Advertise on podcast: IEE 475: Simulating Stochastic Systems

In this lecture, we introduce the detailed process of input modeling. Input models are probabilistic models that introduce variation in simulation models of systems. Those input models must be chosen to match statistical distributions in data. Over this unit, we cover collection of data for this process, choice of probabilistic families to fit to these data, and then optimized parameter choice within those families and evaluation of fit with goodness of fit. In this lecture, we discuss issues related to data collection. The Input Analyzer referenced and used in this lecture can be found at: https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/input_analyzer.html

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.

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 WatermarkingConcept Explorer: Probability Distributions for Input Modeling and StatisticsConcept Explorer: Input Modeling (fully functional Input Analyzer)

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.

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 GalleryInput AnalyzerPseudo-Random Number Generator Explorer

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 WatermarkingConcept Explorer: Monte Carlo Examples for Stochastic ModelingConcept Explorer: Input Modeling (fully functional Input Analyzer)Concept Explorer: Probability Distributions for Input Modeling and Statistics

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

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

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

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.

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.

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."

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.

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

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.