The Gold Standard of Wireless Communication | Why Slot Collision Rates Differ from Attempt Collision Rates

A slot is one fixed time opportunity in our paper model. Four nodes can each attempt at most one transmission in a slot. If nobody attempts, the slot is idle. If exactly one node attempts, that attempt succeeds under the model. If two or more attempt, all attempts in that slot fail through collision.

Packets have equal slot duration, and the model excludes capture effects, bit errors, unequal powers, hidden-node effects and additional protocol overhead. These assumptions are supplied so that counting simultaneous attempts is enough to classify the slot.

A real receiver may behave differently, so a record of busy radio time should not automatically be interpreted through this rule. Begin with the declared collision model rather than infer a complete protocol from the familiar word wireless. We are studying a bounded shared-channel example whose outcomes can be checked with cards and arithmetic.

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Let K be the number of attempts in one slot. In this four-node model, K can be zero, one, two, three or four. Each slot contributes one to the slot count, regardless of K. It contributes K to the attempt count.

This is the key difference. An idle slot contributes no attempts, a successful slot contributes one, and a collided slot contributes at least two. Counting collided slots therefore does not directly count collided attempts.

Imagine a slot containing three simultaneous attempts. It is one collision slot but three failed attempts under the stated all-collide rule. A report can correctly record either quantity if it names what was counted. The mistake is to use the first numerator with the second denominator and call the resulting percentage a packet or attempt failure rate. Accurate labels are part of the calculation, rather than decoration after it.

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The first original ledger has forty idle slots, forty slots containing exactly one attempt and twenty slots containing exactly two attempts. No slot contains three or four attempts. The slot counts sum to one hundred.

The forty single-attempt slots contain forty successful attempts. The twenty double-attempt slots contain forty failed attempts. Total attempts are therefore eighty. Every slot and every attempt has a place in the account, with no attempt counted in an idle slot.

This ledger supplies observed counts for a fictional record. It does not require us to assume independent node decisions or a fixed attempt probability. We can calculate its descriptive proportions directly. A probability model can be introduced later, but it should not replace the actual denominators in a finite record with an unexamined formula chosen because the numbers happen to look plausible.

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In the first ledger, twenty of one hundred slots are collision slots. The collision-slot fraction is 20%. Sixty slots are busy, so the collision fraction among busy slots is 20/60, or one third.

Forty of eighty attempts fail through collision. The failed-attempt fraction is therefore 50%. These three results describe the same ledger: 20% of all slots, one third of busy slots and half of attempts.

There is no contradiction to resolve by averaging the three percentages. They answer different questions. The appropriate number depends on whether a reader samples a clock opportunity, an occupied opportunity or an attempt. A useful report states the numerator and denominator in words before displaying the percentage. Doing so makes it harder for a later summary to turn one legitimate measurement into a claim about a different population.

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The second original ledger also has forty idle slots, forty single-attempt slots and twenty collision slots. However, every collision slot now contains four attempts rather than two. The three broad slot categories are unchanged.

Successful attempts still total forty. Failed attempts now total eighty, giving one hundred and twenty attempts overall. The failed-attempt fraction becomes 80/120 = two thirds, even though the collision-slot fraction is still 20% and the collision fraction among busy slots is still one third.

The two ledgers demonstrate a concrete information gap. Broad slot categories do not record collision multiplicity. They can preserve the number of busy and successful opportunities while leaving the fraction of failed attempts unresolved. The missing detail is not an additional decimal place on the busy percentage. It is how many attempts the collided opportunities actually contained.

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The following table keeps each denominator explicit. The examples are fictional finite records, not measurements of an actual wireless network.

Reading only the first four rows would miss the difference in the last row. A dashboard that stores only idle, successful and collided slot counts cannot later reconstruct the omitted multiplicities without further evidence or a justified model. Keeping that limitation visible is more useful than choosing whichever attempt failure estimate sounds most consistent with a familiar network story.

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Suppose the same broad record reports S successful single-attempt slots and C collision slots, with at most N possible attempts in any slot. Under the all-collide rule, let M be the total attempts inside collision slots. Then 2C ≤ M ≤ NC.

Total attempts are S + M, and the failed fraction is M/(S + M). When S is positive, that fraction increases as M increases. Therefore the broad record gives bounds: 2C/(S + 2C) ≤ failed fraction ≤ NC/(S + NC).

For S = 40, C = 20 and N = 4, the bounds are 1/2 and 2/3. Ledgers A and B attain the endpoints, so the bounds are not merely cautious guesses. They identify exactly what those broad counts permit under the specified maximum and collision rule. Different physical reception behaviour would require a different account.

