17 State Dependent Systems 2 (2)

17 State Dependent Systems 2 (2) - MFG5350 1 2 1 6 5 6 3 6...

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Unformatted text preview: MFG5350 1 2 1 6 5 6 3 6 9 150 175 200 225 250 Frequency Score Mid-term Exam (Fall 2008) MFG5350 2 Lecture Overview • State-dependent systems • Mid-term Survey MFG5350 3 Markov Analysis Loading Sharing Systems Standby Systems Degraded Systems STATE DEPENDENT SYSTEMS MFG5350 4 Markov Analysis 1 1 1 1 2 1 P (t+ t) = P (t)- t P (t)- t P (t) prob of being in state 1 at time t + t P 1 (t) = prob of being in state 1 at time t prob of being in state 1 at time t and transitioning to state i in time t rate out of state 1 into state 2 rate out of state 1 into state 3 MFG5350 5 Differential Equations d P (t) dt 1 1 2 2 2 = P (t) - P (t) d P (t) dt 2 1 1 3 3 = P (t)- P (t) A fourth differential equation is not needed. Why? 1 2 3 4 P (t)+ P (t)+ P (t)+ P (t) = 1 (t) P ) + (- = dt (t) P d 1 2 1 1 MFG5350 6 Solution 1-( + )t P (t) = e 1 2 MFG5350 7 Two-Component System s 1-( + )t R (t) = P (t) = e 1 2 For a series system: (t) P + (t) P + (t) P = (t) R 3 2 1 s = e + e- e- t - t -( + )t 1 2 1 2 For a parallel system: MFG5350 8 • Two-component parallel system • Dependency: if one component fails, the failure...
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17 State Dependent Systems 2 (2) - MFG5350 1 2 1 6 5 6 3 6...

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