A lower bound is permission to stop

If every candidate still needs the same three reference calls, the loop cannot run faster than those calls. A lower bound is permission to stop: remaining spend cannot buy speed if the irreducible per-item work remains. Any tool that claims to beat that floor must name which unit it deletes.

Why these three models

The decision is whether to keep spending on a faster hiring, diligence, or collections loop after the process already sits on its information floor. The person who signs the next invoice has to make it this month, before that invoice is paid.

A vendor bake-off hides the floor. A quality-bar question (maybe the three calls, the field visit, or the signed LPO were never required) is real and sits outside this piece. A schedule question is how the miss arrives: the plan already promised a shorter cycle time, and remaining slack is being spent to hit it. This article takes the information-value framing and uses the schedule as the check. What that hides is theatre. Three reference calls that all go to friends are not a floor.

Three lenses, three outcome types. Value of information: a purchase that cannot change the action has no value as a speed purchase. Queueing: the floor is n times the unit, and congestion only sits on top. Technical-performance burndown: the plan’s margin is being spent on a speed you cannot have. LOOP is folded. The referee, the LP’s counsel, and the customer’s finance clerk all have a next period, but the lever is the unit they still have to do, not a patience inequality. Who may approve a tool that does not name a deleted unit sits in the levers, and the pull to keep buying because the last tool almost worked adds no separate card.

1. The remaining invoice cannot buy a faster call

An upper bound is an algorithm you actually have: a procedure that is correct on every instance, inside a stated time.¹ You already have one. Three reference calls. A signed LPO and a bank statement. The clerk who can release the payment. Those loops run. Their time is ugly and known.

A lower bound is also an algorithm, for a different problem. The input is a claimed faster procedure. The output is one instance on which that procedure is wrong or too slow.¹ If you cannot name an item that no longer needs the unit, you do not have a faster loop. You have a slide.

Edmonds calls a procedure feasible when its time is a polynomial in n, and infeasible when the time is exponential.¹ Do not borrow that word for funding. He means growth rate. The operator fact is smaller. If the per-item work stays, time is at least n times the unit. Spend past that floor is not a better algorithm. It is a description of the one you already run.

Value of information is the price a decision-maker would pay to know something before acting.³ Information that cannot change the action is worth nothing for that action. You will still make the three calls. You will still wait for the signed LPO. You will still sit on the line for the clerk who only picks up after three. A scored CV or a colour-coded ageing report does not change those actions. For “can this loop run faster,” that purchase has zero value. It may help a different decision: fewer no-shows, a cleaner raise file, a list of who is ninety days late. Buy it for that, under that name. Do not buy it as speed.

Tools that claim to beat the floor must name which unit they delete.

The test is one sentence on the invoice. “This removes the second reference call.” “This replaces the field visit with a written confirmation we already trust.” If the sentence will not write, the tool is an upper bound you already possess, with a login. Edmonds’s other stop is for small n. If the input size will not be larger than six, do not waste the time writing an extremely efficient algorithm.² Four hires this quarter is that case. Twelve invoices awaiting a signature is that case. Optimising a loop you will run a handful of times spends the month the calls would have finished.

Value of information, the equilibrium lens

• Assumes: a purchase has value for a decision only if it can change the action.
• Fits because: remaining spend is being sold as speed after the per-item work is already required.
• Breaks when: the tool deletes a named unit, or the unit was never required.
• Evidence: grade A. Definitional; Howard 1966 prices information before the act.
• Counteracts: treating spend and cycle time as the same variable.
• May reinforce: refusing a tool that does delete a real unit.

2. The floor is n times the unit

Work arrives at people with finite hours. If three reference calls, a field visit, or a signed LPO must happen for every item, service time has a lower bound. Little’s law says the long-run number in the system equals arrival rate times time in the system, and it does not care about the arrival distribution, the service distribution, or the service order.⁴ If time cannot fall below the unit, throughput cannot rise above the rate that unit allows. In most queues the service time is the bottleneck that creates the wait.⁴ Software that does not touch the service time cannot touch that wait.

