Statistical Distributions

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.

The Impact of Variation

Consider two scenarios for a task that averages 10 minutes:

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 eight distribution types to model different kinds of variation:

Fixed Distribution

The simplest distribution—always returns the same value.

Parameters

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

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
• Very few take less than 2 or more than 8 minutes

Uniform Distribution

Every value within a range is equally likely—a flat probability distribution.

Parameters

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

Characteristics

• Values cluster around the mode
• Bounded by min and max (no outliers beyond these)
• Asymmetric if mode ≠ (min + max) / 2
• Easy 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.

Poisson Distribution

Models the number of events occurring in a fixed time period—ideal for arrival processes.

Parameters

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

Characteristics

• Always positive (no negative values possible)
• Right-skewed: long tail toward higher values
• Most values cluster near the lower end
• Occasional very 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, sometimes very long” pattern.

Weibull Distribution

A flexible distribution commonly used in reliability engineering and failure analysis.

Parameters

Shape Parameter Effects

When to Use

• Equipment failure times
• Time-to-event analysis
• Reliability modeling
• When you need flexible control over distribution shape

Pearson VI Distribution

An advanced distribution for complex skewed patterns that don’t fit simpler models.

Parameters

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

Quick Reference: Arrival Rates

Best Practices

Start Simple

Begin with Normal or Triangular distributions. They’re easy to understand and parameterize, and often 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:

1. Fit distributions to your data
2. Compare simulated output to actual performance
3. 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
• Complex patterns → Weibull or Pearson VI

Next Steps