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Therefore, this is expressed mathematically:PT¼Pð8:2Þ1148Markov Processes
wherePis the limiting state probability vector andTis the transition matrix. Thismay be rewritten as:PðT±IÞ ¼0ð8:3ÞwhereIis the identity matrix.Substituting the transition matrixTinto the above equation gives the followingexpression.P1P2P3P4½²±ðk1þk2Þk1k20l1±ðl1þk2Þ0k2l20±ðl2þk1Þk10l2l1±ðl1þl2Þ2666437775¼0 0 0 0½ð8:4ÞBy taking the transpose of the equation, the general form of the equation isobtained.±ðk1þk2Þl1l20k1±ðl1þk2Þ0l2k20±ðl2þk1Þl10k2k1±ðl1þl2Þ26643775P1P2P3P426643775¼000026643775ð8:5Þ8.2.4 Step 4: Full Probability ConditionThe sum of all the individual probabilities is equal to 1.P1þP2þP3þP4½¼1This condition is required to be able to solve the above equation as it containsonlyn-1 independent equations and there are four state variables involved.Therefore, any row within the above equation can be replaced with this condition,such as the first row.00110110μ1μ2μ1μ2λ1λ1λ2λ2State 1 State 3 State 2 State 4 Fig. 8.1Diagram for a two-component repairable systemfor availability analyses8.2Systems Availability Analyses115
1111k1±ðl1þk2Þ0l2k20±ðl2þk1Þl10k2k1±ðl1þl2Þ26643775P1P2P3P426643775¼100026643775ð8:6Þ8.2.5 Step 5: Solving the Markov Matrix EquationUsing Linear AlgebraAs equation now contains four independent equations, it may be solved usinglinear algebra. In general, this yields the following expressions.P1¼l1l2ðl1þk1Þðl2þk2Þð8:7ÞP2¼k1l2ðl1þk1Þðl2þk2Þð8:8ÞP3¼l1k2ðl1þk1Þðl2þk2Þð8:9ÞP4¼k1k2ðl1þk1Þðl2þk2Þð8:10Þ8.3 Example of Markov Chains for Reliability AnalysesA simple system with one active component named primary system and onestandby redundant component named secondary system is connected with anothercomponent, i.e., switch . Figure8.2shows this system.When switch fails, it is unable to switch. The failure of the switch matters, if itis required to switch from the primary to the secondary system. If the switch failsafter the standby spare is already in use, then the system can continue operation.However, if the switch fails before the primary unit fails, then the secondarysystem cannot be turned on and the system fails when the primary unit fails. Theorder in which the primary system and switch fail determines whether the systemcontinues operation.