Arrival processes, absorption and honest simulation
Construct a Poisson process from waiting times, prove finite-window independence and thinning, and solve absorbing chains with a termination certificate. Audit bounded simulations without discarding unresolved outcomes or mistaking repeated seeds for new evidence. These four lessons extend existing probability and directed-walk material; they do not constitute a complete stochastic-process course.
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Readiness and preparation
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Connect arrivals, route outcomes and incomplete evidence
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Lesson sequence
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Build arrivals from independent waiting times
Lesson details and objectives
Connect a waiting-time construction, arrival epochs and interval counts before simulating.
- Derive arrival-time densities and count probabilities from independent exponential waits.
- Prove finitely many arrivals occur in a bounded horizon rather than assuming a simulation cap proves it.
- Track rate units, time boundaries and changes to independence or constant-rate assumptions.
6 practice tasks.
Prove what can be separated into independent counts
Lesson details and objectives
Use a joint count law and independent marks instead of inferring independence from a histogram.
- Derive independent Poisson increments on a deterministic partition from the waiting-time construction.
- Derive conditional binomial counts and independent thinning with explicit marking assumptions.
- Construct equal-marginal dependent counts and distinguish logical routing marks from service-time predictions.
6 practice tasks.
Prove a route ends before trusting its equations
Lesson details and objectives
Connect first-step equations to a finite tail bound and inspect what fails for a trapped state.
- Derive absorption probabilities and expected transition counts for a finite substochastic transient block.
- Prove termination, inverse existence and finite-horizon error bounds under an explicit exit lower bound.
- Separate an unresolved capped path, a true nonabsorbing class and a numerical failure.
6 practice tasks.
Audit a simulation without hiding its unfinished paths
Lesson details and objectives
Compare model identities, sampling uncertainty and explicit resource-censored outcomes.
- Derive an exponential inverse sampler and distinguish ideal randomness from seeded numerical replay.
- Retain overflow and unresolved observations in finite reports and bound identifiable outcomes.
- Compare paired-window dependence with frozen predictions, independent checks and explicit limitations.
6 practice tasks.