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If there are no collision slots and at least one successful slot, the failed-attempt fraction from collisions is zero in this model. If there are collision slots but no successful attempts, every observed attempt has collided, giving a fraction of one.

If there are no attempts at all, an attempt-based fraction has a zero denominator and is undefined. It should not be reported as a measured 0% failure rate merely because all slots were idle. No transmission was observed to succeed or fail.

These boundary cases are useful tests of a formula or dashboard definition. A calculation should not quietly turn absence of exposure into evidence of successful delivery. State what was observed, which population exists and which fraction can be calculated. This habit transfers to many other services where an unused opportunity is mistakenly treated as proof that the underlying task works.

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Now move from finite ledgers to a stipulated probability model. There are N = 4 nodes, each attempting in a particular slot with probability p = 1/4. Their decisions are independent within the slot. The probability p is supplied, not estimated from the two earlier ledgers.

The count K follows the corresponding binomial construction: choose which nodes attempt and multiply their independent attempt and non-attempt probabilities. A named pattern with k attempting nodes has probability p^k(1 − p)^(4 − k).

The model lets us calculate expected category frequencies over repeated comparable slots. It does not say that every block of one hundred slots must contain exactly those counts. It also does not establish that actual nodes use this independent rule. The finite descriptive account and the probability model are different kinds of statement, and the explanation should preserve that distinction when moving between them.

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A slot is idle only if all four nodes refrain from attempting. Each non-attempt has probability 3/4, so independence gives P(K = 0) = (3/4)^4 = 81/256, approximately 0.316406.

The busy probability is its complement, 175/256, approximately 0.683594. Busy here means at least one attempt. It includes successful single-attempt slots and collision slots, so it is not automatically a successful-use fraction.

This first calculation gives a clear place to inspect the independence assumption. Multiplying the four non-attempt probabilities is justified by the declared model. Four nodes each having the same individual non-attempt probability would not be enough if their decisions moved together. Later examples preserve the same marginal attempt probabilities while changing that dependence, showing why the assumption needs to remain attached to the formula.

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A successful slot has exactly one attempt. For a particular named node to be that sole transmitter, its own attempt probability is 1/4 and the other three nodes each refrain with probability 3/4. The probability is (1/4)(3/4)^3.

There are four mutually exclusive possibilities for which node is alone. Adding them gives P(K = 1) = 4(1/4)(3/4)^3 = 108/256 = 0.421875. This is the model’s successful-slot probability, not the probability that an arbitrary attempt succeeds.

The four possibilities can be added because they cannot occur in the same slot. Multiplying them instead would ask for an impossible combination of all four different nodes being the only transmitter simultaneously. Explaining this step helps learners distinguish two separate probability tools: multiply independent choices within one specified pattern, then add mutually exclusive successful patterns.

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Every slot in the model is idle, successful or collided. Therefore P(K ≥ 2) = 1 − P(K = 0) − P(K = 1) = 67/256, approximately 0.261719.

Among busy slots, the conditional collision probability is (67/256)/(175/256) = 67/175, approximately 0.382857. The restriction to busy slots changes the denominator. It removes idle opportunities while retaining both successful and collided busy opportunities.

Calling either number simply the collision probability is incomplete when the intended population is unclear. A reader needs to know whether the report concerns all slots or only occupied ones. Neither number yet answers how often a tagged node’s attempted transmission encounters another attempt. That question conditions on a different event and requires a separate calculation, even though it uses the same underlying model.

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Choose one named node and condition on its attempt in a slot. Its attempt succeeds only when the other three nodes do not attempt. Within the independent model, that probability is (3/4)^3 = 27/64, approximately 0.421875.

The tagged attempt’s collision probability is therefore 37/64 = 0.578125. The tagged node’s own probability of attempting is not multiplied again because the question already conditions on that attempt occurring.

This explains a common denominator error. The probability of a node attempting and succeeding in a random slot includes its p factor. The probability of success given that it attempts does not. Both are legitimate quantities. Confusing them makes a model appear either more reliable or less active than its actual assumptions imply. State the conditioning event before using a formula with a familiar-looking success term.

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The expected number of attempts in a slot is Np = 1. Every successful slot contributes exactly one successful attempt, so the expected successful-attempt count per slot is P(K = 1) = 108/256.

The expected failed-attempt count per slot is the difference, 1 − 108/256 = 148/256. Dividing by the expected total attempt count of one gives the attempt-weighted failure fraction 148/256 = 37/64. This matches the tagged-node calculation because the four nodes have the same stipulated attempt behaviour.

