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Valentin Fadeev

Calculus of Warehouses. Part 3

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Edited by Valentin Fadeev, Saturday, 27 Mar 2010, 20:42

Further to the discussion of the cargo traffic through a port started in the previous post, i want to get together some thoughts and results obtained so far. What is missing here is only a leap into probabilistic formulas (i believe Poisson is just inevitable), but i'm still lacking confidence to assemble the whole thing in a working model.

Assume, we have cargo coming in at a daily rate of lamda sub n units, where n is the number of time period concerned (day in our case). Daily gate-out rate shall be mu sub n units. Then daily stock s sub n can be found recursively:

s sub zero equals alpha

equation left hand side s sub n equals right hand side s sub n minus one plus lamda sub n minus one minus mu sub n minus one

inflow

gateout

Mean storage time can be calculated as the total amount of warehouse work (see the previous post) divided by the total amount of cargo put through:

equation left hand side cap t macron equals right hand side n ary summation from i equals one to n over s sub i times open t sub i minus t sub i minus one close divided by n ary summation from i equals zero to n minus one over lamda sub i

In a continuous case it should obviously look like this:

equation left hand side cap t macron equals right hand side integral over zero under cap t open integral over zero under t open lamda of tau minus mu of tau close d tau close d t divided by integral over zero under cap t lamda of t d t

I took this definition form some old textbook and just made it complicated. However, it seems very natural and works perfectly in practice. It looks like formula for the center of gravity of the daily stok graph, but with changed order of summation (integration)

If calculations are made on a daily basis, the formula can be simplified as folows:

equation left hand side cap t macron equals right hand side n ary summation from i equals one to n over s sub i divided by n ary summation from i equals zero to n minus one over lamda sub i

Note, that if at the start of calculations there is some initial stock s sub zero with mean storage time cap t sub zero per unit, the formula needs a small correction:

equation left hand side cap t macron equals right hand side s sub zero times cap t sub zero plus n ary summation from i equals one to n over s sub i divided by n ary summation from i equals zero to n minus one over lamda sub i

It is obvious that on the long run the effect of such correction is negligible (which is also proved by actual observations).

Dwelling time

Applying the Little's formula to the above result we obtain the expression for the required warehouse space at every moment n:

equation left hand side cap e sub n equals right hand side lamda sub n times n ary summation from i equals one to n over s sub i divided by n ary summation from i equals zero to n minus one over lamda sub i

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