Poisson Process — Live Event Arrivals

Arrival Rate

Rate λ (events / sec) 1.0

Simulation speed 1.0×

Process Type

Live Readouts

Events N(t)

Sim time t

0.0s

Estimated rate λ̂ = N(t)/t

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How it works

A Poisson process counts random events arriving independently over continuous time at rate λ. The number of arrivals in an interval of length t follows a Poisson distribution:

P(N(t) = k) = (λt)^k · e^(-λt) / k!
E[N(t)] = λt,   Var[N(t)] = λt

Equivalently, the gaps between consecutive events (interarrival times) are independent and exponentially distributed with mean 1/λ — the process is memoryless: however long you've already waited, the expected remaining wait is always 1/λ.

T_gap ~ Exponential(λ)
P(T_gap > s) = e^(-λs)

Time-varying λ(t) mode drives the rate as λ(t) = λ₀·(1 + 0.6·sin(0.4t)) and samples arrivals with the thinning (Lewis–Shedler) algorithm: draw candidate gaps from the fastest possible rate λ_max, then keep each candidate only with probability λ(t)/λ_max, discarding the rest. What survives is distributed exactly as a Poisson process with the time-varying rate.

The staircase line above the timeline is the counting function N(t) itself — it jumps by exactly 1 at every accepted event and is flat everywhere else. The estimated rate λ̂ = N(t)/t drifts toward the true λ as t grows, by the law of large numbers.

Event arrival N(t) staircase

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