rulebase-co Arrival pattern analysis
Staffing models need two inputs: how many contacts, and how they arrive. Forecasting gets the first; this skill gets the second.
The common failure is assuming Poisson arrivals because the textbook does. Poisson means variance equals the mean — arrivals are independent and evenly random. Support queues routinely violate this: marketing sends, ticket system batch imports, outage piling, and retry behaviour create bursts and correlation that make Erlang C precise and wrong. Analyse the pattern before trusting any closed-form staffing number.
What you are testing
For each channel and interval length you schedule to (usually 15 or 30 minutes):
- Distribution shape — mean, variance, coefficient of variation (CV = σ/μ).
- Independence across intervals — does a hot interval predict the next one?
- Censoring — are "arrivals" only contacts that waited, not those that abandoned or bounced?
- Batch structure — discrete dumps vs steady drip.
Record findings per queue and per interval. Patterns differ by channel; averaging hides the violation that breaks your model.
Poisson and Erlang: when they apply
Poisson arrivals are a reasonable working assumption when:
- Contacts arrive from many independent customers without a shared trigger.
- CV is near 1 (variance roughly equals mean) at your scheduling granularity.
- No systematic batch import or campaign aligns to the clock.
Erlang C (built on Poisson) is appropriate for single-skill voice pools when patience and abandonment behaviour are stable and you are not in a sustained burst regime. It answers: how many agents for a service-level target given Poisson arrivals and exponential service times.
When assumptions hold, Erlang is useful. When they fail, the error direction matters more than the formula.
| Violation | Typical effect on staffing | Direction |
|---|---|---|
| Burstiness (CV > 1) | Peak intervals hotter than mean implies | Under-staffed if staffed to interval average |
| Batch arrivals | All work lands in one interval | Severe under-staff at dump; over-staff after |
| Abandonment censoring | Observed arrivals under-state demand | Under-staff; queues look "manageable" |
| Correlated intervals | Peaks cluster across intervals | Smoothing forecasts hides consecutive misses |
| Heavy tails (outliers) | Few intervals dominate SLA miss | Mean-based planning misses the bad tail |
Do not cite universal CV thresholds from industry slides. Compare CV and miss intervals in your own data — report where Poisson was rejected and by how much.
Burstiness
Burstiness means arrivals clump — variance exceeds the mean. Sources:
- Campaigns and notifications — email, push, SMS driving simultaneous contact.
- Outages and status pages — correlated retries until resolution.
- Billing and payroll cycles — predictable calendar bursts, not random noise.
- Retry behaviour — customers hammer the channel when unanswered.
Diagnosis:
- Plot arrivals by interval for representative weeks; overlay known events.
- Compute CV by interval length — if CV drops when you lengthen the interval, burstiness is within-interval; if it stays high, correlation spans intervals.
- Compare missed service intervals to mean-arrival intervals — bursts often miss when the daily total was correct.
Staffing implication: staffing to the interval mean under-serves burst channels. Options include staffing to a high percentile of interval arrivals (measure which percentile from historical miss data), overlapping shifts at dump times, or pre-positioning flex capacity — not assuming one more Erlang agent fixes a marketing send.
Batch arrivals (especially async)
Email, tickets and back-office queues often arrive as discrete batches:
- Mailbox imports overnight.
- API retries dumping failed creates.
- Partner file drops.
- Auto-routing releasing held tickets at hour boundaries.
Poisson describes drip; batches are a different process. Symptoms:
- Zero or near-zero arrivals then a spike interval.
- Handle time and arrival time decoupled — work arrives at 06:00, SLA clock starts at business open.
- Backlog jumps without a proportional arrival count in customer-facing metrics.
Staffing implication: use backlog and throughput models (Little's Law framing) for steady async work; use ** surge blocks** for known batch times. Erlang C on batch channels is the wrong tool.
Abandonment-censored arrivals
In voice and chat, only contacts who waited appear in handled-arrival counts. High abandonment makes the next interval look lighter than demand was — customers left, they did not disappear.
Failure mode: intraday management cuts staff because "arrivals dropped" after a long queue; the drop was hang-ups, not resolved demand.
Analysis:
- Compare offered contacts (entered queue) to handled arrivals by interval.
- Track abandon rate vs wait time — if abandonment rises as wait rises, censoring is active.
- Reforecast on offered volume when making staffing decisions, not handled alone.
Staffing implication: Erlang C without abandonment over-states requirement; Erlang A or simulation with your measured patience distribution is closer — but only if you feed it uncensored offered load.
Implications for model choice
| Channel / pattern | Prefer | Avoid |
|---|---|---|
| Voice, low CV, stable patience | Erlang C or A with offered load | Ignoring abandonment when it is material |
| Voice/chat, campaign-driven CV high | Simulation or percentile staffing | Plain Erlang on interval mean |
| Chat with concurrency | Concurrency-adjusted capacity; measure AHT by load | Raw Erlang C |
| Async drip | Backlog + throughput target | Erlang |
| Async batch | Scheduled surge capacity + backlog model | Flat hourly staffing from daily total |
| Multi-skill routing | Pool simulation or consolidated planning | Single-queue Erlang on fragmented pools |
When in doubt, simulate with your empirical arrival trace for a sample of peak days rather than debating formula elegance.
Traps
- Testing Poisson on daily totals — burstiness hides in aggregation.
- Using handled contacts as λ in Erlang when abandonment is non-trivial.
- One interval length for analysis and another for scheduling — mismatch invalidates the test.
- Treating marketing calendar as noise — it is the burst generator; model it explicitly.
- Assuming batch async is "low priority" — SLA clocks still run; backlog compounds.
Present results to the user
- Arrival profile by channel and interval — mean, variance, CV — with charts or tables for representative periods.
- Poisson verdict per queue — where it holds, where it fails, and which intervals drive the failure.
- Burst and batch catalogue — named sources (campaigns, imports, billing) tied to clock time.
- Censoring assessment — offered vs handled gap where abandonment matters.
- Staffing model recommendation — Erlang, percentile, simulation, backlog — per channel with assumption list.
- Known limitations — data you did not have (offered vs handled, event calendar gaps) and what to collect next.