The result does not mean that a finite observed fraction must equal its expectation exactly. It gives another route to the probability model’s population account. The agreement between independent derivations is useful: one route conditions on the other nodes, while the second counts expected attempts and successes. They meet only when their populations and model assumptions are aligned.

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For K equal to zero through four, the probabilities are respectively 81/256, 108/256, 54/256, 12/256 and 1/256. The numerators sum to 256, which checks that the possible slot states exhaust the model.

Weighting by K gives the expected attempt count: (108 + 2×54 + 3×12 + 4×1)/256 = 1. Restricting that weighted sum to the collided states gives (2×54 + 3×12 + 4×1)/256 = 148/256.

The corresponding unweighted collision probability is (54 + 12 + 1)/256 = 67/256. These sums display the difference directly. A triple-attempt slot receives weight one when counting collision slots and weight three when counting failed attempts. The changed weight is the mathematical reason the percentages differ, rather than a mysterious feature of radio equipment or a contradiction in the data.

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Keep four independent nodes but change p to 1/2. The idle probability becomes 1/16, the successful-slot probability becomes 4/16 = 1/4 and the collision-slot probability becomes 11/16.

The channel is now busy with probability 15/16, yet successful slots are less frequent than in the p = 1/4 case. Among busy slots, the collision probability is 11/15. A tagged attempt collides with probability 1 − (1/2)^3 = 7/8.

The change illustrates why a busier modelled channel need not carry more successful single transmissions. Additional attempts can overlap rather than produce additional successful opportunities. The example is a comparison of two paper models, not a recommendation for changing a real network setting. Actual protocols have different access rules, dependencies and reception behaviour that need their own technical account.

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For fixed N independent nodes with N greater than one, successful-slot probability is S(p) = Np(1 − p)^(N−1). Differentiating within 0 < p < 1 gives a derivative proportional to 1 − Np, so the interior maximum occurs at p = 1/N for this model.

With four nodes, that is p = 1/4, matching the earlier comparison. The result optimises one quantity: the expected number of successful single-attempt slots per slot under the stated assumptions. It does not optimise every measure of fairness, waiting, energy, deadlines or individual experience.

Do not convert the mathematical optimum into a universal configuration rule. The actual number of active participants may change, and a real protocol need not implement this independent-attempt model. The value of the derivation is to show how an objective and assumptions produce a conditional result, while leaving other objectives and physical constraints visible.

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An attempted transmission is not necessarily a distinct application message. A packet may be retried, an acknowledgement may be lost or later protocol stages may change what the receiver accepts. Counting attempts does not independently establish unique payload delivered.

The existing retry-time goodput lesson develops that separate account. It distinguishes useful payload, repeated attempts and elapsed time. This article isolates how slots and attempts weight the same collision events differently.

A report should preserve both boundaries where needed. It can describe the fraction of attempts that collide under the paper model without calling that an application’s final message-loss probability. Likewise, a successful single-attempt slot in this simplified model does not establish a particular deadline or end-to-end service outcome. Those conclusions need their own events, records and conditions.

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Four nodes can each have attempt probability 1/4 without making independent decisions. To construct a different model, choose exactly one of the four nodes uniformly in every slot. Each node still attempts with probability 1/4, but K is always one.

Every slot is busy and successful in this construction. There are no collisions. The marginal attempt probabilities match the independent model while the joint behaviour differs completely. The rule that coordinates the choices is part of the model, not information that can be recovered from the four marginal probabilities alone.

This comparison does not prescribe a scheduling protocol. It is a mathematical counterexample to a particular inference. Knowing the chance that each node attempts does not by itself justify multiplying their non-attempt probabilities. Independence is an additional relationship among their decisions, and changing that relationship can change every slot category while preserving the individual marginals.

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For another construction, all four nodes attempt together with probability 1/4, and otherwise all remain idle. Again, each node has marginal attempt probability 1/4 and the expected total attempts per slot is one.

Now three quarters of slots are idle and one quarter contain a four-way collision. There are no successful attempts. Every attempt that occurs collides, even though the channel is busy much less often than in the independent model or the exactly-one construction.

The three models therefore share the same four marginal attempt probabilities and the same expected total attempt count, while producing very different success and collision outcomes. The missing information is dependence. A low busy fraction cannot establish a low attempt collision fraction when attempts concentrate together. This is why observations of when traffic overlaps matter alongside totals and individual activity summaries.

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An observed high failure fraction can have several causes in a real service. Our model classifies simultaneous attempts as collisions by assumption. A field measurement may also involve reception thresholds, interference, device differences, errors, timing and protocol behaviour.

The right evidence question is how the reported category was identified. Did a record directly count simultaneous attempts, infer collisions from a missing acknowledgement or combine several kinds of failure? Those methods do not necessarily measure the same event.

