How the Simulation Engine Works
Understand the mechanics behind ProcessMind's discrete event simulation engine and how it models your process.
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Statistical Distributions
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Why Use Distributions?
In the real world, processes exhibit variation. One customer service call takes 5 minutes, another takes 25 minutes. One day brings 50 orders, the next brings 120. This natural variability is a fundamental characteristic of business processes.
Fixed values (like “every activity takes exactly 10 minutes”) create unrealistic simulations. Distributions capture variability mathematically, producing simulations that behave like real processes. You can try them on your own process with a free trial.
The Impact of Variation
Consider two scenarios for a task that averages 10 minutes:
| Scenario | Distribution | Effect on Simulation |
|---|---|---|
| Fixed 10 min | No variation | Unrealistic queuing, predictable patterns |
| Normal (mean=10, stdDev=3) | Realistic variation | Natural queuing behavior, realistic delays |
The second scenario better represents reality; some tasks are quick, some take longer, and this variation creates the queuing effects you see in real processes.
Available Distributions
ProcessMind provides nine distribution types to model different kinds of variation:
| Distribution | Best For | Key Parameters |
|---|---|---|
| Fixed | Constant, unchanging values | value |
| Normal | Symmetric variation around average | mean, stdDev |
| Uniform | Equal probability within a range | min, max |
| Triangular | Range with a most likely value | min, mode, max |
| Poisson | Random event arrivals | lambda, rateUnit |
| Lognormal | Right-skewed (usually quick, sometimes long) | mean, stdDev |
| Weibull | Reliability and failure analysis | scale, shape |
| Beta | Flexible shape between min and max | min, max, alpha, beta |
| Pearson VI | Complex skewed patterns | alpha1, alpha2, beta |
Fixed Distribution
The simplest distribution: always returns the same value.
Parameters
| Parameter | Description |
|---|---|
| value | The constant value to return |
Characteristics
- No variation whatsoever
- Every sample returns exactly the specified value
- Useful for modeling system-controlled or automated steps
When to Use
- Automated system responses with consistent timing
- Regulatory timeouts or deadlines
- Initial simulation setup before adding variation
- Modeling SLAs or contractual time limits
Example
A system-generated email always sends in exactly 5 seconds.
Normal (Gaussian) Distribution
The familiar “bell curve”: values cluster symmetrically around an average, with decreasing probability as you move away from the center.
Parameters
| Parameter | Description |
|---|---|
| mean | The average value (center of the curve) |
| stdDev | Standard deviation (spread of values) |
Characteristics
- Symmetric around the mean
- 68% of values fall within 1 standard deviation
- 95% of values fall within 2 standard deviations
- 99.7% of values fall within 3 standard deviations
- Can theoretically produce negative values (simulation handles this)
When to Use
- Processing times that vary symmetrically around an average
- Measurements with random error
- Any quantity influenced by many small, independent factors
Example
A data entry task averages 5 minutes with standard deviation of 1 minute:
- 68% of entries take 4-6 minutes
- 95% take 3-7 minutes
- Few take less than 2 or more than 8 minutes
Uniform Distribution
Every value within a range is equally likely, creating a flat probability distribution.
Parameters
| Parameter | Description |
|---|---|
| min | Minimum possible value |
| max | Maximum possible value |
Characteristics
- Flat probability: no value is more likely than another
- Sharp cutoffs at min and max
- Mean is exactly (min + max) / 2
When to Use
- When you only know the range, not the typical value
- Random selection from a range
- Time spent waiting for a scheduled event
- Modeling uncertainty when you have no historical data
Example
An approval takes somewhere between 2 and 8 minutes, with no information about what’s typical. All durations in this range are equally likely.
Triangular Distribution
A simple distribution with minimum, maximum, and most likely (mode) values, forming a triangle shape.
Parameters
| Parameter | Description |
|---|---|
| min | Minimum possible value |
| mode | Most likely value (peak of triangle) |
| max | Maximum possible value |
Characteristics
- Values cluster around the mode
- Bounded by min and max (no outliers beyond these)
- Asymmetric if mode ≠ (min + max) / 2
- Straightforward to estimate from expert knowledge
When to Use
- When you know “typically X, but can range from Y to Z”
- Expert estimation scenarios
- When Normal might produce unrealistic negative values
Example
An invoice review:
- Best case (min): 2 minutes
- Typical (mode): 5 minutes
- Worst case (max): 15 minutes
Most reviews cluster around 5 minutes, with a tail toward 15 for complex invoices.
Expert Estimation
The triangular distribution maps perfectly to expert estimates. Ask: “Best case time? Typical time? Worst case time?” You’ll get min, mode, and max directly.
Poisson Distribution
Models the number of events occurring in a fixed time period; it is ideal for arrival processes.
Parameters
| Parameter | Description |
|---|---|
| lambda | Average rate of events |
| rateUnit | Time unit for the rate (perHour, perDay, perWeek, perMonth, perYear) |
Characteristics
- Discrete values (whole numbers: 0, 1, 2, 3…)
- Variance equals the mean
- Events are independent
- Models “random arrivals” well
When to Use
- Case arrivals into the process
- Customer arrivals
- Order generation
- Any “events per time period” scenario
Example
Lambda=20, rateUnit=perDay models ~20 cases arriving per day. Some days might see 15, others 25: the natural variation of random arrivals.
