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Math Help - long run distribution for a continuous time Markov chain

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    long run distribution for a continuous time Markov chain

    3 state; 1,2,3
    v1=2, v2=3,v3=5
    embedded markov chain has transition matrix P

    0 ; 1/4 ; 3/4
    2/5 ; 0 ; 3/5
    1/3 ; 2/3 ; 0

    a. determine the long run distribution of this chain
    b. determine the mean first passage time from state 2 to state 3
    my solution
    v1P1=q21P2+q31P3
    v2P2=q12P1+q32P3
    v3p3=q13P1+q23P2
    P1+P2+P3=1
    with q21=v2P21=3.2/5=6/5
    similarly q31=v3P31=5/3
    q12=v1P12=.5
    q32=vpP32=10/3
    q13=v1P13=1.5
    q23=v2P23=9/5

    I solve P1=30/73, P2=25/73, P3=18/73
    b. mean time from state 2 to 3
    m23=1+P21m13+P22m23
    =1+2/5 m13
    m13=1+P12m23+P11m13
    = 1+1/4 m23
    I solve m23=14/9
    I would like to know whether or not my solution is corrrect. if it's not correct, hope someoone give me some comments to correct it
    thank you
    Last edited by johnma; June 9th 2014 at 09:04 PM.
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