What is Mathematical Simulation?
As with most business decisions, there are risks involved when implementing new processes, especially those that require expensive investments. Mathematical simulations can test scenarios and answer vital questions such as “How would the new fast track process operate if we doubled the full time equivalent employees?” or “Would adding a third MRI machine reduce the patient wait time? If so, by how much?"
Mathematical simulations use historical data (such as duration times, probabilities of events, etc.) and process flow mapping to reflect actual processes. This allows the decisionmaker to test scenarios in the model then compare before and after outcome measures.This can help guide the stakeholder in advantageous business decisions.
As with most business decisions, there are risks involved when implementing new processes, especially those that require expensive investments. Mathematical simulations can test scenarios and answer vital questions such as “How would the new fast track process operate if we doubled the full time equivalent employees?” or “Would adding a third MRI machine reduce the patient wait time? If so, by how much?"
Mathematical simulations use historical data (such as duration times, probabilities of events, etc.) and process flow mapping to reflect actual processes. This allows the decisionmaker to test scenarios in the model then compare before and after outcome measures.This can help guide the stakeholder in advantageous business decisions.

DiscreteEvent Simulation Demonstration
To the left is a simulation of a medical clinic using ProccessModel. The objective is to determine if adding more patient rooms would reduce patient wait time. The moving dots on the screen represent individual patients moving through a process. The white squares are discrete steps in a process (such as a patient evaluation). The times and routes the dots travel are randomly drawn from historical data. At the end of the model, the statistics (such as wait times) are recorded for further analysis. 
Types of Simulation Models

DiscreteEvent

Monte Carlo

AgentBased

Markov Chain Monte Carlo

Queuing
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DiscreteEvent Simulation
Discreteevent simulation models reflect the individual operations (discreteevents) of a system in time. This tool is often used to help understand bottlenecks in multistage processes with variations in arrivals, service times and utilizes shared resources. Example: Here is a simulation of an assembly line using Arena software Video Source: https://www.youtube.com/watch?v=gj_vfPLvr7Y&app=desktop 

Monte Carlo Simulation
Monte Carlo simulations are a simple calculation that relies on repeated random sampling from predefined distributions (much like repeatedly spinning a roulette wheel and then recording each outcome). After sampling the simulation thousands of times, the analyst can then quantify the risk in terms of probabilities. Example:

AgentBase Modeling
Agentbased modeling is used for simulating the actions and interactions of autonomous agents (such as people) to understand the system as a whole. A common use of agentbased modeling is to quantify the growth rate of a flu outbreak or a wild forest fire. Unlike a discreteevent simulation, the agents need to be free to create their own paths. Example: To the right is a demonstration of a wild forest fire. The yellow is grass, green are trees, purple is empty space and red is fire. Watch the rate at which the grass and trees regenerate versus the rate at which the fire diminishes the landscape over the years. Video Source: https://www.youtube.com/watch?v=2hmmcI0kd5M 

Markov Chain Monte Carlo Simulation
The Markov Chain Monte Carlo simulations is a mix between Markov Chain and Monte Carlo simulation. A Markov Chain is a random process that undergoes transitions from one state to another in a “memorylessness” fashion (i.e. the probability distribution of the next state depends only on the current state or location and not on the sequence of events that precedes it). Example: The graph to the right is a simple example of a Markov Chain. Circles E and A are discrete events with a probability of their next destination. The cycle continues until simulation is stopped. 
Queuing Simulation
Queuing simulations can be viewed as simplified versions of discreteevent simulations. This tool is used to evaluate waiting time in a line or a queue. Most queuing models involve linear paths of unshared resources (such as a single checkout line). Example:
