Call the path sets of(c) above ‘units, a, b, and c. Using the numerical value fo

Question

Call the path sets of(c) above ‘units, a, b, and c. Using the numerical value for , determine the MTTF, of each nit. I. 2. Ignore the unit with the least reliability (a judgment call may be involved here You now have an active parallel system with two units. Assume that both these units can be repaired as and when they occur with a constant repair rate of i. Draw the state transition diagram for this system. ii. Determine the transition matrix Min P - MiP iii. Determine numerically the steady-state probability state vector P(oo Determine the steady-state probability of the system's operation. iv. 2 F ) Unit 3 t3N3 13.6231

Explanation / Answer

Assume that both the transition probabilities from state 1 A L 1 j to 2 and vice versa (Fig. la) are required to double.

It is y= [I - Al]-' = 1/22 [ 1p 2X I 0 1 0 K= I_ -1 T=2[0 1 0 =(3.+ p)/2Xt2 T = [1/2 1i8 ], H= z-1 = L2 fl]

ii) Assume that the quantity 2X (associated with the transi. tion from state 1 to state 2) is to be doubled; result in sponds to the new situation.

The determination of MTTF proceeds as follows: 0' = (5X + 21i)/4X2.

The advantage of i over ii & iii is that no additional transiF1 -1] tions are introduced and thus the total number of states of the [0 1 1 iK=1 I subdivisions equals that of the original transition diagram. 1L-1 lj However, when the numbers of states and the transitions between states are very large, ii & iii can be less expensive than i 2X 0 0 due to the fact that the number of transitions between the subT= O ,p O. divisions is kept small compared to i. Figure 2a-b shows the -0 0

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