The Wireless Communication root connects coverage, capacity, delay, delivery and system conditions. A clear classroom model can improve the questions asked of a technical report without becoming a diagnosis of an unseen network. Preserve the distinction between an observed event, a modelled mechanism and a proposed explanation for the actual service.

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There are two different sampling procedures that can both sound like choosing a transmission. One chooses a busy slot uniformly, then chooses one of its attempts. Another chooses uniformly from all attempts in the complete record. They need not produce the same collision fraction.

In Ledger A, forty busy slots contain one attempt and twenty contain two. Choosing a busy slot first gives a one-third chance of selecting a collision slot. Selecting one of the attempts inside that already chosen slot does not change whether the selected slot collided. The resulting attempt is not uniformly sampled from all eighty attempts, because attempts in single-attempt slots each receive more sampling weight.

Choosing uniformly from the complete eighty-attempt list instead gives forty failed choices out of eighty, or one half. That procedure weights a double-attempt slot twice because it contains two eligible attempts. Neither sampling procedure is inherently impossible or dishonest. They answer different questions. A report should state the actual selection rule before treating a conditional result as the experience of a uniformly sampled attempt.

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In the four-node independent model, a uniformly attempt-weighted account assigns multiplicity k a weight proportional to k times P(K = k). The normalising denominator is E[K], which equals one in our chosen p = 1/4 case.

The attempt-weighted probabilities for multiplicities one through four are therefore 108/256, 108/256, 36/256 and 4/256. They sum to one. The last three categories sum to 148/256, again giving the collision probability among attempts. By contrast, conditioning a uniformly chosen slot on being busy divides the unweighted category probabilities by 175/256.

The arithmetic explains the changed perspective without assigning a physical cause. Slots with more simultaneous attempts appear more often when the sampling unit is an attempt. A learner can check this by writing each slot number once per attempt before sampling. The list becomes longer for more crowded slots, exactly as the weighted formula requires. This sampling rule is part of the modelled claim and should remain explicit when results are compared.

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Return to the fifty-slot transfer record with twenty successful slots and ten collision slots. Suppose an additional record establishes forty-six total attempts. Then twenty-six attempts belong to collision slots, and the failed-attempt fraction is 26/46 = 13/23.

With at most three attempts per slot, write n2 and n3 for the numbers of double- and triple-attempt slots. The equations are n2 + n3 = 10 and 2n2 + 3n3 = 26. Subtracting twice the first equation gives n3 = 6, leaving n2 = 4.

If up to four attempts were possible, the same totals would not identify every multiplicity. The collision counts could contain four double and six triple slots, or seven double and three quadruple slots, among other possibilities. Both produce twenty-six failed attempts. The extra attempt total resolves the requested failure fraction even when finer details remain ambiguous. An evidence review should ask which missing quantity is actually needed for its conclusion, rather than demand a complete reconstruction of every event before answering any bounded question.

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MIT OpenCourseWare’s 2012 lecture on shared media and MAC protocols provides the teaching context for contention and slotted Aloha. It discusses performance measures and the independent-attempt model used to analyse shared access.

The source was checked on 26 October 2026. The two one-hundred-slot ledgers, their bounds and the four-node comparisons in this article are original worked teaching accounts. They are not measurements from MIT’s slides or reports of an actual wireless deployment.

Use the source to continue into the protocol model, while retaining the conditions under which a formula applies. A historical teaching lecture is not a current radio-configuration or regulatory manual. This article gives no operating permissions, transmission settings or compliance conclusion; its practical learner task is to explain and check the selected populations.

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A new fictional record has fifty slots: twenty idle, twenty successful single-attempt slots and ten collision slots. At most three nodes can attempt in a slot, but the record does not state whether the collided slots contained two or three attempts.

Calculate the busy fraction, the collision fraction among all slots and the collision fraction among busy slots. Then bound the failed fraction among attempts. Construct one complete multiplicity ledger attaining each bound.

The problem changes the observation length, successful count, collision count and maximum number of participants. Return to the definitions rather than copying a percentage from the earlier example. A correct answer should make clear why the broad slot fractions are fully determined while the attempt fraction remains an interval. That explanation is the transferable result, rather than a memorised collision formula detached from its population.

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Thirty of fifty slots are busy, giving 60%. Ten of fifty are collision slots, giving 20%. Ten of thirty busy slots collide, giving one third. These broad percentages are determined by the supplied slot categories.

If all ten collision slots contain two attempts, there are twenty failed and twenty successful attempts, giving a failed fraction of 1/2. If all contain three attempts, there are thirty failed and twenty successful attempts, giving 3/5. Both ledgers satisfy the same fifty-slot broad record.

