Calculate occupancy indicated in speed and speed

You have a bucket. You don't know how big the bucket is. You don't know how much water is currently in the bucket, but you know that the bucket is not overfilled and it never overflowed.

You know the rate at which water flows into the bucket, which varies at different times of the day, but is never zero. You also know the rate at which water flows out of the bucket from the sleeve, which also changes throughout the day, but is never exactly zero.

There is a certain level of error with meters that measure inflow and outflow. We know this because the total inflow is greater than the outflow, but the bucket does not overflow.

Which algorithmic approach is best for approximating bucket water at any given time?

What I have so far:

t: current time period    
Occ(t):  occupancy     
In(t): period rate in    
Out(t): period rate out    
Occ(t) = In(t) – Out(t) + Occ(t-1)   

      

        
        
        
      

    

Problems:

• When there is never really a period when we can expect a bucket to be zero-filled, we need to figure out a way to estimate occupancy when at = 0 (where t can be arbitrarily chosen from past data.) If there is some pattern as to how flow rates vary throughout the day, is there any statistical way to calculate it?

• Accounting for the error ratio of the In / Out ratio. This usually boils down to finding the difference between an input and an output, dividing it by two, and adding the result to the missing side while subtracting from the excess side. But this is like a rough approximation that obeys the start and stop condition and any increase will be appreciated.