Lecture F (2026-10-01): Midterm Review

During this lecture, we review the topics covered up to this point in the course as preparation for the upcoming midterm exam. Students are encouraged to bring their own questions to class so that we can focus on the topics that students feel like they need the most help with.

Lecture E2 (2026-09-29): Random-Variate Generation

In this lecture, we review pseudo-random number generation and then introduce random-variate generation by way of inverse-transform sampling. In particular, we start with a review of the two most important properties of a pseudo-random number generator (PRNG), uniformity and independence, and discuss statistically rigorous methods for testing for these two properties. For uniformity, we focus on a Chi-square/Chi-squared test for larger numbers of samples and a Kolmogorov–Smirnov (KS) test for smaller numbers of samples. For independence, we discuss autocorrelation tests and runs test, and then we demonstrate a runs above-and-below-the-mean test. We then shift to discussing inverse-transform sampling for continuous random variates and discrete random variates and how the resulting random-variate generators might be implemented in a tool like Rockwell Automation's Arena.

Concept explorers relevant to content in this lecture:

• Concept Explorer: Pseudo-Random Number Generators and AI Watermarking
• Concept Explorer: Probability Distributions for Input Modeling and Statistics
• Concept Explorer: Input Modeling (fully functional Input Analyzer)
  
  

Lecture E1 (2026-09-04): Random-Number Generation

In this lecture, we first cover some discrete distributions (and the Poisson process) that we ran out of time for during the previous lecture. We then launch into a discussion of how to generate pseudo-random numbers distributed uniformly between 0 and 1 (which are necessary for us to easily generate random variates of any distribution). We talk about the two most important properties of a pseudo-random number generator (PRNG), uniformity and independence. We then talk about desirable properties. Some examples are given of some early PRNG's, and then we introduce the linear congruential generator (LCG) and its variants (like the Combined Linear Congruential Generator, CLCG), which represent a much more modern PRNG that has a number of good properties. We close with a discussion of tests of uniformity. We will continue this discussion and add on tests for independence during next lecture (which will primarily cover random-VARIATE generation).

Students may wish to follow up by checking out the Pseudo-Random Number Generators and AI Watermarking concept explorer, which covers LCG's, CLCG's, and the more modern MRG approach to PRNG. It also covers methods for AI watermarking of stochastic outputs, like the ones that are now used in large language models like Gemini and Claude.

Lecture D2 (2026-09-22): Probabilistic Models

In this lecture, we review basic probability fundamentals (measure spaces, probability measures, random variables, probability density functions, probability mass functions, cumulative distribution functions, moments, mean/expected value/center of mass, standard deviation, variance), and then we start to build a vocabulary of different probabilistic models that are used in different modeling contexts. These include uniform, triangular, normal, exponential, Erlang-k, Weibull, and Poisson variables. We will finish the discussing next time with the Bernoulli-based discrete variables and Poisson processes.

Interactive widgets referenced or related to this lecture:

• Probability Distribution Gallery
• Input Analyzer
• Pseudo-Random Number Generator Explorer

Lecture D1 (2026-09-17): Probability and Random Variables

In this lecture, we introduce the measure-theoretic concept of a random variable (which is neither random nor a variable) and related terms, such as outcomes, events, probability measures, moments, means, etc. Throughout the lecture, we use the metaphor of probability as mass (and thus probability density as mass density, and a mean as a center of mass). This allows us to discuss the "statistical leverage" of outliers in a distribution (i.e., although they happen infrequently, they still have the ability to shift the mean significantly, as in physical leverage). This sets us up to talk about random processes and particular random variables in the next lecture.

Concept Explorers linked from this lecture:

• Concept Explorer: Pseudo-Random Number Generators and AI Watermarking
• Concept Explorer: Monte Carlo Examples for Stochastic Modeling
• Concept Explorer: Input Modeling (fully functional Input Analyzer)
• Concept Explorer: Probability Distributions for Input Modeling and Statistics

Lecture C2 (2026-09-15): Beyond DES Simulation – SDM, ABM, and NetLogo

This lecture provides some historical background and motivation for System Dynamics Modeling (SDM) and Agent-Based Modeling (ABM), two other simulation modeling approaches that contrast with Discrete Event System (DES) simulation. In particular, in this lecture, we briefly introduce System Dynamics Modeling (SDM) and Agent-Based/Individual-Based Modeling (ABM/IBM) as the two ends of the simulation modeling spectrum (from low resolution to high resolution). The introduction of ABM describes applications in life sciences, social sciences, and engineering (Multi-Agent Systems, MAS)/operations research. 

This lecture is also coupled with notes discussing the Lab 3 (Monte Carlo simulation) results and general experience. These comments focus on interval estimation (which is right 95% of the time, as opposed to point estimation that is right 0% of the time) and the role of non-trivial distributions of random variables (as opposed to just their means).

