Calculator Inputs
Example Data Table
This example uses an Erlang waiting-time model with λ = 0.8 per minutes and k = 3. It represents the waiting time until the third event.
| Time (minutes) | CDF | Survival | |
|---|---|---|---|
| 0.50 | 0.0429 | 0.0079 | 0.9921 |
| 1.00 | 0.1150 | 0.0474 | 0.9526 |
| 1.50 | 0.1735 | 0.1205 | 0.8795 |
| 2.00 | 0.2067 | 0.2166 | 0.7834 |
| 3.00 | 0.2090 | 0.4303 | 0.5697 |
| 4.00 | 0.1670 | 0.6201 | 0.3799 |
Formula Used
This calculator models waiting times under a Poisson arrival process. For the first event, the waiting time is exponential. For the kth event, the waiting time follows the Erlang distribution.
Probability density function
For k ≥ 1, the density is: f(t) = (λk tk−1 e−λt) / (k−1)!, for t ≥ 0.
Cumulative distribution function
The cumulative probability is: F(t) = 1 − e−λt Σ[(λt)n / n!] from n = 0 to k − 1.
Key summary measures
Mean: E(T) = k / λ
Variance: Var(T) = k / λ²
Standard deviation: √k / λ
Mode: (k − 1) / λ for k > 1, otherwise 0
Interpretation
A larger λ compresses the waiting time distribution. A larger k shifts the waiting time farther right and usually reduces relative variation.
How to Use This Calculator
- Choose whether you want to enter a direct event rate or an average time to the first event.
- Enter the shape value k. Use 1 for the first event, 2 for the second event, and so on.
- Provide the time point for pointwise probability measures and the interval bounds for interval probability.
- Enter a percentile target such as 90 or 95 if you want a waiting-time threshold.
- Set the time-unit label so outputs match your context.
- Leave the chart range blank for automatic scaling, or specify a custom upper limit.
- Press the calculate button to display results above the form and below the header.
- Use the CSV and PDF buttons to export summary outputs and chart data.
FAQs
1. What does this calculator measure?
It measures waiting-time probabilities for event arrivals. You can estimate point probabilities, cumulative chances, survival values, interval probabilities, and percentile thresholds for first-event or kth-event waiting times.
2. When should I use k = 1?
Use k = 1 when you need the waiting time until the first event. That case follows the exponential model and is common for interarrival times in Poisson processes.
3. What does a larger λ mean?
A larger event rate means events arrive faster on average. That shifts the distribution left, lowers expected waiting time, and usually increases the density near earlier times.
4. Why is the kth-event model useful?
It helps when the question is not about the first arrival. For example, you may want the time until the third customer, fifth failure, or fourth message.
5. What is the difference between PDF and CDF?
The PDF describes local density around a time point. The CDF gives the probability that the waiting time is less than or equal to that time.
6. What does the survival value show?
The survival value gives the probability that waiting continues beyond the chosen time. It is useful for service-level checks, delay-risk analysis, and timeout planning.
7. Can I use this for queueing and reliability work?
Yes. It is useful for arrivals, failures, response events, queue delays, and other Poisson-process timing problems when the exponential or Erlang assumptions are reasonable.
8. Why are exports included?
Exports help you save results for reporting, audits, classroom work, or operational reviews. The CSV file is spreadsheet-friendly, and the PDF file is convenient for sharing.