--- name: math-thumb-stochastic-processes description: "Apply Chapter 14 (Stochastic Processes: Rates, Waiting, and Dependence) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to recognize when a steady-state queue estimate exists and why delay rises near capacity." --- # Stochastic Processes: Rates, Waiting, and Dependence Use this chapter to help the reader make a checked mathematical decision. All 20 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Rate and time units; arrival/service model; stationarity; dependence; system boundary; whether count, wait, inventory, or mixing is sought. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Do you have a stable event rate and an exposure?** Use $\lambda t$ for a Poisson count scale. Use the zero-count complement for event risk and $1/\lambda$ for an exponential mean wait. - **Are event streams being combined or split?** Add rates only for independent Poisson streams. Multiply by label probabilities only for eventwise-independent thinning. - **Is the total count fixed?** Simulate homogeneous event locations as sorted uniforms; otherwise retain the exponential-gap view. - **Is randomness accumulating through independent centered increments?** Expect root-time or root-step spread. Add drift separately and inspect dependence. - **Is a time series persistent?** Translate $\phi$ into half-life and stationary variance only after checking $|\phi|<1$ and the adequacy of an AR(1) model. - **Is a smoother or renewal process being summarized?** Convert EWMA weight to a variance-equivalent window, or wait moments to a long-horizon count approximation, while keeping their regimes explicit. - **Is a Markov chain being run?** Estimate its slowest dependence scale, then use multiple-chain evidence and estimand-specific ESS rather than nominal iterations. - **Is a queue or flow system involved?** Apply Little's law with consistent boundaries. For M/M/1 calculations, establish $\rho<1$ before evaluating delay and preserve capacity headroom. - **Is $0.234$ proposed as a tuning target?** Confirm high-dimensional random-walk Metropolis assumptions; otherwise use diagnostics appropriate to the actual sampler. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Little's law uses compatible system boundaries and long-run averages; M/M/1 delay adds specific queue assumptions. Nominal simulation draws are not automatically independent draws. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 14.3.1, 14.3.2, 14.3.3 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C14-D01: Approach the queue capacity boundary**: rule 14.3.3. - **C14-D02: Follow a correlated shock through time**: rule 14.2.3. - **C14-D03: Turn exposure into a Poisson count**: rule 14.1.1. - **C14-D04: Watch a random walk spread like a square root**: rule 14.2.1. - **C14-D05: Check whether waiting makes the end nearer**: rule 14.1.4. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.