The Walmart checkout manager flow equation

He is in charge of a flow of customers and  flow of clerks.
Start with our abstract algebra tree, we want the trunk round. When round, or oval, there is a math between the graph representing roots and branches, the twp expansions are homomorphic. So we can match branch to root directly and uncover the equivalent continuous equations.

We have a bound, the variance of interarrival at the counter must be bound, no customer of clerk waits too long, and none exceed the bound unless they want  counters open or closed.

How are the two queues matched? Here we apply the stability requirement, the staged queues must be stable, typically; we will use a compact generator that is maximum entropy. The queues managet he flow of:  frequency * package size, package size being the number of items in their basket. We can, for completeness, say the clerk actually assembles the bag,to model the inventory input, but letus skip that..

Thus, we see, each checkout counter is a solution to a flow condition defined by stable queues, a finite Poisson model..  The net queue, customer var - clerk var,  is a variance difference, bound to normal. This is a description of a matched queue, a double sided Shannon equation.  Otherwise it is a model of Guassian agents running around inside Walmart and exiting with quantized bags of goods. If they have a large number of goods, nearly filling the basket, then they take more steps through the staged queue. 

We expect a semi - repeatable sequence, all of them solutions to the hyperbolics, I would think, as the 'rank' of the queue went to infinity. But the Walmart manager only sequences through a semi-repeatable sequence, rarely opening or closing counters; rarely changing items per basket.  Those changes do not occur unless counters exceed the mutual arrival variance, a clerk is gone too long, or customers partially dry up.

The banker is working with S/L, or better, L/S depending on the circumstances, and maximizing the second derivative, liquidity.  He could actually superimpose the currency function last set of solutions resulting in swaps, and those prior points should always point the second derivative to better value than before. The above is true generally, more specifically, I suspect the flow is hyperbolic, then he can use the tanh flow conditions.  But he is still doing finite, stage queues, and will exhibit a set of solutions, almost repeatable, and seemingly uniform random. Simply using interest swaps, as needed, to select the next best solution.