Outlines Examples of CTMCs 2 Examples of CTMCs Example: The - PowerPoint PPT Presentation

Markov Chains (3)

Outlines  Examples of CTMCs 2

Examples of CTMCs  Example:  The following CTMC have been solved in the class and   1,2 ( ) t transient state probabilities have been computed. i  1 2    (0) (1,0) 3

Examples of CTMCs …  Example_2:  Consider a component with a constant failure rate . On a failure, it is   repaired with an exponential repair time distribution of parameter . Thus 1  1  the MTTF is and the MTTR is . 4

Examples of CTMCs …  Solution  0 1   1     , Steady state availability:     0 1 MTTF  A  MTTF MTTR 5

Examples of CTMCs …  Example_3: The two state model of component failure-repair assumed that the failure  and the repair time distributions are both exponential. Assume now that the exponential failure law is reasonable, but the repair process can be broken down into two phases: (1) fault detection and location and, (2) actual repair. 1  1  These two phases have exponential distributions with means and , 1 2 respectively. 6

Examples of CTMCs …  Solution   1 1 2 0  2 7

Examples of CTMCs …  Example_4:  Consider a two-component system, each component with failure rate .  Suppose there is only a repair facility in the system which services a failed component. The system is unavailable to users if both components fail. 8

Examples of CTMCs …  Solution 2   1 2 0   9

Examples of CTMCs …  Example_5: We now introduce detection delay that is exponentially distributed with  1  1  mean . Suppose that it takes time units in average to detect a fault occurred in a component. 10

Examples of CTMCs …  Solution  2    1D 1 2 0   11

Examples of CTMCs …  Example_6: Consider another variation of two-component system in which the failure is  detected and handled with probability c and is not detected with probability ( 1-c ). If the system can not be able to coverage the failure, the whole system is rebooted. 12

Examples of CTMCs …  Solution  2 c  1 2 0     2 (1 c )  1C 13

Examples of CTMCs …  Exercise:    0.04     0.09    0.3  1     2.5 l 14