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we can set-up and perform a correct analysis based on the actual process. When all radios have failed, one radio is in service, so the queue length is 6. Our interest is in predicting the time to total failure, the waiting time from state 0 to state 6, denoted by Z. Starting in state 0, the waiting time for one radio to fail is the minimum of ve independent Exp( ) distributions, which is Exp(5 ). By the assumptions for the M/G/1 queue, from state 1 the queue can proceed directly to any of the states 0, . . . , 5. From any state i {2, . . . , 5} the queue can go to states i 1, . . . , 6. The branch transmittances among states are summarized as 0 T10 (s) 0 T (s) = 0 0 0 0 T01 (s) 0 0 0 T11 (s) T12 (s) T13 (s) T14 (s) T21 (s) T22 (s) T23 (s) T24 (s) 0 T32 (s) T33 (s) T34 (s) 0 0 T43 (s) T44 (s) 0 0 0 T54 (s) 0 0 0 0 0 T15 (s) T25 (s) T35 (s) T45 (s) T55 (s) 0 0 0 T26 (s) T36 (s) , T46 (s) T56 (s) 0

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In this section we introduce three types of traditional case adaptation strategies: Reinstantiation is the simplest form of adaptation, in which the solution of the new problem is simply copied from the case retrieved and used directly, without modi cation. Substitution replaces parts of the old solution attributes that are invalid because they con ict with or contradict the new problem requirements. For example, in medical diagnosis, parts of a drug prescription in the past may need to be replaced and updated with more effective medicine for the treatment of new types of illnesses. Transformation is used when no appropriate substitute item is available. A tailored solution will be derived based on the constraints and the characteristics of the required solution. A constraint describes or de nes the properties of a component(s) of the solution. It speci es what properties the solution should or should not have. The solution component must conform to the constraints, and no contradiction or con ict is allowed. To identify the constraints, some prede ned expert knowledge or heuristic must be available. In the following subsections we provide examples to illustrate three approaches. Table 4.1 compares the three methods. Both substitution and transformation are triggered or based on constraints and feedbacks from the problem and solution characteristics, while reinstantiation can be performed simply by applying the old solution directly.

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where the last condition is true for an open neighborhood containing s = 0. An alternative way to conceptualize this loop is to consider the two branches, 0 R and 0 1, exiting state 0 as branches in parallel. Recall that parallel owgraphs lead to nite mixture distributions. In this case, each time the feedback loop is taken, another term in the mixture is generated so that we have in nite parallel branches. This is an in nite mixture with weights p, p(1 p), p(1 p)2 , . . . . This direct approach of (2.18) is the least elegant of the three feedback loop derivations. Example 2.14: Engineering Application: Hydraulic Pump with Feedback. We return to the full hydraulic pump system of Example 2.8. To solve this owgraph (Figure 2.10) for the MGF of T , MT (s), we set up a balance equation at state 0. A balance equation equates the inputs into a state with the outputs of the state. The idea is based on conservation laws applied to systems and originates from Kirchhoff s laws governing circuit behavior, which are derived from conservation laws applied to circuits. In our case, the input into state 0 is p10 M10 (s). The output from state 0 is M01 (s). This is magni ed by the factor MT (s), which is the MGF of the loop that is fed back into itself. In systems analysis, this factor is called the gain of the system. The balance equation for state 0 in Figure 2.10 is MT (s) = M01 (s)p10 M10 (s)MT (s) + M01 (s)p12 M12 (s). (2.19)

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