Congestion sits on top of the floor. It does not replace it. In the elementary single-server model, mean response time is the service time divided by one minus utilisation.^{5, 9} That quantity is always at least the service time. At moderate load the extra wait is small. Near capacity it climbs sharply. A founder reads the climb as proof that a faster procedure exists, and buys a tool. The tool can shave the extra. It cannot shave the unit. Thirty signed LPOs, each needing a call to the same clerk, are waiting on one person’s afternoon, not on a dashboard.

Two repairs, neither a new login. Add servers who can actually do the unit: a second person allowed to make the reference call, a second staffer who can sit in the customer’s accounts office. Or delete a unit in writing, because you have decided it was never load-bearing. Parallelising the three calls on one candidate helps that candidate. It does not change n times the unit across the book. Twelve candidates times three calls is thirty-six conversations. A scored inbox does not make that twelve.

If utilisation is genuinely high, the unit is already as small as you will allow, and items wait because there are not enough hours, then people are the answer. Hire collectors. Do not buy a collector-shaped screen while the same conversations still have to happen.

Queueing, the complex lens

• Assumes: each item has a service time that cannot go below the unit, and arrivals meet finite servers.
• Fits because: the floor is n times that unit, busy board or quiet.
• Breaks when: arrivals feed on the wait, or the unit can be parallelised to nothing.
• Evidence: grade A. Little’s law is distribution-free; the blow-up near capacity is structural.
• Counteracts: reading a long cycle time as proof that a faster algorithm exists.
• May reinforce: calling an hour-shortfall a floor and refusing the hire.

3. The plan is burning margin on a number it cannot hit

NASA tracks technical performance by comparing what has been achieved with the value anticipated for this date, and with values projected for later dates. A reading outside the expected range is supposed to force a correction.⁶ Margins are allowances put into budget, schedule, and technical parameters up front, and they are typically consumed as the programme proceeds.⁷ Consuming margin faster than planned is a leading indicator. The end date has not slipped yet. The slack that was supposed to absorb the slip is already gone.

Your hiring plan, diligence calendar, or collections forecast already has this shape. Cycle time was supposed to be twenty-eight days by month end, twenty-one by the end of the next. The remaining days in the band are the margin. The invoice in front of you is how that margin is being spent. If the floor is three reference calls plus a field visit, and those still take most of three weeks because other people have calendars, then the descent from twenty-eight to twenty-one is a number no procedure you own can hit. The plan’s margin is being spent on a speed you cannot have.

If the existing trend will produce an unfavourable outcome, corrective action is supposed to start as soon as practical, not at the deadline.⁸ Look at the last six closed files. Plot days against the plan. If the line is flat and spend on acceleration is rising, the band is already breached. Stop the spend. Rewrite the profile around the floor, or delete a unit in writing and rewrite around the new floor. Do not keep the old profile and buy another tool to defend it.

The grade is B: institutional practice, not an experiment. A profile set to please a board is theatre. A cycle-time number people are paid on will be hit by dropping a unit quietly, and a floor you still needed comes back as a bad hire or an unpaid invoice you cannot prove.

Technical-performance burndown, the cycle lens

• Assumes: a dated planned profile, a tolerance band, and a margin allocated up front.
• Fits because: a cycle-time number and a date already exist, and remaining slack is being spent on forbidden speed.
• Breaks when: the profile was political, the measure is gamed, or there is no intermediate observable.
• Evidence: grade B. NASA handbook practice, not a trial.
• Counteracts: waiting for the end date to admit the plan will miss.
• May reinforce: treating a band you set yourself as independent evidence.

GEER: cut the unit, or stop buying around it

Draw the boundary first. Inside: the unit on each item, remaining spend aimed at speed, and the planned cycle-time profile. Outside: whether the quality bar is right, reference rules in your market, and the customer’s payment policy. Those three will surprise you. You agreed to that when you left them out.

The stock you are trying to move is cycle time. You cannot set a stock. You can change the inflow of items, the people who can do the unit, or the unit itself. The lag is why the last tool is still credited. An ATS bought in March produces a “faster” June because two quiet roles closed. Size the correction as if that lag were real.

