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TheoryIntermediate03.01
Uncertainty & Stochastic Systems

Stochastic Processes

Systems that evolve randomly over time.

Overview

Stochastic Processes focuses on systems that evolve randomly over time. In the map of OR, it connects Poisson, Renewal, Markov chains to decisions that must be modeled, solved, explained, and revised as evidence changes.

Probability foundations for OR: Poisson processes, renewal processes, Markov chains, MDPs, and reliability models. The practical use case is clearest in Call centers, Inventory, Finance, Maintenance, Healthcare, where the method helps turn constraints and tradeoffs into a decision artifact someone can inspect.

Core ideas

Poisson

Poisson is a core checkpoint for Stochastic Processes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Renewal

Renewal is a core checkpoint for Stochastic Processes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Markov chains

Markov chains is a core checkpoint for Stochastic Processes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

MDPs

MDPs is a core checkpoint for Stochastic Processes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Brownian motion

Brownian motion is a core checkpoint for Stochastic Processes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

How to use it

  1. 1Start with Call centers: write the decision, time horizon, actors, and objective in operational language.
  2. 2Translate the problem into Poisson, Renewal, and Markov chains; define units and data sources for each one.
  3. 3Build a small instance of Stochastic Processes that can be solved or simulated by hand inspection before using full production data.
  4. 4Compare the recommendation against a baseline policy, not just against mathematical optimality.
  5. 5Document assumptions, sensitivity results, and the conditions under which the recommendation should be revisited.

Applications

Call centersInventoryFinanceMaintenanceHealthcare
  • Call centers: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Inventory: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Finance: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Maintenance: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Healthcare: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.

Common pitfalls

  • Applying Stochastic Processes because the label sounds appropriate while leaving the actual decision boundary vague.
  • Treating Poisson as a technical detail instead of a modeling choice that affects the recommendation.
  • Reporting one answer without showing sensitivity to demand, capacity, costs, or behavioral assumptions.
  • Ignoring implementation details such as data quality, explainability, ownership, and how users will override bad recommendations.

Resources