Why are Poisson interarrival times exponential?

The independence and stationarity of increments force the waiting time to the next event to be memoryless, and the only continuous memoryless distribution is the exponential, with rate equal to the process rate.

What does the rate parameter mean?

The rate lambda is the average number of events per unit time; the expected count in an interval is lambda times its length, and the mean interarrival time is one over lambda.

Homogeneous Poisson Process

The homogeneous Poisson process counts events occurring at a constant average rate, with the number of events in any interval Poisson distributed and counts in disjoint intervals independent.

Definition

A homogeneous Poisson process of rate lambda is a counting process starting at zero with independent stationary increments in which the number of events in an interval of length t is Poisson distributed with mean lambda times t, equivalently a process whose interarrival times are independent exponential random variables with rate lambda.

Scope

This topic covers the rate parameter, the Poisson distribution of counts, independent and stationary increments, the exponential distribution of interarrival times and the gamma distribution of arrival times, the order statistics property of event times conditioned on the count, and the memoryless property underlying these results.

Core questions

How is the homogeneous Poisson process defined and parameterised by its rate?
Why are the interarrival times exponential and independent?
How are the arrival times distributed given the number of events?
What is the role of the memoryless property?

Key theories

Equivalence of the counting and interarrival descriptions: A counting process has Poisson increments with stationary independent increments if and only if its successive interarrival times are independent exponentials with the same rate, so the process can be built either by counting or by summing waiting times.
Order statistics property: Conditioned on the number of events in an interval, the event times are distributed as the order statistics of independent uniform points in that interval, which makes many conditional computations and simulations straightforward.

Clinical relevance

The homogeneous Poisson process is the standard model for arrivals in queueing, radioactive decay counts, photon detection, and rare-event occurrences, and it serves as the arrival mechanism in the elementary M/M/1 and M/G/1 queues and as the null model of randomness in event-time data.

History

Bortkiewicz's 1898 analysis of rare events and Erlang's 1909 study of telephone traffic established the Poisson process empirically, while Rutherford and Geiger's 1910 counts of alpha particles gave a classic physical confirmation; the rigorous theory followed from the general study of processes with independent increments.

Key figures

Simeon Denis Poisson
Agner Krarup Erlang
Ernest Rutherford

Seminal works

kingman1993

Frequently asked questions

Why are Poisson interarrival times exponential?: The independence and stationarity of increments force the waiting time to the next event to be memoryless, and the only continuous memoryless distribution is the exponential, with rate equal to the process rate.
What does the rate parameter mean?: The rate lambda is the average number of events per unit time; the expected count in an interval is lambda times its length, and the mean interarrival time is one over lambda.