Ruin sits on skipping the unit, not on refusing the tool. A hire without the calls, a facility without the LPO, a large invoice released on a verbal, can end you with payroll theft, a lender, or a customer who never meant to pay. Keep the unit in place until someone writes that it is no longer required.

1. Name n, and name the unit. Candidates this quarter, files in the raise, invoices past due. Then the work on each: the three calls, the field visit, the signed LPO, the clerk. One hour, one page.
2. Write the floor as n times the unit. Thirty-six conversations. Twelve LPO chases. That number is permission to stop, and the hours you still have to staff.
3. Put one sentence on every speed invoice. Which unit is deleted. Blank sentence, no payment. You can still buy the tool later as filing, under that name.
4. Add a server before you add a screen. A second person allowed to make the call beats a dashboard of the first person’s queue.
5. If n is small, stop. Six or fewer items is Edmonds’s own cut. Finish the calls.
6. Do not skip the unit to hit the profile. That is the ruin path. The floor is a safety rail as well as a stop.

RADAR: what to freeze before the next vendor invoice

Do now, by T+3, one afternoon, effect visible the same day. Write n, the unit, and the floor for the loop you are being asked to accelerate. Reversible. Dominant across every story about why this vendor is different. The effect arrives when the next demo meets a number.

Hedge, by T+14, premium is a week of the old process, cover live before the invoice date. Require the deletion sentence on any speed purchase already in flight. If the vendor cannot name the unit, keep running the calls, the visit, or the LPO chase in parallel. The premium is the double work for one cycle. The cover has to be live before you switch the old process off.

Defer and trigger, size declared when the trigger is set. A new ATS, a paid data room, a collections platform, is spend you will not unwind cleanly. Pre-commit the observable: six closed items under the new tool, and a count of units actually skipped. If the skip count is zero, reclassify as filing or cancel. Set that trigger now, while the demo still sounds like speed.

CHAIN: what a faster-loop purchase usually does next

Name the shape before explaining the last tool. Cycle time flat, spend on acceleration rising, is a stock that is not draining next to a flow you keep increasing. The other shape is goal-seeking that never arrives: the plan still points at twenty-one days, actuals bounce between twenty-six and thirty, and each bounce funds the next invoice.

Has a rule changed, has an actor entered or left, has a measurement become a target? The third fires often. Time-to-hire or days-sales-outstanding becomes the thing the team is judged on, and the unit starts to leave the file. That is a different process from the one whose history you are about to quote.

Match the class on structure: any loop whose remaining work is a conversation with a person you do not employ. A former supervisor. A customer’s finance clerk. A landlord who must stamp the LPO. Software rearranges your side of the desk. It does not rearrange theirs. Those loops do not get faster from a login. Subtract the counterfactual before crediting the vendor. A quarter with four hires instead of ten will look fast. A collections month after a supermarket pays its January LPOs will look like the dashboard worked.

If a lender or a lead investor now requires a visit or a document you used to skip, the floor rose. The old profile describes a process you are no longer allowed to run.

What a floor cannot see

These three models can tell you that remaining spend cannot buy speed, that the wait cannot fall below n times the unit, and that the plan is burning slack on a number it cannot hit. None of them can tell you whether the unit you named is the right quality bar. A field visit that always confirms the CV may be deletable. A third reference call that has caught two payroll disasters is not. Score the last twelve files for what the unit actually caught.

One property no member models. Once a floor is named, people treat every step as sacred, including the ones that were always cargo cult. The informal skip on a candidate everyone already knew, or on a repeat customer whose LPO has cleared twenty times, gets written back in for consistency. The floor becomes thicker than the work. You notice it when a role you could have closed in a week is still open because the ritual now has a page.

A different person in the same seat, with the same demo and the same board date, would often sign. They are trapped by the plan. Change the profile and the invoice rule. Do not lecture them.

The one action that survives the ignorance: before Friday, write the irreducible units for the loop you are about to accelerate. Then attach one sentence to every speed invoice in the next twenty-eight days naming the unit that will disappear. If the sentence is blank, do not pay.

Who signs, and the test that stops them

The person who needs this is the one holding the vendor invoice, or the one who put a shorter cycle time into this quarter’s plan. The cheapest first test is an afternoon: n, the unit, the floor, and last six actuals against the plan. If the actuals sit on the floor and the plan sits below it, the next payment is not a speed purchase. Call it filing.

