# Queueing Models > Choosing, parameterising and falsifying queueing models: M/M/1, M/M/c, M/G/1, finite/loss and closed networks; Erlang C/B, Pollaczek–Khinchine, Kingman/Allen–Cunneen, variability, queue topology and what model assumptions permit. Use when a predicted wait time disagrees with the measured one, when latency is far worse than utilisation suggests, when service times are bimodal or GC-spiked, when arrivals are retries or cron bursts rather than independent users, when Erlang C must be computed for a large number of servers, when routing or partitioning changes the queue topology, or when deciding whether a measured tail can be inferred from an analytical model. Does not cover the `L = λW` conservation law or operational pool sizing (littles-law-and-queueing), the alpha/beta scalability model (universal-scalability-law), or the statistics of the measured numbers themselves (latency-statistics). - Skill: `robsonkades/queueing-models` (Agent Skill, multi-file: 5 files) - Install (CLI): `npx skillmds@latest add robsonkades/queueing-models` - Raw SKILL.md: https://api.skillmd.com/api/skills/robsonkades/queueing-models/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: AI & ML - Author: robsonkades (https://skillmd.com/u/robsonkades) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/robsonkades/queueing-models --- # Queueing Models ## Purpose Use the smallest model that answers the decision, expose its assumptions, and try to falsify it on held-out operating points. The failure this skill prevents is a precise Erlang/Kingman output whose queue boundary, population, routing, service process or outcome policy does not match the system. Every model here is one formula plus a set of assumptions. The formula is the cheap part. Which parameters you feed it, and which assumption you have quietly broken, is the work. ## Workflow Inspect the project's compiler release, resolved executor/framework configuration and runtime before applying Java examples. The lifecycle snippet is partial Java 8+ code; management/JFR observations are scoped to JDK 25 and need verification on the deployed build. No model choice authorizes changing Java, dependencies or runtime configuration. 1. **Write down the Kendall notation you are claiming** — `A/S/c` at minimum, plus `K` if the system rejects and `N` if the population is fixed. Naming the arrival distribution, the service distribution and the number of servers forces each assumption into the open before any number is produced. 2. **Decide population and topology.** Open arrivals, finite-source/closed users, loss systems and semi-open sessions are different. Determine whether work waits in one shared queue, is routed among per-server queues, is partition-affine, or traverses a queueing network. Do not infer this from pod/thread counts. See `references/production-behaviour.md`. 3. **Characterise arrivals.** "M" requires a stationary Poisson process, not merely many HTTP users or `c_a≈1`. Inspect time-varying intensity, count dispersion, autocorrelation, batches, retries and state dependence at the selected boundary. 4. **Characterise service.** Separate queue wait from server-occupancy/resource demand; inspect empirical distribution, second moment, modality, autocorrelation and dependence on load/class. `c_s=1` does not prove exponential service, and a coefficient of variation does not determine a tail distribution. 5. **Choose the model from those answers**, not from familiarity. See `references/model-selection-and-formulas.md`. 6. **Parameterise from boundary-consistent measurements**, with uncertainty and censoring. Model servers are simultaneous service positions with the assumed service process—not “threads not currently blocked”. See `references/measuring-the-parameters.md`. 7. **Calibrate and validate separately.** Predeclare acceptable error from the decision, fit on some operating points, predict held-out loads/topologies, and inspect residuals. A 30% rule has no universal meaning; direction of error suggests hypotheses but does not identify one. 8. **Infer only metrics the model supplies.** M/M/c gives a point mass at zero plus an exponential queue-wait tail. P–K/Kingman primarily give means. Do not turn a mean correction or `c_s` into a p99, and keep queue wait distinct from total response and terminal failures. ## Rules - State the model as `A/S/c` before quoting any number from it. A wait time with no declared model is not a prediction. - In M/G/1, P–K makes mean queue wait proportional to `(1+c_s²)/2` relative to M/M/1 at the same mean/utilisation; M/D/1 is exactly half. This does not generalise unchanged to multiple servers, non-Poisson arrivals, percentiles or load-dependent service. - Use Erlang C for the probability of waiting in M/M/c. It is `C(c, a)` with `a = lambda/mu` in Erlangs — not an ad-hoc ratio, and not rho. - For large `c`, compute Erlang C through the Erlang B recursion rather than the direct sum; `c!` overflows a double at `c = 171` and the running sum overflows past `a ≈ 700` even with the term recurrence. Cross-check any published number by both methods where both run. - Pooling benefit depends on arrival splitting, server equivalence, load and discipline. Under M/M/c a shared queue reduces wait versus balanced independent M/M/1 queues, but not by a universal factor; isolation, affinity and head-of-line effects are competing objectives. - Do not confuse Kingman (G/G/1, carries `c_a`) with Pollaczek-Khinchine (M/G/1, which assumes `c_a = 1` and has no such term). Setting `c_a = 1` in Kingman reproduces P-K. For a shared homogeneous pool, an Allen–Cunneen-style mean approximation is one candidate when arrival/service assumptions and held-out validation support it; non-unit `c_s` alone does not select a model. - Kingman is a heavy-traffic mean approximation, not a general upper bound. Validate it over the load range; no `c_s` converts an exponential percentile formula into a general-service tail. - Retries, hedges, health checks and fan-out are visits/arrivals at their respective boundaries. Count them by class; retry probability may depend on queue state, invalidating a stationary exogenous-arrival model. - Expected residence across sequential stages adds for the same cohort by linearity, but stages may overlap and tail quantiles do not add. Model feedback/blocking networks explicitly. - Never insert a percentile of utilisation into a stationary formula. For changing load/capacity, use transient/fluid/simulation models or short quasi-stationary regimes only when timescale separation is demonstrated. - A bounded queue can be approximated by M/M/c/K only under Markovian arrival/service and fixed FCFS capacity assumptions. Its finite state has a stationary loss distribution even for offered `ρ≥1`, but retries, abandonment and state-dependent service need another model. Read loss and completion latency together. - Priority conservation results require their stated Poisson, service, discipline and work-conserving assumptions. Priority moves risk between classes and can starve low classes; validate per-class SLOs and aging/admission policy. - The exponential tail formulas here do not apply to M/D/1 or arbitrary bimodal service. Specific non-exponential models can have analytical distributions, but two moments alone cannot select one. Use a justified distributional solution, numerical method, simulation or measurement for their tail decisions. ## Required model card ```text Decision/metric: mean wait, wait probability, loss, tail, staffing or sensitivity Boundary/cohort: arrival, admission, departure, classes and terminal outcomes Topology: shared/per-server/partition queues; routing; stages; open/closed population Kendall claim: A/S/c/K/N/D plus patience, priorities and vacations where relevant Parameters: sources, units, uncertainty, censoring; time/load dependence Fit/validation: calibration points, held-out points, residuals and acceptance criterion Alternatives: analytical model, simulation, trace replay or direct measurement Decision limits: what the model cannot infer and conditions requiring re-fit ``` ## References - [Model selection and formulas](references/model-selection-and-formulas.md) — explicit formula contracts for M/M/1, M/M/c, M/G/1, Kingman/Allen–Cunneen, Erlang B and M/M/c/K; numeric stability, topology, tails and the boundary where simulation is required. - [Measuring the parameters](references/measuring-the-parameters.md) — arrival-process evidence, service/occupancy boundaries, model-server capacity, censoring, task-queue instrumentation and calibration/held-out validation. - [Production behaviour](references/production-behaviour.md) — structural mappings for executors, pools, partitions and routed/autoscaled fleets; open/closed/semi-open populations, retry feedback, transient fluid bounds, residual diagnosis and failure tests.