A simple view of hyperbolics as a dual queueing problem

I was looking at yet another economist problem, hiring workers who later leave. What is the best way to model this.

An input queue and an output queue.
The firm has a queue of workers leaving and people entering. Thwe firm has a rule, there should be one more person in the input queue than the output queue, that way it never has a shortage. It can change the arrival rate in and out by raising or lowering wages.
What is the equilibrium? Poisson queue where mean equals variance. so, using deviation, I (inout) and O (output), we get:
I^2-O^2 = 1
And waddya know, the hyperbolic condition. I and O are functions of wages paid. This conserves labor and so works in finite systems.
If you want some slack, let the input queue grow and the constant on the right become larger than one, more liquidity.
Labor is conserved, the condition meets not just the envelope theorm but meets Ito's calculus, so the economist can compute the probability distribution of labor over firms. This also connects wages across firms because at equilibrium the input and output rates will segregate the variances up to the optimum overlap. determined by the selected precision of the overall labor market.
It is time for the economist to move on to complete, Ito compatible, likelihood macro models. Leave time out of the system, use relative rates. Find equilibrium, then go and time the hiring process with some real firms and you can add time back in.