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

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

Oct 21, 2023 146 likes •289 views

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)

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Markov Chains (3)

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Outlines

 Examples of CTMCs

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Examples of CTMCs

 Example:

 The following CTMC have been solved in the class and

transient state probabilities have been computed.

1 2

 

(0) (1,0)  

1,2( ) i

t  

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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 the MTTF is and the MTTR is .

  1  1 

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Examples of CTMCs …

 Solution

Steady state availability:

1 , 1        

1

  MTTF A MTTF MTTR  

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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. These two phases have exponential distributions with means and , respectively.

1

1 

2

1 

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Examples of CTMCs …

 Solution

1

2

2

1

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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.

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Examples of CTMCs …

 Solution

2 1

2   

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Examples of CTMCs …

 Example_5:

We now introduce detection delay that is exponentially distributed with mean . Suppose that it takes time units in average to detect a fault

  • ccurred in a component.

1  1 

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Examples of CTMCs …

 Solution

2 1

2   

1D

 

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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.

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Examples of CTMCs …

 Solution

2 1

2 c    

1C

2 (1 ) c  

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Examples of CTMCs …

 Exercise:

1

0.04 0.09 0.3 2.5

l

              