Concept explorers referenced in this lecture material (particularly for Lab 3 material):

• https://tpavlic.github.io/asu-simulating-stochastic-systems/monte_carlo/mc_explorer.html
• https://tpavlic.github.io/asu-simulating-stochastic-systems/monte_carlo/mc_examples.html
• https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/prob_models.html

Lecture C1 (2026-09-10): Basic Simulation Tools and Techniques

This lecture covers content related to implementing simulations with spreadsheets and the motivations for the use of special-purpose Discrete Event System Simulation tools. In particular, we discuss different approaches to implementing Discrete Event System (DES) simulations (DESS) with simple spreadsheets (e.g., Microsoft Excel, Google Sheets, Apple Numbers, etc.). We cover inventory management problems (such as the newsvendor model) as well as Monte Carlo sampling and stochastic activity networks (SAN's). Although we show that spreadsheets can be very powerful for this kind of work, we highlight that this approach is cumbersome for systems with increasing complexity. So this motivates why we would use more sophisticated tools specifically built for simulation (but perhaps not so great for data analysis by themselves), like Arena, FlexSim, Simio, and NetLogo.

Some parts of this lecture make use of an interactive widget that includes several Monte Carlo method examples for stochastic modeling. It can be found at:

• https://tpavlic.github.io/asu-simulating-stochastic-systems/monte_carlo/mc_examples.html

Lecture B3 (2026-09-08): Discrete-Event Simulation Examples, Part II

In this lecture, we close out our review of DES fundamentals and hand simulation. After going through a hand-simulation example one last time, we show how to implement a Discrete Event System (DES) simulation using a spreadsheet tool like Microsoft Excel without any "macros" (VBA, etc.). This involves defining relationships ACROSS TIME that allow the spreadsheet to (in a declarative fashion) reconstruct the trajectory that is the output of the simulation.

If time permits, this lecture slot will be used to pivot to discussing the previous "Lab 2 (Muffin Oven Simulation)", which lets us introduce common random numbers (CRNs), statistical blocking, requirements of 2-sample and paired t-tests, and more sophisticated statistical methods that better characterize PRACTICAL significance (and take into account the multiple comparisons problem). Thus, the post-lab2 reflections are largely a preview of future topics in the course.

As an introduction to next week's lab, this lecture also links to an interactive Monte Carlo method concept explorer for students to engage with offline. It can be found at:

https://tpavlic.github.io/asu-simulating-stochastic-systems/monte_carlo/mc_explorer.html

Lecture B2 (2026-09-03): Discrete-Event Simulation Examples, Part I

In this lecture, we cover fundamentals of discrete-event system (DES) simulation (DESS). This involves reviewing basic simulation concepts (entities, resources, attributes, events, activities, delays) and introducing the event-scheduling world view, which provides a causality framework on which an automatic simulation of a DES system can be built. We also discuss briefly how the stochastic modeling inherent to DESS means that outputs will be variable and thus will require rigorous statistics to make sense of.

Lecture B1 (2026-09-01): Fundamental Concepts of Discrete-Event Simulation

In this lecture, we cover fundamentals of discrete-event system (DES) simulation (DESS). This involves reviewing basic simulation concepts (entities, resources, attributes, events, activities, delays) and introducing the event-scheduling world view, which provides a causality framework on which an automatic simulation of a DES system can be built. We also discuss briefly how the stochastic modeling inherent to DESS means that outputs will be variable and thus will require rigorous statistics to make sense of.

Lecture A2 (2026-08-27): Introduction to Simulation Modeling

In this lecture, we introduce the three different simulation methodologies (agent-based modeling, system dynamics modeling, and discrete event system simulation) and then focus on how stochastic modeling is used within discrete-event system simulation. In particular, we define terms such as system, dynamic system, state, state variable, activity, delay, resource, entity, and the notion of "input modeling."

Lecture A1 (2026-08-25): Introduction to Modeling

In this lecture, we introduce Industrial and Systems Engineering as a blend of science and engineering that necessitates model building. We then define model (as something that answers a "What If" question) and different types of models. This gives us an opportunity to discuss how modeling is less about describing reality and more about generating tools to do useful things/make useful predictions. We end with a comparison of mental and quantitative models, as well as a comparison of different types of quantitative models (including simulation modeling).

Lecture 0 (2026-08-20): Introduction to the Course and Its Policies

This short lecture introduces the IEE 475 course (Simulating Stochastic Systems) for the Fall 2026 semester. Due to issues with the projector and the camera in the room, the video feed only shows the slides that are being shared (but there is audio that narrates them, and any pointing to the slides is done with a pointer reflected on the video). Future videos this semester will be of higher quality.

Lecture M (2025-12-02): Final Exam Review

In this lecture, we prepare for the final exam and give a brief review of all topics from the course.

