Try
A program stops tracing a route after 64 transitions. It has not reached C or D. Is the outcome a C failure, a D success, or unresolved? Record unresolved. Changing the label to make a summary table look complete would change the experiment.
Similarly, a window with a detected 65th arrival exceeds the 64-arrival storage cap. Its count is at least 65, not exactly 64 and not zero. Censoring means the desired outcome is only partly observed under the resource limit.
Predict
Before running the workbench, write three predictions at rate λ = 2 per hour and window duration h = 1 hour. Under independent windows, the chance that both are empty is exp(−4). Under copied counts it is exp(−2). For reference routing, the eventual D chance from A is 2/5 and the mean exit time is 9/5 transitions. Predict which of these quantities should change when only the “copy first window” setting changes.
The experiments are separate: arrival timestamps use hours, while the routing chain uses logical transitions. The widget does not simulate a queue, service durations or completion deadlines. Return to independent increments and absorption bounds for the analytical predictions.
Experiment
Run the reference, then change only the dependence setting and record the changed joint prediction. Replay exactly the same settings and seed: the result should reproduce, not become an additional independent experiment. Change the seed only as part of a declared replicate plan; do not search for a seed that makes a preferred prediction look accurate.
Next switch the route to persistent routing, and then to the closed-state counterexample. Compare empirical outcomes, the full finite-state probability trace and the available mathematical conclusion. In the closed-state model, unresolved paths are expected even as the cap grows: half the theoretical paths remain trapped forever. In the regular models, a geometric bound makes the unresolved probability tend to zero. A finite simulation by itself does not prove either statement.
The workbench runs 128 replicate records, stores at most 64 arrivals per window and follows each route for at most 64 transitions. Arrival rate and duration each range from one through four in the displayed units. Controls and results are temporary. Download the JSON evidence before changing controls or resetting; it retains the settings, trials, analytical trace, caps, provenance and version. Keep those files with the predictions and relevant rows in a retained investigation report. The widget does not keep a saved run history, and resetting controls does not create a new independent validation set.
Understand
A sampler implements a mathematical transformation
For an ideal Uniform(0,1) variable U, W = −ln(1 − U)/λ has exponential survival: W greater than w means U greater than 1 − exp(−λw), an event of probability exp(−λw). The alternative −ln(U)/λ has the same law, although it does not give the same wait for the same individual U.
The software uses a finite seeded pseudorandom generator, interior values strictly between zero and one, and a floating-point logarithm. It is a reproducible numerical approximation to the ideal experiment. Excluding the endpoints avoids an undefined logarithm; it does not prove that a finite generator produces truly independent continuous variables. With ideal continuous waits, a deterministic time boundary has zero hit probability. The implementation still uses the declared windows (0,h] and (h,2h] consistently.
Keep denominators and unresolved rows
Suppose 128 route trials yield 50 D labels, 70 C labels and eight unresolved paths. The observed D-by-cap fraction is 50/128. The possible eventual D fraction among those very same generated paths is between 50/128 and 58/128, because the eight unfinished paths may or may not reach D. This range describes missing outcomes; it is not a 95% confidence interval for a population probability.
Using 50/120 instead conditions on being resolved by the cap. That may be a useful secondary question if named explicitly, but it cannot silently replace the all-path calculation. Likewise, if one of 128 arrival windows overflows and the fully counted windows sum to S, the full generated-sample mean is at least (S + 65)/128. The known records do not supply a finite upper bound for that mean without further information. Retain the overflow flag.
An overflowed window is definitely nonempty, so the both-empty indicator is still known. This permits a particular statistic to remain complete even when an exact count statistic is not. Completeness must be checked for the chosen quantity, not asserted for the whole record.
Four different kinds of uncertainty
- Analytical truncation: under the row-sum condition q less than one, omitted absorption probability is at most qᵐ after m transitions.
- Monte Carlo variation: finite independent simulated trials fluctuate around their model expectation.
- Numerical approximation: logarithms, arithmetic and printed decimals are not exact real operations.
- Model uncertainty: real arrival or route observations may violate the stipulated independence, stationarity or transition rules.
Increasing the replicate count addresses only the second item under its assumptions. It does not repair a shared hidden cause or turn a trapped chain into a terminating one.
For a concrete uncertainty calculation, let Z mark a pair of empty windows. Under a fixed model Z is Bernoulli(p), with p = exp(−2λh) for independent windows and p = exp(−λh) for copied counts. For R ideal independent replicate pairs, the mean Z̄ has variance p(1 − p)/R, at most 1/(4R). Chebyshev therefore gives
This can be a loose bound; a small standard error is not a guarantee for a particular run. Repeating the same seeded record does not meet the independent-replication assumption. Return to finite-variance bounds and limit assumptions to distinguish a guarantee from a normal approximation.
Apply
Create a short protocol before a fresh run: process version, rate and duration, dependence construction, chain case, seed schedule, caps, predicted quantities, retained rows and uncertainty method. Independently check one arrival calculation with the joint mass formula and one chain calculation by substituting into the first-step equations. Preserve a result that challenges your prediction instead of replacing it with a more convenient seed.
For a changed case on paper, choose a hidden rate of one or three per hour once per pair, each with probability one half; conditional on it, generate independent counts in the two one-hour windows. Each marginal mean is two and variance three, while pair covariance is one. Redrawing the hidden rate independently per window keeps both marginal laws but makes covariance zero. Neither is the widget’s fixed-rate Poisson or literal-copy construction. Derive the difference using total variance and conditional expectation before proposing what a real observation protocol could distinguish.
The source context remains the dated MIT arrival-process notes and absorption recitation. The finite caps, paired experiments and evidence protocol are original teaching choices, not source-mandated completion standards.
Continue
Use the unit’s mixed assessment to connect arrival counts, labels, finite routes and incomplete evidence without a lesson title giving away every method. You can also revisit calculus-based probability and the public cumulative probability form. Public solutions support practice; independent mathematical review and learner transfer evidence remain separate.
