The Poisson Process

Calls hitting a switchboard, emails landing in an inbox, customers pushing through a turnstile, clicks on a website, atoms decaying in a lump of uranium — all of these are random arrivals scattered along time. They come in bursts and lulls, with no schedule, yet at a steady average rate. The Poisson process is the canonical model of exactly this: pure randomness in continuous time, governed by a single number \lambda, the average arrivals per unit time. It is the beating heart of queueing theory and the arrival engine behind nearly every stochastic model in operations research.

The counting process

Let N(t) count how many arrivals have occurred by time t. It starts at zero and jumps up by one at each arrival — a staircase that only ever climbs. The steepness of the staircase reflects the rate: crowd the arrivals together and it rises fast; spread them out and it creeps.

The jumps here are drawn at fixed illustrative instants, but in a true Poisson process their timing is random — and the beauty is that a single parameter \lambda pins down everything about that randomness.

Three ways to say the same thing

A Poisson process of rate \lambda can be defined from three angles that turn out to be completely equivalent — a rare and lovely fact. Any one implies the other two.

Independent, stationary increments. The number of arrivals in disjoint time intervals are independent, and the count in an interval depends only on its length, not its position. In a tiny interval h the chance of one arrival is \lambda h and of two or more is negligible.
Exponential inter-arrival times. The gaps between successive arrivals are independent exponential random variables with mean 1/\lambda.
Poisson counts. The number of arrivals in any interval of length t follows the Poisson distribution with mean \lambda t: P(N(t) = k) = e^{-\lambda t}(\lambda t)^{k}/k!.

In particular \mathbb{E}[N(t)] = \lambda t: on average the count grows linearly at rate \lambda.

Memorylessness — the defining magic

Because the gaps are exponential, the Poisson process is memoryless: if a bus-stop's arrivals are Poisson and you have already waited ten minutes, the distribution of your remaining wait is exactly the same as if you had just arrived. The process keeps no record of how long it has been since the last event. This is the continuous-time cousin of the Markov property, and it is precisely what makes queues driven by Poisson arrivals so clean to analyse.

Merging and splitting

Two properties make Poisson processes wonderfully composable:

Merging (superposition). Pool several independent Poisson streams with rates \lambda_1, \lambda_2, \dots and the combined stream is again Poisson, with rate \lambda_1 + \lambda_2 + \cdots. Three tills each seeing 20 customers an hour, funnelled into one queue, give a single Poisson stream at 60 an hour.
Splitting (thinning). Randomly label each arrival "type A" with probability p and "type B" otherwise; the two sub-streams are independent Poisson processes with rates p\lambda and (1-p)\lambda.

These closure rules are why the Poisson assumption propagates so easily through networks of servers, routers and machines — the arithmetic of rates just adds and scales.

There is a deep reason the Poisson process shows up everywhere. Take a huge number of independent sources — thousands of phone subscribers, millions of atoms — each of which very rarely triggers an event. The superposition of all those rare, independent streams converges to a Poisson process, regardless of the details of the individual sources. This "law of small numbers" is the arrival-time analogue of the central limit theorem: whenever many independent, individually-negligible contributions add up, a universal shape emerges. That universality is why one parameter \lambda so often suffices.

The Poisson distribution and the Poisson process are related but not the same thing, and confusing them is a classic slip. The Poisson distribution is a formula for a single random count — how many events in a fixed window, a number like 0, 1, 2, …. The Poisson process is the whole random pattern of events unfolding in time, an object indexed by t. The process uses the distribution (its count over any window is Poisson) but also carries far more: the timing of events, the exponential gaps, independence across intervals. Say "distribution" for the count in one window; "process" for the arrivals through time.