 , respectively. Assume that states 1 and 2 do not communicate, so instant transition rates 1 2 and 2 1 are both equal to zero. 1. Derive the steady-state distribution of X(t), as t o 2. Determine network availability, lim P (X(t) = 0) 3. Assuming that = M1 = 2 and 12 = 12 = 10, find what network availability would be 4. Telecommunications technicians decided to combine two types of failure into a single one, by setting X = 11 + 12 and y = i + H2. Evaluate network availability under their assumptions and check whether it coincides with what you have already found before. Problem 6: 20 points (= 5 + 5 + 5 + 5) Telecommunications engineers consider a continuous time Markov chain with three states, so the state space is S = {0,1,2} The process is at state 0 means that network is functioning, while states 1 and 2 indicate two different types of failure. Instant transition rates, 0 1 and 0 2 are denoted as 11 and 12, respectively. They are viewed as failure rates. Instant restoration (or repair) rates denoted as wi and M2, determine transitions 1 0 and 2 0 , respectively. Assume that states 1 and 2 do not communicate, so instant transition rates 1 2 and 2 1 are both equal to zero. 1. Derive the steady-state distribution of X(t), as t o 2. Determine network availability, lim P (X(t) = 0) 3. Assuming that = M1 = 2 and 12 = 12 = 10, find what network availability would be 4. Telecommunications technicians decided to combine two types of failure into a single one, by setting X = 11 + 12 and y = i + H2. Evaluate network availability under their assumptions and check whether it coincides with what you have already found before