Ratios and structured queues

In the pits we are dealing with structured queues,queues whose structures are homomorphic, they have the same shape but are scaled one relative to the other.  So the sandbox pits are all about structuring the bid and ask queues such that queue structureshave the same shape.  This happens after the pit boss adds in the adjustments to the queues to get them shaped.  The pit boss is accumulating marginal gains and losses to get the match between bids and asks.

My visualization of this is quite direct, the check out counter at an imaginary grocery store. The pit boss is a runt time 'wencoder' encoding the typical number of items in a basket for each checkout clerk.  The instructions of the pit boss is to alter the number of items each clerk handles, the 5 items or lass, 5-7 items, 8 and above, and so on.  Each customer having the range of items is sent to the specific checkout clerk, and the goal is to keep all queue equal length.

The two queues being that arrival of clerks and customers independently.  When the two queues are 'matched' (each individual queues is equal and stable),  then we can compute a stable customers/clerks ratio.

Consider the hyperbolic system as a relationship between queuing processes, var(C)-var(s) = unit variance.   This is the grocery store equation,. it is the condition we just specified, the queues are stable.  The pit boss function is a random process that marginally allocates sample space, it literally changes the number of samples (things in the basket), it is a process that packs an integer, and bound index space.

So, the tanh ratio is a ratio of deviations, and is should be valid to apply the differential conditions, namely that elasticity is always greatest then there are about 1.5 ask then bid.  This result arrives because there is a maximum in the second differential.  Unsurprisingly, the maximum involves Phi, the golden ratio. And we can hand wave that connection, or search prior posts.

But, this came up again because of this post:

Elasticities and the Inverse Hyperbolic Sine Transformation

He talks about using Arcsinh functions, functions that take us to index space for reasons having to  do with the  rational approximation theorem, and the inverse hyperbolic solution  to the diophantine equations,which are restrictions on integer index space.

In conclusion, Marc has a time series relation:

y = a + x + error

The typical shotastice ergodic process.   We need to treat x and y as random functions and do ratios with them, they are arrival processes, structure the entire data set such that they are guassian arrivals and departures entering the checkout counters.  Then taking the arcsinh makes a whole lot of sense, it is exactly what the pit boss does.

Arc functions

I define these are the class of operators taking a system to index space.  Antilog log, arcsinh both examples.  Taking the log of prices, for example, implicitly means there is an index space allocated to this pricer, the pricer exists in a value chain, a structured queue. The pricers are allocated trade space according to the significance of the price posted.