Lognormal Distribution
Right-skewed distribution where most values are small, but occasional large values occur. The logarithm of the values follows a normal distribution.
Parameters
| Parameter | Description |
|---|---|
| mean | Mean of the generated values (not of their logarithm) |
| stdDev | Standard deviation of the generated values (not of their logarithm) |
Characteristics
- Always positive (no negative values possible)
- Right-skewed: long tail toward higher values
- Most values cluster near the lower end
- Occasional large values
When to Use
- Tasks that usually complete quickly but sometimes take much longer
- Financial data, income distributions
- Response times with occasional delays
- Bug fixing times
Example
Technical support tickets:
- Most resolve in 1-2 hours
- Some take a full day
- Rare complex issues take multiple days
The lognormal captures this “usually quick, occasionally long” pattern.
Weibull Distribution
A flexible distribution commonly used in reliability engineering and failure analysis.
Parameters
| Parameter | Description |
|---|---|
| scale | Scale parameter (characteristic life) |
| shape | Shape parameter (determines distribution form) |
Shape Parameter Effects
| Shape Value | Distribution Behavior |
|---|---|
| shape below 1 | Decreasing failure rate (infant mortality) |
| shape = 1 | Constant failure rate (exponential distribution) |
| shape above 1 | Increasing failure rate (wear-out) |
When to Use
- Equipment failure times
- Time-to-event analysis
- Reliability modeling
- When you need flexible control over distribution shape
Beta Distribution
A flexible distribution that generates values between a minimum and maximum, with shape controlled by two parameters. This is the generalized (4-parameter) Beta distribution.
Parameters
| Parameter | Description |
|---|---|
| min | Minimum possible value |
| max | Maximum possible value |
| alpha | First shape parameter (α > 0) |
| beta | Second shape parameter (β > 0) |
Characteristics
- Values are always between min and max
- Extremely flexible shape depending on α and β
- When α = β, the distribution is symmetric around the midpoint of [min, max]
- When α < β, values skew toward min (left-skewed)
- When α > β, values skew toward max (right-skewed)
- When α = β = 1, it becomes a uniform distribution on [min, max]
- Mean = min + (max − min) × α / (α + β)
Shape Parameter Effects
| Parameters | Distribution Behavior |
|---|---|
| α = 1, β = 1 | Uniform (flat) across [min, max] |
| α = β | Symmetric bell shape centered at midpoint |
| α < β | Left-skewed (values cluster near min) |
| α > β | Right-skewed (values cluster near max) |
| α < 1, β < 1 | U-shaped (values cluster at both extremes) |
When to Use
- Bounded quantities where you know the range and want to control the shape
- Task durations that cluster near one end of a known range
- Modeling probabilities or success rates (use min=0, max=1)
- Quality scores, completion percentages, or yield rates
- Expert estimates with a known range and a sense of where values tend to cluster
Example
A review task takes between 2 and 15 minutes, with most reviews finishing closer to the shorter end: use Beta(min=2, max=15, α=2, β=5). The mean is about 2 + 13 × 2/7 ≈ 5.7 minutes, with values clustering toward the lower end but never below 2 or above 15.
Pearson VI Distribution
An advanced distribution for complex skewed patterns that don’t fit simpler models.
Parameters
| Parameter | Description |
|---|---|
| alpha1 | First shape parameter |
| alpha2 | Second shape parameter |
| beta | Scale parameter |
When to Use
- Complex distributions derived from data analysis
- When simpler distributions don’t fit your historical data
- Advanced statistical modeling scenarios
Choosing the Right Distribution
Quick Reference: Processing Times
| Your Situation | Recommended Distribution |
|---|---|
| Times vary symmetrically around an average | Normal |
| You only know the range (min to max) | Uniform |
| You know typical, best case, and worst case | Triangular |
| Usually quick, sometimes much longer | Lognormal |
| Time is constant (rare) | Fixed |
| Proportions or probabilities (0 to 1) | Beta |
Quick Reference: Arrival Rates
| Your Situation | Recommended Distribution |
|---|---|
| Random, independent arrivals | Poisson |
| Arrivals at a constant rate | Fixed |
Best Practices
Start Simple
Begin with Normal or Triangular distributions. They are straightforward to understand and parameterize, and usually work well enough. Add complexity only if needed.
Use Expert Knowledge
Subject matter experts can provide excellent estimates:
- “Best case?” → minimum
- “Typical?” → mean or mode
- “Worst case?” → maximum
Validate Against Data
If you have historical data:
- Fit distributions to your data
- Compare simulated output to actual performance
- Refine distribution parameters
Consider Outliers
Real processes often have outliers. Lognormal and Weibull can capture these better than Normal or Triangular.
Match to Process Behavior
- Symmetric variation → Normal
- Bounded variation → Triangular or Uniform
- Right-skewed → Lognormal
- Proportions (0 to 1) → Beta
- Complex patterns → Weibull or Pearson VI
Generate a Custom BPMN Model
Model the variation you want to simulate
Describe your process and its timings and the AI creates a BPMN model you can simulate with realistic variation.
Next Steps
Periodicity
Learn how distributions can vary by time period (business hours, seasons, etc.).
Running Simulations
Apply distributions when configuring your simulation.
How It Works
Understand how the simulation engine uses distributions.
Related Documentation
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