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# chap8_p2 - Probability and Statistics with Reliability...

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Copyright © 2003 by K.S. Trivedi 1 Probability and Statistics with Reliability, Queuing and Computer Science Applications Second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 8 (Part 2) :Continuous Time Markov Chain Availability Modeling Dept. of Electrical & Computer engineering Duke University Email: [email protected] URL: www.ee.duke.edu/~kst

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Copyright © 2003 by K.S. Trivedi 2 2-State Markov Availability Model 1) Steady-state balance equations for each state: Rate of flow IN = rate of flow OUT • State1: • State0: 2 unknowns, 2 equations, but there is only one independent equation. UP 1 DN 0 µ λ MTTR MTTF = = µ λ 1 1 1 0 π λ π µ = 0 1 π µ π λ =
Copyright © 2003 by K.S. Trivedi 3 2-State Markov Availability Model (Contd.) Need an additional equation: 1 1 0 = + π π • Downtime in minutes per year = * 8760*60 µ λ π π µ λ π + = = + 1 1 1 1 1 1 min 356 . 5 10 1 99999 . 0 5 = = = DTMY A A ss ss MTTR MTTF MTTR + MTTR MTTF MTTF A ss + = + = + = + = = µ λ λ µ λ µ µ λ π 1 1 1 1 1 1 MTTR MTTF MTTR A ss + == 1

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Copyright © 2003 by K.S. Trivedi 4 2-State Markov Availability Model (Contd.) 2) Transient Availability for each state: Rate of buildup = rate of flow IN - rate of flow OUT This equation can be solved to obtain assuming 1 (0)=1 ) ( ) ( 1 0 1 t t dt d π λ π µ π = have we t t 1 ) ( ) ( since 1 0 = + π π ) ( )) ( 1 ( 1 1 1 t t dt d π λ π µ π = µ π λ µ π = + + ) ( ) ( 1 1 t dt d t e t A t ) ( 1 ) ( ) ( µ λ µ λ λ µ λ µ π + + + + = = π
Copyright © 2003 by K.S. Trivedi 5 2-State Markov Availability Model (Contd.) 3) 4) Steady State Availability: t e t R λ = ) ( µ λ µ + = = ss t A t A ) ( lim

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Copyright © 2003 by K.S. Trivedi 6 DTMC vs. CTMC Many books on fault tolerant or dependable computing unnecessarily restrict themselves so as to view a CTMC through the limited prism of a DTMC like in this state diagram. Instead, by using the rich theory of CTMC directly, we can gain efficiency in expression and solution. 0 1 λ t μ t 1- λ t 1-μ t
Copyright © 2003 by K.S. Trivedi 7 Example-Defective Distribution We now consider task-oriented measures for the two-state availability model. Consider a task that needs x amount of time to execute in absence of failures. Let T ( x ) be the completion time of the task. First consider λ = 0 so that there are no failures. In this case T 1 ( x ) = x , and the distribution function of T 1 ( x ) is the unit step function at x Next consider a nonzero value of λ but set μ = 0. Assuming that the server is up when the task arrives, the task will complete at time x provided the server does not fail in the interval (0 , x ). Otherwise, the task will never complete, hence

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Copyright © 2003 by K.S. Trivedi 8 Example-Defective Distribution (contd.) T 2 ( x ) is a defective random variable with a defect at infinity equal to 1-e - λ x , the probability that a task will never finish.
Copyright © 2003 by K.S. Trivedi 9 Example-Defective Distribution (contd.) Third case which is relatively complex has been analyzed If a server failure occurs before the task is completed, we need to consider two separate cases.

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