Lecture K2 (2025-11-25): Variance Reduction Techniques, Part 2 (Antithetic Variates and Importance Sampling)

In this lecture, we review four different Variance Reduction Techniques (VRT's). Namely, we discuss common random numbers (CRNs), control variates, antithetic variates (AVs), and importance sampling. Each one of these is a different approach to reducing the variance in the estimation of relative or absolute performance of a simulation model. Variance reduction is an alternative way to increase the power of a simulation that is hopefully less costly than increasing the number of replications.

We wrap up with a summary of the four VRT's that we have discussed in the class – Common Random Numbers (CRN's), Antithetic Variates (AV's), Importance Sampling, and Control Variates.

Lecture K1 (2025-11-20): Variance Reduction Techniques, Part 1 (CRNs and Control Variates)

In this lecture, we start by reviewing approaches for absolute and relative performance estimation in stochastic simulation. This begins with a reminder of the use of confidence intervals for estimation of performance for a single simulation model. We then move to different ways to use confidence intervals on mean DIFFERENCES to compare two different simulation models. We then move to the ranking and selection problem for three or more different simulation models, which allows us to talk about analysis of variance (ANOVA) and post hoc tests (like the Tukey HSD or Fisher's LSD). After that review, we move on to introducing variance reduction techniques (VRTs) which reduce the size of confidence intervals by experimentally controlling/accounting for alternative sources of variance (and thus reducing the observed variance in response variables). We discuss Common Random Numbers (CRNs), which use a paired/blocked design to reduce the variance caused by different random-number streams, and then we introduce control variates (CVs), which allow for reducing the output variance of a measurement by leveraging information about other sources that may be corrupting the output.

Lecture J4 (2025-11-19): Estimation of Relative Performance

In this lecture, we review what we have learned about one-sample confidence intervals (i.e., how to use them as graphical versions of one-sample t-tests) for absolute performance estimation in order to motivate the problem of relative performance estimation. We introduce two-sample confidence intervals (i.e., confidence intervals on DIFFERENCES based on different two-sample t-tests) that are tested against a null hypothesis of 0. This means covering confidence interval half widths for the paired-difference t-test, the equal-variance (pooled) t-test, and Welch's unequal variance t-test. Each of these different experimental conditions sets up a different standard error of the mean formula and formula for degrees of freedom that are used to define the actual confidence interval half widths (centered on the difference in sample means in the pairwise comparison of systems). Next time, we will generalize to the case of more than 2 systems, particularly for "ranking and selection (R&S)." That will let us review the multiple-comparisons problem (and Bonferroni correction) and how post hoc tests (after an ANOVA) are more statistically powerful ways to do comparisons.

Lecture J3 (2025-11-13): Estimation of Absolute Performance, Part III (Non-Terminating Systems/Steady-State Simulations)

In this lecture, we start by further reviewing confidence intervals (where they come from and what they mean) and prediction intervals and then use them to motivate a simpler way to determine how many replications are needed in a simulation study (focusing first on transient simulations of terminating systems). We then shift our attention to steady-state simulations of non-terminating systems and the issue of initialization bias. We discuss different methods of "warming up" a steady-state simulation to reduce initialization bias and then merge that discussion with the prior discussion on how to choose the number of replications. In the next lecture, we'll finish up with a discussion of the method of "batch means" in steady-state simulations.

Lecture J2 (2025-11-06): Estimation of Absolute Performance, Part II (Terminating Systems/Transient Simulations)

In this lecture, we review estimating absolute performance from simulation, with focus on choosing the number of necessary replications of transient simulations of terminating systems. The lecture starts by overviewing point estimation, bias, and different types of point estimators. This includes an overview of quantile estimation and how to use quantile estimation to use simulations as null-hypothesis-prediction generators. We the introduce interval estimation with confidence intervals and prediction intervals. Confidence intervals, which are visualizations of t-tests, provide an alternative way to choose the number of required replications without doing a formal power analysis.

Lecture J1 (2025-11-04): Estimation of Absolute Performance, Part I (Introduction to Point and Interval Estimation)

In this lecture, we introduce the estimation of absolute performance measures in simulation – effectively shifting our focus from validating input models to validating and making inferences about simulation outputs. Most of this lecture is a review of statistics and reasons for the assumptions for various parametric and non-exact non-parametric methods. We also introduce a few more advanced statistical topics, such as non-parametric methods and special high-power tests for normality. We then switch to focusing on simulations and their outputs, starting with the definition of terminating and non-terminating systems as well as the related transient and steady-state simulations. We will pick up next time with discussing details related to performance measures (and methods) for transient simulations next time and steady-state simulations after that. Our goal was to discuss the difference between point estimation and interval estimation for simulation, but we will hold off to discuss that topic in the next lecture.