Poisson Process — Mathematics Knowledge System

Overview

The Poisson process is the canonical model for counting random events occurring over time. A homogeneous Poisson process with rate λ > 0 is a counting process N(t) such that: N(0) = 0, it has independent and stationary increments, and N(t) - N(s) ~ Poisson(λ(t-s)) for t > s. Equivalently, the inter-arrival times between events are i.i.d. Exponential(λ) random variables. The Poisson process arises naturally in queuing, reliability, insurance, and physics as the unique counting process with stationary independent increments.

Intuition

Think of cars passing a highway sensor. If arrivals are rare relative to any interval, memoryless (how long you have waited tells you nothing about when the next car arrives), and can't pile up simultaneously, then the count over any window is Poisson distributed. The exponential distribution arises automatically: given the memoryless property and continuity, the only waiting-time distribution is the exponential. Superimposing two independent Poisson streams (rates λ₁ and λ₂) simply produces a Poisson stream at rate λ₁ + λ₂ — a powerful thinning/superposition calculus.

Formal Definition

Definition

A counting process {N(t), t ≥ 0} is a Poisson process with rate λ > 0 if it satisfies the following axioms.

N(0) = 0

Starts at zero

N(t)-N(s) \perp N(s) \text{ for all } 0 \le s < t

Independent increments

N(t)-N(s) \sim \operatorname{Poisson}(\lambda(t-s))

Stationary Poisson increments

P(N(t+h)-N(t)=1) = \lambda h + o(h)

Single arrival in small interval

P(N(t+h)-N(t) \ge 2) = o(h)

No simultaneous arrivals

T_k = S_k - S_{k-1} \sim \operatorname{Exp}(\lambda) \text{ i.i.d.}

Exponential inter-arrival times

Notation

Notation	Meaning
$N(t)$	Number of events in [0, t]
$\lambda$	Rate (intensity) of the process, events per unit time
$T_k$	k-th inter-arrival time, i.i.d. Exp(λ)
$S_n$	Time of n-th arrival; S_n = T₁ + ... + Tₙ ~ Gamma(n, λ)

Theorems

Theorem 1: Theorem 1

\text{ If } N_1,N_2 \text{ are independent Poisson processes with rates } \lambda_1,\lambda_2, \text{ their superposition } N_1+N_2 \text{ is Poisson }(\lambda_1+\lambda_2). \text{ If each arrival of a rate- }\lambda \text{ process is independently kept with probability } p, \text{ the thinned process is Poisson }(p\lambda).

Theorem 2: Theorem 2

\text{ The inter-arrival times } T_k \sim \operatorname{ Exp }(\lambda) \text{ satisfy } P(T > s+t \mid T > s) = P(T > t) \text{ for all } s,t \ge 0. \text{ The exponential is the unique continuous memoryless distribution. }

Theorem 3: Theorem 3

\text{ If } N(t) \text{ is Poisson }(\lambda) \text{ and } Y_1,Y_2,\ldots \text{ are i.i.d. independent of } N, \text{ then } X(t) = \sum_{ k=1 }^{ N(t) } Y_k \text{ has mean } \lambda t\,\mathbb{ E }[Y] \text{ and variance } \lambda t\,\mathbb{E}[Y^2].

Worked Examples

1
In 30 minutes = 0.5 hours, the expected number of calls is λt = 5 × 0.5 = 2.5.
$\lambda t = 5 \times 0.5 = 2.5$
2
N(0.5) ~ Poisson(2.5), so P(N = 3) = e^{-2.5} · 2.5³ / 3!.
$P(N(0.5)=3) = \frac{e^{-2.5}\cdot 2.5^3}{3!} = \frac{e^{-2.5}\cdot 15.625}{6} \approx 0.2138$

✓ Answer

The probability is approximately 0.214.

Practice Problems

Easyapplication

Buses arrive at rate 3 per hour and taxis at rate 2 per hour, independently. What is the rate of the combined arrival process? What fraction of arrivals are buses?

Mediumfree response

For a Poisson process with rate λ, derive E[S_n] and Var(S_n) where S_n is the time of the n-th arrival.

Common Mistakes

Common Mistake

Using the homogeneous Poisson formula for a non-homogeneous process

For a non-homogeneous process with rate lambda(t), the expected count in [s,t] is the integral of lambda(u) from s to t, not lambda*(t-s).

Common Mistake

Forgetting the memoryless property of exponential inter-arrivals

P(T > s+t | T > s) = P(T > t): having waited s seconds tells you nothing about when the next event will arrive.

Common Mistake

Confusing Poisson counts with exponential inter-arrival times

N(t) ~ Poisson(lambda*t) counts events; inter-arrival times T_k ~ Exp(lambda) measure gaps. They describe the same process but are different random variables.

Quiz

Historical Background

Siméon Denis Poisson introduced the Poisson distribution in 1837 in his work on the probability of judicial errors. The Poisson process as a continuous-time model was formalized by Agner Krarup Erlang in the 1900s through his work on telephone traffic engineering. The connection to exponential inter-arrivals and the axiomatic characterization were developed in the early 20th century, and the process became a cornerstone of queuing theory, reliability theory, and stochastic processes.

1837
Poisson introduces the Poisson distribution in a study of judicial errors
Siméon Denis Poisson
1909
Erlang uses the Poisson process to model telephone traffic
Agner Krarup Erlang
1940s
Axiomatic characterization of the Poisson process developed; connection to exponential inter-arrivals established

Summary

A Poisson process with rate λ has N(t) ~ Poisson(λt) and independent, stationary increments.
Inter-arrival times are i.i.d. Exponential(λ), the unique continuous memoryless distribution.
Superposition of independent Poisson processes with rates λ₁ and λ₂ yields a Poisson process with rate λ₁ + λ₂; thinning at probability p yields rate pλ.
Conditionally on N(t) = n, the arrival times are the order statistics of n i.i.d. Uniform[0, t] random variables.

References

BookRoss, S. — Introduction to Probability Models, Chapter 5
BookDurrett, R. — Probability: Theory and Examples, 4th ed.
WebsiteWikipedia — Poisson process