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A fitted arrival-rate curve should do more than follow the busy parts of a day. This guide turns observed event times into a scale on which a correct model has a simple, testable pattern.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Build the transformation
Route 2 · Work the numbers
Route 3 · Diagnose the fit
Route 4 · Respect the limits
Route 5 · Check and continue
CHAPTER 1 OF 12 · BUILD THE TRANSFORMATION
1. A rate curve is a claim about where events should land
Suppose a clinic, server or help desk has a quiet opening hour and a busy second hour. A graph of counts per interval may look plausible even when the proposed intensity puts individual arrivals in the wrong places. Visual agreement is useful, but it is not a complete goodness-of-fit check.
Time rescaling asks a sharper question. If the fitted conditional intensity is correct, does the accumulated model intensity between successive observed arrivals behave like the gaps of a unit-rate Poisson process? The transformation converts a changing-rate clock into a standard clock.
This is a diagnostic for the arrival model. It does not by itself validate service times, queue discipline, abandonment, capacity or any performance conclusion built on those other parts.
CHAPTER 2 OF 12 · BUILD THE TRANSFORMATION
2. Accumulate intensity instead of elapsed minutes
For a nonhomogeneous Poisson process with intensity λ(t), define the cumulative intensity Λ(t)=∫₀ᵗλ(s)ds. Λ(t) is dimensionless: rate multiplied by time gives an expected count.
If observed arrival times are t₁,…,tₙ, transform them to τᵢ=Λ(tᵢ). The transformed interarrival increments are zᵢ=τᵢ−τᵢ₋₁, taking τ₀=0 only when the observation window begins at a genuine process origin or when the initial condition is handled explicitly.
Under the fitted model and its assumptions, the zᵢ values should behave as independent exponential variables with mean one. Equivalently, uᵢ=1−exp(−zᵢ) should behave as independent Uniform(0,1) variables. Both views contain the same information.
CHAPTER 3 OF 12 · WORK THE NUMBERS
3. Work a two-rate day from the original clock to model time
Use a deliberately simple fitted intensity: 2 arrivals per hour from time 0 to 1 hour, then 6 arrivals per hour from 1 to 2 hours. The cumulative intensity is Λ(t)=2t before one hour and Λ(t)=2+6(t−1) afterwards.
Suppose the observed arrivals occur at 0.20, 0.75, 1.10, 1.40 and 1.85 hours. Substitution gives transformed times 0.40, 1.50, 2.60, 4.40 and 7.10. Their successive transformed gaps are 0.40, 1.10, 1.10, 1.80 and 2.70.
| Arrival time t | Fitted region | Λ(t) | Transformed gap z |
|---|---|---|---|
| 0.20 h | 2 per h | 0.40 | 0.40 |
| 0.75 h | 2 per h | 1.50 | 1.10 |
| 1.10 h | 6 per h | 2.60 | 1.10 |
| 1.40 h | 6 per h | 4.40 | 1.80 |
| 1.85 h | 6 per h | 7.10 | 2.70 |
Five events are far too few for a verdict. The value of the example is mechanical: each interval is weighted by the intensity the model assigned to that part of the day.
CHAPTER 4 OF 12 · WORK THE NUMBERS
4. Convert exponential gaps into a uniform check
Applying u=1−exp(−z) gives approximately 0.330, 0.667, 0.667, 0.835 and 0.933. A correct model does not require these five numbers to be evenly spaced. It predicts that a long sequence from repeated comparable windows has a uniform marginal distribution and no remaining serial structure.
A probability plot or quantile–quantile plot compares ordered transformed gaps with exponential quantiles. A Kolmogorov–Smirnov-style uniform check can compare the u values with the Uniform(0,1) distribution. Residual autocorrelation or runs checks ask whether small and large gaps still cluster.
Distribution and dependence checks answer different questions. A sample can have a plausible histogram while retaining time order, or weak serial dependence while its marginal distribution is badly wrong.
CHAPTER 5 OF 12 · WORK THE NUMBERS
5. Changed context: a constant rate leaves the daily shape behind
Now ignore the quiet–busy structure and fit a constant 4 arrivals per hour. The same event times transform to 0.80, 3.00, 4.40, 5.60 and 7.40, giving gaps 0.80, 2.20, 1.40, 1.20 and 1.80.
One small sample cannot prove which model is better, but the comparison shows what changed. The piecewise model assigns fewer units of model time to quiet-clock minutes and more to busy-clock minutes. The constant model cannot remove systematic clock-time variation, so repeated days may show residual patterns around the one-hour boundary.
Transfer check: keep the observed arrivals but move the rate change from 1.00 to 1.25 hours. Recompute Λ before looking at a plot. A diagnostic must follow the exact fitted intensity, not the story attached to it.
CHAPTER 6 OF 12 · DIAGNOSE THE FIT
6. Use more than one residual view
A dependable review combines several views: an exponential or uniform probability plot, transformed-gap sequence versus event number, transformed gap versus clock time, and residual summaries by day, weekday or operating regime. Each view can reveal a different misspecification.
