Definition and Basic Properties
Definition
Poisson process: counting process {N(t), t ≥ 0} with properties: N(0)=0, independent increments, stationary increments, and increments follow Poisson distribution with parameter λt (λ > 0 rate).
Key Properties
Independent increments: counts in disjoint intervals independent. Stationarity: distribution depends only on interval length. N(t) ~ Poisson(λt). Process is simple: no simultaneous events.
Notation
Intensity (rate): λ. Counting function: N(t). Event times: {T₁, T₂, ...}. Interarrival times: {S₁, S₂, ...} with Sᵢ = Tᵢ - Tᵢ₋₁, T₀=0.
Historical Background
Origins
Named after Siméon-Denis Poisson (1837) for Poisson distribution. Poisson process concept developed by formalizing random event arrivals over time.
Development
Poisson process formalized in early 20th century by mathematicians including William Feller, Andrey Kolmogorov. Used to model random arrivals in queueing theory, telephony, and physics.
Modern Applications Evolution
Expanded to fields: finance, biology, reliability engineering, telecommunications. Basis for studying counting processes and jump processes.
Construction of Poisson Processes
Counting Process Definition
Defined via interarrival times: Sᵢ ~ Exponential(λ) i.i.d. Arrival times Tₙ = ΣSᵢ. N(t) = max{n: Tₙ ≤ t}.
Alternative Construction
Limit of Bernoulli processes: divide time into n intervals of length 1/n, events occur with probability λ/n independently; as n → ∞, process converges to Poisson process.
Superposition and Thinning
Superposition: sum of independent Poisson processes with rates λ₁, λ₂ is Poisson(λ₁+λ₂). Thinning: each event retained with probability p independently creates Poisson(λp) process.
Interarrival Times and Exponential Distribution
Distribution of Interarrival Times
Interarrival times Sᵢ are i.i.d. Exponential(λ): density f_S(s) = λe^{-λs}, s ≥ 0. Memoryless property: P(S > t+s | S > s) = P(S > t).
Waiting Time Distribution
Waiting time for nth event Tₙ = sum of n exponentials → Gamma distribution with shape n, rate λ.
Implications
Exponential interarrival times imply lack of memory, continuous-time Markov properties, and facilitate analytical tractability.
Markov Property and Memorylessness
Markov Property
Poisson process is Markov: future evolution depends only on current state N(t), not past. Transition probabilities depend on increments.
Memorylessness of Exponential
Exponential interarrival times cause Markov property. For s,t ≥ 0, P(N(t+s)-N(t) = k | history) = P(N(s) = k).
Consequences
Enables use of Markov chains, differential equations to analyze event counts and waiting times.
Distribution of Counts and Events
Counting Distribution
N(t) ~ Poisson(λt). Probability mass function: P(N(t) = k) = e^{-λt}(λt)^k/k!, k = 0,1,...
Joint Distributions
Counts in disjoint intervals independent. For intervals (t₀,t₁], (t₁,t₂], ..., counts follow independent Poisson distributions.
Order Statistics of Event Times
Conditioned on N(t) = n, event times T₁,...,T_n are distributed as order statistics of n i.i.d. Uniform(0,t) random variables.
| Parameter | Distribution | Remarks |
|---|---|---|
| N(t) | Poisson(λt) | Number of events by time t |
| Sᵢ | Exponential(λ) | Interarrival times |
| Tₙ | Gamma(n, λ) | Time of nth event |
Extensions and Variants
Nonhomogeneous Poisson Process
Rate λ(t) varies with time. N(t) has increments Poisson distributed with parameter ∫₀^t λ(u) du.
Compound Poisson Process
Each event carries random jump magnitude independent of counting process. Used in risk theory, finance.
Spatial Poisson Process
Events distributed randomly in space, intensity measure replaces λt. Basis for point processes in multiple dimensions.
Applications in Probability and Other Fields
Queueing Theory
Models arrival of customers, packets with Poisson arrivals; fundamental in M/M/1 and related queues.
Reliability Engineering
Failure events modeled as Poisson process; used to estimate lifetimes and maintenance schedules.
Telecommunications and Physics
Model call arrivals, photon counts, radioactive decay events.
Simulation Methods
Interarrival Time Method
Generate exponential interarrival times. Sum to get event times. Simple, exact method.
Thinning Algorithm
Simulate Poisson process with higher rate, accept events with probability p for nonhomogeneous cases.
Inverse Transform Sampling
Uses uniform random variables U ~ Uniform(0,1), transform via T = -ln(U)/λ for interarrival times.
Algorithm: Simulate homogeneous Poisson process with rate λ up to time T_maxInitialize t ← 0While t < T_max: Generate U ~ Uniform(0,1) Set S = -ln(U)/λ t ← t + S If t ≤ T_max, record event at tEndMathematical Formulations and Proofs
Probability Mass Function
P(N(t) = k) = e^{-λt} (λt)^k / k!, k = 0,1,2,...Interarrival Time PDF
f_S(s) = λ e^{-λs}, s ≥ 0Order Statistics Property
Conditioned on N(t) = n, event times T₁,...,T_n have joint pdf:
f(t₁,...,t_n) = n! / t^n, 0 < t₁ < t₂ < ... < t_n < tProof Sketch of Independent Increments
Using definition via exponential interarrivals and memorylessness, increments over disjoint intervals depend on independent exponential variables.
Limitations and Assumptions
Assumption of Stationarity
Constant rate λ often unrealistic; nonhomogeneous processes needed for time-varying phenomena.
No Simultaneous Events
Poisson process excludes multiple events at exact time; unsuitable for bursty phenomena.
Independence Assumptions
Independence of increments may not hold in dependent or clustered event scenarios.
References
- Kingman, J.F.C. "Poisson Processes." Oxford University Press, 1993.
- Daley, D.J., Vere-Jones, D. "An Introduction to the Theory of Point Processes." Springer, 2003.
- Ross, S.M. "Stochastic Processes." Wiley, 2nd Edition, 1996.
- Grimmett, G., Stirzaker, D. "Probability and Random Processes." Oxford University Press, 2001.
- Asmussen, S. "Applied Probability and Queues." Springer, 2003.