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Math Help - Tank Problem

  1. #1
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    Tank Problem

    A tank has some stochastic inflow of water, a water level and some deterministic outflow for different time steps. What I need is the probability density function for the water level in the tank at time step t+1, T_{t+1}(n).

    The probability density function for the inflow is given by  G_t(i) , the outflow by u and the tank level at time t_0=n_0. The inflow is assumed to be independent between the time steps.

    Does anyone have some suggestions?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Robert Hall View Post
    A tank has some stochastic inflow of water, a water level and some deterministic outflow for different time steps. What I need is the probability density function for the water level in the tank at time step t+1, T_{t+1}(n).

    The probability density function for the inflow is given by  G_t(i) , the outflow by u and the tank level at time t_0=n_0. The inflow is assumed to be independent between the time steps.

    Does anyone have some suggestions?
    What happens when there is insufficent water in the tank to support the outflow?

    If we allow a water debt when there is insufficient to support the outflow (so the outflow is maintained ) then the water content of the tank at epoc t is the original level plus the inflows minus the outflows. Thus we would only need to know the distribution of the RV representing the total inflow up to epoc t:

    C_t=\sum_{i=1}^t I_i

    where I_i is the RV representing the inflow between t=i-1 and t=i

    CB
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  3. #3
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    I was thinking something like this:
    Say that X is the water level at time t=1 and that Y is the water level at time t=0. Then

     f_{X \mid Y=n_0}(n) = \frac{f_{X,Y}(n,n_o)}{f_Y(n_0)}

     f_{X,Y}(n,n_o)=f_Y(n_0) f_{X \mid Y=n_0}(n)

     f_{X}(n)=\int f_Y(n_0) f_{X \mid Y=n_0}(n) dn_0

    Where  f_{X}(n) is the probability density function for the water level at time t+1.

     f_Y(n_0) should be given by
     f_Y(n_0) = G_0(i)-u+n_0

    But what is  f_{X \mid Y=n_0}(n)  ?
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