Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Deriving Equations! (differentiate)as !t " 0lim( ) ( )( ) ( )!!!ti iii jji jP t t P ttP t P t"# #+ $= $ +% %0& &ij ji1Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Deriving Equations!With m states => m differential equations!m-1 independent equations!mth equation2Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Deriving Equations!Matrix NotationdP tdtdP tdtdP tdtPPPPimjmi ij imimj mm mimm11121 31 11 21 1111111 1 1 1( )( )( )( ) ( )!""!" !!!#$%%%%%%%&'(((((((=!!!#$%%%%%%&'(((((()#$%%%%&'((((***+ + + ++ + + ++ + +1jij(m-1)j……!!3Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Steady State Solutions!Steady state solution:!Steady state solution = Availability–set of linear alg. equations rather than linear differential equationslim( )tjdP tdt!"= 04Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Steady State Solution!Example: Simplex system with repair! = failure rate µ = repair rate10!µ5Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9 6Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Steady State Solution!Simplex with Repair!Solution:!Steady State Availability!e.g. Availability:The prob. thatsystem is up!10µP A tt1=+=!"µµ #lim ( )7Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Transient Solution!Simplex with Repair! is a first order diff. equationwith we get!10µ8Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Transient Solution! has general solution!Get C by setting t=0!SolutionP t P et1 10( ) ( )( )=++ !+"#$%&'! +µµ (µµ (µ (9Page: © 2011 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 9Transient Solution!with we get our steady state solution(steady state availability)t0t ! "P t P et1 10( ) ( )( )=++ !+"#$%&'=+! +µµ (µµ (µµ (µ
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