The Loop That Will Not Settle, and the Loop That Settles on Something Wrong

The service answers a customer's written question about an order in four steps.

It sorts the request, finds the matching order and the policy that applies,

drafts the reply, and checks the draft against that policy. The first three

steps run once each. The fourth is different, because it can say no, and the

only sensible thing to do when it says no is to draft again.

That turns a chain into a loop, and a loop has a property the chain did not.

Where it stops is no longer a fact about the system. It is a decision somebody

has to make, and if nobody makes it the loop stops when the money runs out.

Write $p$ for the chance that one pass through draft and check produces a draft

the checker accepts. Measured on this service, $p$ is 0.6. Write $q = 1 - p$,

which is 0.4, for the chance a pass does not settle it. Write $N$ for the cap,

the number of passes allowed before the loop gives up, and let $N = 3$.

The prices are the ones every step in this service is charged at. Each step

sends 1,500 words of instruction and context and gets 300 back, input is 3 units

per million words and output is 15, so a step costs $1500 \times 3 / 10^6 =

0.0045$ for what it sends and $300 \times 15 / 10^6 = 0.0045$ for what it

returns, which is 0.009 a step, and takes 0.8 seconds. Drafting and checking are

two steps, so one pass of the loop is 0.018 and 1.6 seconds. Sorting and finding

are the other two, also 0.018 and 1.6 seconds, and they happen once.