Generating a Correct “Time to First Event” Distribution

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When modeling arrivals or failures, it is common to reuse the inter-arrival time distribution as the one used to sample the time to first event.
Except for memoryless processes (e.g., negative exponential), this is not statistically correct and introduces a left-bias where early events occur sooner than they should.

The user command in the attached library generates a statistically valid time-to-first-event distribution from a definition of inter-arrival times.

Problem Being Addressed

• Inter-arrival distributions describe time between events, not time to the first event
• Reusing the same distribution biases early arrivals/failures
• Bias is most visible in:
• Short simulation runs
• Models without warm-up periods
• Models initialized with work-in-progress

What the Command Does

The command generates a sample set for time to first event using inter-arrival data provided as:

• A FlexScript distribution expression (string)
• An array of inter-arrival samples
• An existing Empirical Distribution object

The process:

1. Generates a timeline of events using inter-arrival samples
2. Selects random observation points along the timeline
3. Measures time to the next event
4. Stores these values in a new Empirical Distribution object
5. Runs the Fit function to suggest a best-fit distribution and parameters

Output

• A new Empirical Distribution representing time to first event
• A fitted distribution expression (name + parameters) suitable for direct use
• Option to sample empirically or via the fitted distribution

Typical Use Cases

• Time to first failure fields
• Arrival processes where warm-up is undesirable
• Short-term forecasting models
• Models requiring statistically defensible early-run behavior

Why Use This Approach

• Avoids bias introduced by reusing inter-arrival distributions
• Produces statistically defensible results
• Reduces reliance on warm-up periods
• Especially beneficial when early statistics matter

How to Use the Command

Loading the attached user library will auto-install the required user commands. The command createTTFeventDist creates an Empirical Distribution that represents the time to first event, derived from an inter-arrival time definition.

Parameters

• P1 (string)
  Name of the new Empirical Distribution object to create.
• P2 (overloaded)
  Definition of the inter-arrival times. One of:
• A FlexScript distribution expression (string)
• An array of inter-arrival samples
• The name of an existing Empirical Distribution object
• P3 (optional integer)
  Number of time-to-first-event samples to generate and store in the new Empirical Distribution.
  (Default is suitable for most use cases.)
• P4 (optional integer)
  Number of inter-arrival samples used to construct the event timeline.
  (Larger values improve stability at the cost of run time.)

Example: Time to First Failure from a Weibull Inter-Arrival Process

Assume failures follow a Weibull inter-arrival distribution, but the model starts at an arbitrary point in time and does not use a warm-up period.

Instead of reusing the Weibull distribution directly for the first failure, generate a statistically correct time-to-first-failure distribution.  Open a script window and run this command with the example parameters:

createTTFeventDist(
	"FirstFailureDist",
	"weibull(0,120, 1.8)",
	5000,
	100000);

This command:

• Generates a large inter-arrival timeline from 100000 samples of  the Weibull distribution
• Samples valid times to the next failure
• Creates an Empirical Distribution named FirstFailureDist
• Fits a theoretical distribution and reports suggested parameters (eg.  beta(-0.08, 356.54, 1.20, 4.89))

Using the Result in the Model

You can now use the generated distribution in either of the following ways:

• Use the fitted distribution expression
  Copy the suggested distribution name and parameters into a
  Time to First Failure or arrival field.
• Reference the Empirical Distribution directly
  Sample from the Empirical object where a time-to-first-event value is required, choosing either:
• Fitted distribution sampling, or
• Empirical data sampling

Both approaches ensure the first event is statistically consistent with the defined inter-arrival process.