Some stochastic processes study not “position” but “event count”: how many customers arrive in an hour, how many requests a server receives in a minute, how many radioactive decays occur in an interval. The classic model for this kind of process is the Poisson process. Probability and Statistics 2 already introduced the Poisson distribution itself; this article places it in the perspective of a process that keeps happening over time.

1. From Counting Process to Poisson Process

1.1 Definition

Define N(t)N(t) as the total number of events that have occurred from time 0 to tt. Then {N(t)}t≥0\{N(t)\}_{t\ge0} is a counting process, and a single sample path is a step function: the count increases by 1 when an event occurs and stays flat otherwise. A homogeneous Poisson process with rate λ>0\lambda>0 requires N(0)=0N(0)=0, independent increments over non-overlapping time intervals, and the event count in an interval of length tt following N(t)∼Poisson⁡(λt)N(t)\sim\operatorname{Poisson}(\lambda t), where λ\lambda is the average number of events per unit time.

1.2 The Poisson distribution describes one moment; the Poisson process describes an entire trajectory

The Poisson distribution describes “how many events occur in a fixed interval” — for example, N(10)N(10) is a random variable. The Poisson process describes how the cumulative count N(t)N(t) changes over time, that is, the entire process {N(t)}t≥0\{N(t)\}_{t\ge0}. Take λ=4\lambda=4 (an average of 4 events per minute) as an example: the probability of exactly 10 events in two minutes is P(N(2)=10)=e−8810/10!≈0.0993P(N(2)=10)=e^{-8}8^{10}/10!\approx0.0993 — this calculation only uses the distribution at the single time point N(2)∼Poisson⁡(8)N(2)\sim\operatorname{Poisson}(8). What the Poisson process asks is how such counts evolve jointly across all time points, not merely what the distribution looks like at one moment.

2. Increment Structure: Independent and Stationary

If the intervals [0,1][0,1] and [2,3][2,3] do not overlap, the standard Poisson process assumes that the event counts N(1)−N(0)N(1)-N(0) and N(3)−N(2)N(3)-N(2) are independent of each other — this is called independent increments. The Poisson process also has stationary increments: for any s,t≥0s,t\ge0, N(s+t)−N(s)∼Poisson⁡(λt)N(s+t)-N(s)\sim\operatorname{Poisson}(\lambda t) depends only on the interval length tt, not on when the interval starts. An hour of the same length, whether it occurs in the morning or at night, follows the same statistical law in a homogeneous model.

3. Inter-Arrival Times and Memorylessness

3.1 Inter-arrival times follow an exponential distribution

Let T1T_1 be the waiting time before the first event. The probability that no event has occurred before tt is P(T1>t)=P(N(t)=0)=e−λtP(T_1>t)=P(N(t)=0)=e^{-\lambda t}, so T1∼Exponential⁡(λ)T_1\sim\operatorname{Exponential}(\lambda), and the waiting times between successive arrivals are likewise i.i.d. exponential. Still with λ=4\lambda=4: the probability of waiting more than 0.5 minutes for the next event is P(T1>0.5)=e−4×0.5=e−2≈0.1353P(T_1>0.5)=e^{-4\times0.5}=e^{-2}\approx0.1353. The Poisson process therefore has two equivalent views: how many events occur in a fixed time is the Poisson distribution; how long until the next event is the exponential distribution — the two describe the same random mechanism from different angles.

3.2 Memorylessness

The exponential distribution has the memoryless property: P(T>s+t∣T>s)=P(T>t)P(T>s+t\mid T>s)=P(T>t). How long you have already waited does not change the distribution of how much longer you still need to wait. This is closely related to the Markov property of the Poisson process: knowing only that “no event has occurred so far” reveals no additional information about “how much longer.”

4. Combining, Splitting, and Generalizing the Process

4.1 Superposition and thinning, worked out

If two independent Poisson processes have rates λ1,λ2\lambda_1,\lambda_2, merging their events still gives a Poisson process, with rate λ1+λ2\lambda_1+\lambda_2 — this is called superposition. For example, two independent streams of user requests combine into a total request stream with rate 3+5=83+5=8 (per minute). Conversely, if each event in a Poisson process of rate λ\lambda is independently labeled A with probability pp, the A-labeled events themselves form a new Poisson process with rate pλp\lambda, and the remaining events form one with rate (1−p)λ(1-p)\lambda — this is called thinning. For example, an original request stream has rate 10 (per minute); if 30% of requests come from mobile devices, the mobile-device requests form a Poisson process of rate 0.3×10=30.3\times10=3, and the remaining device requests form one of rate 0.7×10=70.7\times10=7, with the two remaining independent of each other.

4.2 The nonhomogeneous Poisson process

Real event rates often change over time — a restaurant’s midday and midnight foot traffic clearly differ. The intensity can then be written as λ(t)\lambda(t), giving a nonhomogeneous Poisson process, and the average event count in an interval [a,b][a,b] becomes ∫abλ(t) dt\int_a^b\lambda(t)\,dt. The other assumptions (independent increments, a Poisson distribution given the average count) remain unchanged; only stationary increments no longer hold — the statistical law can differ across different time periods.

5. Model Limitations and Summary

The Poisson process assumes no long-range dependence between events, a simple arrival-rate structure, and inter-arrival times that are exactly exponentially distributed. Together, these three assumptions amount to assuming that the tendency for an event to occur never changes based on what has happened recently. Many real systems do not satisfy this: user visits often cluster in bursts (one event triggers a chain of related events, violating independent increments), and machine failure lifetimes are usually not exponential (wear effects make failure probability increase with usage time, violating memorylessness). Relaxing the requirement that inter-arrival times must be exponential, while keeping the core structure of independent and identically distributed inter-arrival times, naturally leads to the more general renewal process.

The Poisson process is the most basic model of random arrivals: a fixed average rate, independent increments, and counts that follow a Poisson distribution. It ties together the Poisson distribution, exponential waiting times, the Markov property, and counting processes, and is the starting point for understanding more general arrival models.