Simulation Lecture Notes "Regular"

Flashcards covering key vocabulary and concepts from the lecture notes on simulation.

Simulation

A model that attempts to imitate a process or system.

Types of Simulation

Physical or mathematical, static or dynamic, deterministic or stochastic, and discrete or continuous.

Advantages of Simulation

Ability to study complex systems, observe the effects of changes, recommend improvements, provide insights on variable interactions, and verify analytic solutions.

Appropriate Use of Simulation

Studying a complex system, when changes can be simulated, when it can provide knowledge for improvements, when it can provide insights on variables, when used as a teaching method, to experiment with new policies or designs, and to verify analytic solutions.

Components of a System

System, entity, attribute, activity, event, and state variable.

Problem Formulation

Define the problem to be studied in the simulation.

Setting Objectives

Clarify what the simulation is intended to achieve.

Model Conceptualization

Create a simplified logical model that reflects the real-world system, including assumptions.

Data Collection

Gather real-world data required to drive the simulation inputs.

Model Translation

Implement the conceptual model in simulation software.

Verification

Ensure the simulation behaves as intended logically.

Validation

Compare simulation output with actual system performance to check model accuracy.

Experimental Design

Plan scenarios or parameter changes to test in the simulation.

Production Runs and Analysis

Execute the simulation multiple times and analyze the results statistically.

More Runs (If Needed)

Perform additional simulations if results are inconclusive or further refinement is required.

Documentation and Reporting

Document model assumptions, logic, results, and interpretations.

Implementation

Use the simulation insights to make real-world decisions or changes.

ERG Basics

ERG is a visual depiction modeling the association(s) between events in a simulation. A state variable of the system is a variable to be tracked.

ERG Notation and Symbology

Node A and Node B are events, not locations. The arc/edge between events A and B means event A schedules event B.

Manual Simulation Techniques

Event calendars are a manual simulation technique.

Simultaneity of Simulations

Events can occur at the same time during a simulation.

Event Calendar Understanding

Event calendars track events, time, and system state.

Event Calendar Creation and Updating

Event calendars are created and updated by tracking events and changes in system state over time.

Queueing Terms

Queueing terms include inter-arrival time, service time, waiting time, queue length, etc.

Kendall’s Notation

Kendall's notation is used to classify queueing systems (e.g., M/M/1, M/G/1).

Measures of Performance

Average delay in queue, average time in system, average number of customers in queue/system, and server utilization.

Little’s Formula

Little's formula relates entity-based averages to temporal averages.

Steady State Calculations

Steady-state calculations determine long-run average behavior of queueing systems.

Types of Distributions

Trace-driven, empirical, parametric, and non-parametric.

Common Parametric Distributions

Poisson, Binomial, Negative Binomial, Discrete Uniform, Exponential.

Random Number Properties and Their Generation

Random numbers should appear statistically independent and uniformly distributed. psuedo random numbers are generated with closed form mathematics, not truly random, but repeatable

Inverse Transform Method

The inverse transform method is used to generate random variables from a desired distribution.

Distribution Fitting

Distribution fitting involves selecting a distribution and estimating its parameters to match observed data.

Acceptance/Rejection Sampling

Alternative to Inversion method for generating random variables X from U.Acceptance/Rejection (A/R) may be used when: X has density f(x) with bounded range of values. If FX is hard (or impossible) to invert then A/R may work.

Input modeling

Selecting appropriate probability distributions for system inputs (e.g., arrival times, service times).

Random variate generation

Mechanism to generate random values based on the input distributions.

Validation

Ensuring the model's results are close enough to observed real-world data.

Output analysis

Interpreting simulation results to inform decision-making.

Input Uncertainty

Errors due to incorrect or imprecise input distributions.

Modeling Error

Oversimplification or inaccuracies in the model logic.

Estimation Error

Arises due to finite sample size; addressed using confidence intervals.

Convergence in Probability

Wₙ → µ in probability if for any ε > 0, P(|Wₙ - µ| > ε) → 0 as n → ∞

Almost Sure Convergence

Wₙ → µ almost surely if P(limₙ→∞ Wₙ = µ) = 1

Convergence in Distribution

If Fₙ(x) → F(x) at all continuity points of F, then Wₙ converges in distribution to W.

Weak Law of Large Numbers (WLLN)

Sample mean converges in probability to the true mean.

Strong Law of Large Numbers (SLLN)

Sample mean converges almost surely to the true mean.

Central Limit Theorem (CLT)

Standardized sample mean approaches a normal distribution as n increases.

Independent Replications

Multiple replications of the simulation using independent random number seeds.

Confidence Intervals

Used to estimate the error in the sample mean, CI = Ȳ ± t_(α/2, n-1) * (S / sqrt(n))

Prediction Intervals

Used to estimate the range where a new replication's outcome will fall, PI = Ȳ ± t_(α/2, n-1) * S * sqrt(1 + 1/n)

Independent Sampling

Uses separate random number streams for each system being simulated. Leads to larger variance in comparison metrics.

Correlated Sampling

Also known as Common Random Numbers (CRN). Uses the same random numbers across systems to reduce variance. Helps detect small differences in system outputs more reliably.

Model Output Comparison

Used to determine whether differences in outputs between models are statistically significant. Tool: Two-sample t-test for equal means