Poisson process
= Poisson process
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= Homogeneous Poisson process
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A <Homogeneous Poisson process> models arrivals at a constant rate $\lambda>0$. Its <counting process> $N_t=N(t)$ counts arrivals in $(0,t]$, with $N_0=0$. Its paths are <right-continuous> <nondecreasing functions>, take nonnegative <integer> values, and have unit jumps <almost surely>. The defining <Poisson distribution> of each increment is
$$
\mathbb P(N_{s+t}-N_s=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!},\qquad s\geq0,\quad t>0,\quad k=0,1,2,\ldots.
$$
It has <stationary increments>: the <probability distribution> above depends on the duration $t$, not the starting time $s$. It also has <independent increments>: counts in any finite collection of disjoint time intervals are <independent random variables>. Together these conditions specify the joint count laws, not merely the <Poisson distribution> at a single time.<Discrete Stochastic Processes (draft of 2nd edition, 31 January 2011) by Robert Gallager/Poisson Processes (Chapter 2)>[pp. 78-79: counting-process definition and its equivalence to exponential interarrival times]{ref}
The rate has units of arrivals per unit time; $\lambda t$ is dimensionless. The <Poisson distribution> gives
$$
\mathbb E[N_{s+t}-N_s]=\operatorname{Var}(N_{s+t}-N_s)=\lambda t.
$$
For a toy rate $\lambda=2$ per minute and a duration of $1.5$ minutes, the <expected value> and <variance> of the count are both $3$. The <probability> of no arrival is $e^{-3}\approx0.049787$, and the <probability> of exactly three is $3^3e^{-3}/3!=\tfrac92e^{-3}\approx0.224042$. In two consecutive one-minute intervals, the <probability> of exactly two arrivals in the first and exactly one in the second is $(2e^{-2})(2e^{-2})=4e^{-4}\approx0.073263$, using <independent increments>. These are exact model calculations with rounded decimals, not measured frequencies.
\Image[/poisson-arrival-events-counting-path.png]
{title=Synthetic arrival times and their right-continuous counting path, compared with the exact mean at two arrivals per minute}
The diagram's nine hand-chosen arrivals illustrate a path; the line $\lambda t$ is an <expected value>, not a prescribed trajectory. Constant rate does not imply evenly spaced arrivals. <Poisson interarrival times> connect the count description to waiting, while <Poisson thinning> explains how independently routing arrivals produces separate streams. This makes the <Poisson process> useful as a simple arrival model for a <queue>, provided constant rate and <independent increments> are plausible. For a deterministic changing rate, use an <Inhomogeneous Poisson process>; bunching or history-dependent arrivals need additional assumptions. Conditional on $N_t=n$, the arrival times are the <order statistics> of $n$ <independent> <uniform distributions> on $[0,t]$, as developed in <Poisson process conditional arrival times>.