Sources and notes

1. Jeff Edmonds, How to Think About Algorithms, Cambridge University Press, 2008. Introduction, printed p. 2: a procedure is feasible if its time is a polynomial such as Time(n) = Θ(n²), and infeasible if the time is exponential such as Time(n) = Θ(2ⁿ). Chapter 7, printed p. 85: “An Upper Bound Is an Algorithm,” obtained by constructing a procedure that is correct on every input inside the bounded time; “A Lower Bound Is an Algorithm” for a different problem, whose input is a claimed faster procedure and whose output is an instance on which that procedure is wrong or too slow. Printed pages follow the book’s pagination (PDF page minus 16). Book record: cambridge.org. Chapter 7 landing (abstract only on the open page): Chapter 7. Edmonds’s “feasible” is polynomial time. It is not used here as a claim about funding-feasibility.
2. Edmonds, ibid., chapter 23, printed p. 366. “If the input size won’t be larger than six, don’t waste your time writing an extremely efficient algorithm.” Verified in the open chapter extract. cambridge.org chapter 23.
3. “Value of information,” Wikipedia. VOI is the amount a decision-maker would be willing to pay for information prior to making a decision. The value cannot be less than zero, because the extra information can be ignored. The operator corollary used in section 1, that information with no power to change the action has no value for that action, follows from the definition. Howard 1966 is the named source on the page for information value theory. en.wikipedia.org/wiki/Value_of_information. Ronald A. Howard, Information Value Theory, IEEE Transactions on Systems Science and Cybernetics 2(1), 1966, pages 22 to 26. The IEEE copy is not used as a click-through here.
4. “Little’s law,” Wikipedia. In a stationary system the long-run average number of customers L equals the long-run average effective arrival rate λ times the average time a customer spends in the system W, written L = λW. The relationship is not influenced by the arrival-process distribution, the service distribution, or the service order. “In most queuing systems, service time is the bottleneck that creates the queue.” en.wikipedia.org/wiki/Little’s_law. Original proof: John D. C. Little, A Proof for the Queuing Formula: L = λW, Operations Research 9(3), 1961, pages 383 to 387.
5. Raj Jain, Analysis of a Single Queue, Washington University in St. Louis, CSE 567 lecture notes, 2017. For the M/M/1 queue, mean response time E[r] = (1/μ) / (1 − ρ), which is at least the mean service time 1/μ for every utilisation ρ below 1. cse.wustl.edu/~jain/cse567-17/ftp/k_31asq.pdf.
6. NASA Systems Engineering Handbook, NASA SP-2016-6105 Rev 2. Appendix B glossary: technical performance measures are “monitored by comparing the current actual achievement of the parameters with that anticipated at the current time and on future dates.” Values that fall outside an expected range around the anticipated values “indicate a need for evaluation and corrective action.” Section 4.2.1.3, handbook p. 61: TPMs are monitored by comparing current actuals or best estimates with values anticipated for the current time and projected for future dates. science.nasa.gov handbook PDF.
7. NASA SP-2016-6105 Rev 2, Appendix B, handbook p. 193. “Margin: The allowances carried in budget, projected schedules, and technical performance parameters (e.g., weight, power, or memory) to account for uncertainties and risks. Margins are allocated in the formulation process based on assessments of risks and are typically consumed as the program/project proceeds through the life cycle.” Same PDF as note 6.
8. NASA SP-2016-6105 Rev 2, section 6.7.1.2.2, handbook p. 159. Key reviews “are points at which technical measures and their trends should be carefully scrutinized for early warning signs of potential problems. Should there be indications that existing trends, if allowed to continue, will yield an unfavorable outcome, corrective action should begin as soon as practical.” Same PDF as note 6.
9. “M/M/1 queue,” Wikipedia. The model is stable only if the arrival rate is below the service rate. Average response time can be computed using Little’s law as 1/(μ − λ), equivalently the service time divided by one minus utilisation. en.wikipedia.org/wiki/M/M/1_queue.

Joshua Agonya Pi’Rwot, Founder.