# Tank Problem

• February 9th 2009, 04:13 AM
Robert Hall
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?
• February 9th 2009, 11:02 PM
CaptainBlack
Quote:

Originally Posted by Robert Hall
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
• February 11th 2009, 01:19 AM
Robert Hall
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 $f_{X \mid Y=n_0}(n) ?$