Poisson Process Simulator
If a shop averages four customers an hour, how many hours have none at all? More than people expect. The Poisson distribution describes counts of independent events, and its defining oddity is that the variance equals the mean — so the moment you state a rate, you have also stated how variable the counts will be. You cannot ask for an average of four and a reliably steady four.
What this generator does
Draws counts from a Poisson process using Knuth's method and histograms them. Because the variance of a Poisson equals its mean, the theoretical spread is simply the square root of the rate — a fact that surprises people and is worth seeing confirmed by simulation rather than asserted.
How to use this tool
- Set the average number of events per interval.
- Choose how many intervals to simulate.
- Run it and look at how often you get far more or far fewer than the average.
- Compare the observed spread against the theoretical one.
Understanding the controls
- Average events per interval
- The rate. Everything else about the distribution follows from this single number, including how spread out the counts are.
- Repetitions
- How many intervals to simulate. More intervals give a smoother picture of the tail, which is usually the interesting part.
- Seed
- Any word reproduces the identical set of interval counts, which is what a worked example or a teaching slide needs.
Common use cases
- Understanding staffing when arrivals are genuinely random
- Seeing how often a quiet period happens purely by chance
- Checking whether an unusual count is actually unusual
- Teaching why averages hide variability
- Sizing capacity for something bursty rather than steady
How this generator works
Each interval's count is drawn by multiplying uniform random numbers until their product falls below e to the minus lambda, which is Knuth's standard method. The counts are then histogrammed and the observed mean and spread compared against the rate and its square root.
Randomness and fairness
Each interval's count is random and independent of the others. Without a seed this uses the browser's cryptographically secure generator; with a seed a deterministic one, which is reproducible and so explicitly not cryptographically secure.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- It assumes events are independent and the rate is constant. Real arrivals cluster around lunchtimes and paydays, which breaks both.
- Rates above about 100 per interval are not offered; at that point a normal approximation is more useful anyway.
- It counts events, and says nothing about how long each takes to handle. The queue simulator does that.
- Counts are not kept once you leave, so seed a run you intend to come back to.
Privacy and your data
Everything is simulated in the page. No rates, seeds or generated counts leave your browser or persist after you close the tab.
Related generators
- Binomial Experiment SimulatorRepeat a fixed run of trials thousands of times and watch the count of successes form its distribution, with theory shown beside it.
- Queue SimulatorSimulate a queue from arrival and service rates and see the waiting times — including why waiting explodes near capacity rather than rising smoothly.
- Sampling Distribution SimulatorTake a sample, take its mean, repeat — and watch means from even a badly skewed population pile up symmetrically.
- Dice Probability CalculatorThe exact distribution of any dice total — every outcome counted, not simulated, with the odds of rolling at least a given number.