If short gaps cluster during a known surge, the model may smooth the peak too aggressively. If long gaps follow service outages, the arrival record may contain exposure gaps rather than true absence of demand. If the distribution is heavy-tailed after rescaling, unmodelled heterogeneity or self-exciting behaviour may remain.
| Residual signal | Possible explanation | Next check |
|---|---|---|
| Too many very small z | Unmodelled bursts or duplicate events | Inspect event identity and short-lag dependence |
| Too many large z | Missing exposure or rate too high | Audit closures, sensors and observation windows |
| Pattern versus clock time | Intensity shape still wrong | Refit time-of-day effects on training data |
| Serial clustering | Dependence beyond NHPP | Test renewal, state-dependent or self-exciting alternatives |
CHAPTER 7 OF 12 · DIAGNOSE THE FIT
7. Fitting and testing on the same events changes the benchmark
When λ(t) is estimated from the same arrivals being checked, transformed residuals are not a free independent test of a fully specified model. Flexible fitting can absorb random variation, making a probability plot look better than genuine out-of-sample performance.
Use held-out days when enough comparable data exist. Otherwise use a parametric bootstrap: simulate new event sets from the fitted procedure, refit each simulated set in the same way, and compare the observed diagnostic with that full workflow. The refitting step matters because it reproduces estimation uncertainty and optimism.
For operational forecasting, time-split validation is often clearer than random event splitting. The intended use is usually a future shift or future day, so the test should preserve that direction.
CHAPTER 8 OF 12 · DIAGNOSE THE FIT
8. Window edges and missing exposure are part of the model
The first observed gap may be left-censored because the previous event occurred before observation began. The final quiet period is right-censored by the window end. Treating only event-to-event gaps and discarding the closing interval can hide an intensity that predicts too many late events.
Observation exposure must enter the cumulative intensity. If a detector was offline for 20 minutes, integrating λ across that period and then interpreting no recorded events as evidence against the model is invalid. Either model the observation process or exclude the unavailable exposure transparently.
Repeated windows need consistent origins and end rules. A store opening, system restart and midnight are not interchangeable merely because each is labelled time zero.
CHAPTER 9 OF 12 · RESPECT THE LIMITS
9. Passing the diagnostic does not prove a Poisson mechanism
A non-rejection means the chosen checks did not find a clear incompatibility at the available sample size. It is not proof that arrivals are truly Poisson, that the intensity is unique, or that every relevant dependence has been tested.
Aggregation can conceal heterogeneity. Separate sources may each be structured while their total looks smooth. Timestamp rounding can create ties. Batch arrivals violate the one-event jump convention unless event and job counts are represented correctly. Retries can make attempts differ from unique demand.
State-dependent arrivals require a conditional intensity that includes the relevant history or system state. Using only a deterministic clock-time curve answers a narrower question.
CHAPTER 10 OF 12 · RESPECT THE LIMITS
10. Connect model fit to queue decisions cautiously
A well-calibrated arrival model is an input to a queue analysis, not the finished decision. Service-time variation, staffing delays, priority rules, blocking, abandonment and unfinished work can dominate waits even when the arrival process is described well.
Report what was fitted, the observation population, time resolution, exposure rules, training and test windows, diagnostic plots, and the operational metric that follows. Then vary plausible arrival models in simulation to see whether the staffing conclusion is robust to residual uncertainty.
The Gold Standard habit is to separate two claims: “this model reproduces the observed event timing reasonably” and “this capacity plan meets the chosen service objective.” Evidence for one does not automatically establish the other.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers should ask before trusting the fit
Does every transformed gap need to be near one?
No. An exponential distribution has substantial variation. The prediction concerns the distribution and independence of many gaps, not a narrow band around the mean.
Can I rescale with a smoothed count chart?
Only if the chart defines a nonnegative intensity that can be integrated consistently, with its estimation and edge effects acknowledged.
What if two arrivals share a timestamp?
Check the recording resolution and whether they are a batch. Arbitrarily adding tiny offsets creates information that was not observed.
Is a uniform histogram enough?
No. Check order, clock-time structure, subgroup behaviour, window edges and the effect of fitting the same data.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Continue through the verified queueing ecosystem
Start with the Queueing Theory root, the Gold Standard of Education and the How X Works hub. For the closest mathematical bridge, read Time Averages and Arrival Averages Under Nonhomogeneous Arrivals and When Poisson Arrivals See Time Averages.
For operational alternatives, use Time-Varying Arrivals and Capacity That Arrives Too Late, Batch Arrivals, Event Rates and Job Counts, and Retries, Attempt Rates and Unique Jobs.
Authoritative technical reading: Brown and colleagues’ time-rescaling theorem paper develops the transformation and goodness-of-fit use; the PubMed record supplies the publication details. Apply the theorem to the fitted point-process assumptions actually used.
Practical evidence check: preserve the raw event times, exposure calendar, fitted intensity code, transformed residuals and held-out results so another reader can reproduce both the clock-time and model-time views.
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