The bounds are therefore 1/2 to 3/5 under the three-node limit and all-collide rule. An answer of one third has used the busy-slot denominator instead. An answer of 20% has used all slots. Those values are not arithmetically wrong; they answer different questions from the requested fraction among attempts. Diagnosing the denominator is more useful than merely replacing the final percentage.

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For a separate transfer exercise, use three independent nodes, each attempting with probability 1/2 in a particular slot. Derive the idle, successful and collision-slot probabilities, then the conditional collision probability among busy slots and the tagged attempt’s collision probability.

The multiplicity probabilities for K = 0, 1, 2, 3 are 1/8, 3/8, 3/8 and 1/8. Thus the collision-slot probability is 1/2 and the busy probability is 7/8. Their ratio is 4/7.

Given that one named node attempts, the other two both refrain with probability 1/4, so the tagged attempt collides with probability 3/4. Independently, expected attempts per slot are 3/2 and successful attempts per slot average 3/8. The failed-attempt ratio is (3/2 − 3/8)/(3/2) = 3/4. The two routes agree because they preserve the same independent, symmetric model and attempt-based population.

A heterogeneous example adds another population check. Suppose two independent nodes attempt with probabilities 1/2 and 1/10. The first node’s attempt collides with probability 1/10; the second node’s attempt collides with probability 1/2. Averaging those two conditional probabilities equally gives 3/10, but that is an equal-node average, not an attempt-weighted fraction.

Expected attempts per slot are 3/5. A joint attempt occurs with probability 1/20 and contributes two failed attempts, giving an expected failed-attempt count of 1/10. The attempt-weighted failure fraction is therefore (1/10)/(3/5) = 1/6. The more active node contributes more attempts to that denominator. A report should distinguish a typical node selected uniformly from a typical attempt selected from traffic, especially when participants have different activity levels. Neither sampling unit automatically represents every user’s experience.

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For a slot-based descriptive account, retain the observation period and the counts of idle, single-attempt and collided slots under the stated classification method. For an attempt-based conclusion, retain the number of attempts within relevant slots or another legitimate record that supports that denominator.

For a probability model, state the participant count, attempt assumptions, dependence, success rule and conditions. Mark which inputs were observed, estimated or stipulated. A formula can calculate a conditional consequence without establishing that its assumptions describe the actual network.

The record should also identify the task left unresolved. Slot utilisation, attempt collision, unique payload goodput, delay and end-to-end deadline delivery belong to related but distinct accounts. Keeping them separate helps a learner choose the next useful article and helps a reviewer see which evidence is missing, rather than letting one attractive percentage stand in for an entire service evaluation.

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For repair, place one card for every slot, then place zero, one or several attempt counters on each card. Count collided cards and failed counters separately. The physical representation makes the change of weighting visible before probability notation is introduced.

For stabilisation, reconstruct the two one-hundred-slot ledgers and their attempt fractions. Then derive the finite-record bounds and explain why the endpoint constructions attain them. The learner should be able to say which fact is missing when only broad slot counts are supplied.

For extension, use the independent four-node model and then change its dependence while preserving marginals. The education parent connects that progression with independent transfer. The useful achievement is recognising which assumptions justify a calculation and which changed conditions require a different account, without borrowing authority from a familiar formula.

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Is a collision-slot fraction an attempt failure fraction?

Not generally. A collided slot can contain several failed attempts, while each slot contributes only one to the slot denominator.

Does a busy channel necessarily deliver more successful attempts?

No. In the independent examples, increasing attempt probability can increase collisions while reducing successful single-attempt slots.

Do equal marginal attempt probabilities establish independence?

No. The exactly-one and all-together constructions preserve individual probabilities while changing the joint decisions and collision outcomes.

Does this model recommend a setting for my network?

No. It is a stipulated teaching model. Actual protocol, equipment, reception conditions and legitimate technical assessment determine the relevant service account.

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Rebuild the complete communication task

Return to Wireless Communication for coverage, capacity, conditions and end-to-end service evidence. A slot model answers one bounded question inside that wider account.

Distinguish attempts from delivered work

Use Accounting for Retry Time in Unique Payload Goodput when repeated attempts and elapsed time affect useful delivery. Use Deadline Delivery, Burst Loss and Honest Reliability Claims when timing and clustered failures affect the actual task.

Carry the sampling-unit check elsewhere

The Gold Standard education parent develops independent checking. The How X Works hub provides adjacent subjects. Carry forward one question: am I counting opportunities, occupied opportunities, attempts or completed work, and does my denominator describe the claim I am making?

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