Poisson Processes

📊Poisson Process Definition

Two Equivalent Characterizations

A Poisson process is a continuous-time counting process that describes the occurrence of rare events. Let $N(t)$ denote the number of events in the interval $(0, t]$ , with state space $I = \{0, 1, 2, \ldots\}$ .

Definition 1: Based on Increment Probability

Initial condition: $N(0) = 0$
Independent increments: For any $0 = t_0 < t_1 < \cdots < t_n$ , the increments $N(t_1) - N(t_0), \ldots, N(t_n) - N(t_{n-1})$ are independent
Rarity: For any $t \geq 0$ , as $h \to 0^+$ :
- $P\{N(t+h) - N(t) = 1\} = \lambda h + o(h)$ (intensity $\lambda > 0$ )
- $P\{N(t+h) - N(t) \geq 2\} = o(h)$ (almost no multiple events)

Definition 2: Based on Increment Distribution

Initial condition: $N(0) = 0$
Independent increments: Same as Definition 1
Increment distribution: For any $t > s \geq 0$ , $N(t) - N(s) \sim \text{Poisson}(\lambda(t-s))$

Equivalence Proof Sketch

Divide $(s, t]$ into $n$ subintervals of length $h = (t-s)/n$ . By rarity, each subinterval has approximately $\lambda h$ probability of one event. The total increment $N(t) - N(s)$ approximately follows $B(n, \lambda h)$ . As $n \to \infty$ , this converges to $\text{Poisson}(\lambda(t-s))$ .

🔍Core Properties

Mean and Variance

E[N(t)] = \lambda t, \quad D[N(t)] = \lambda t

Both the mean and variance grow linearly with time. For intensity $\lambda = 2$ events per minute, in 10 minutes we expect 20 events with variance 20.

Covariance and Autocorrelation

\text{Cov}[N(s), N(t)] = \lambda \min(s, t)

R_N(s, t) = \lambda \min(s, t) + \lambda^2 st

The correlation between two time points depends only on the shorter time interval. For $s = 3, t = 5$ , correlation = $\lambda \times 3$ .

Conditional Distribution 1

Given total events $N(t) = n$ , the distribution of events in $(0, s]$ :

P\{N(s) = m \mid N(t) = n\} = \binom{n}{m}\left(\frac{s}{t}\right)^m\left(1-\frac{s}{t}\right)^{n-m}

This follows a binomial distribution, indicating events are uniformly distributed in time.

📈Key Distributions: Event Timing and Time Intervals

Event Occurrence Times $W_n$

The $n$ -th event occurs at time $W_n$ :

W_n \sim \Gamma(n, \lambda)

Properties: $E[W_n] = \frac{n}{\lambda}$ , $D[W_n] = \frac{n}{\lambda^2}$

Application: Find probability that 5th customer arrives within 10 minutes.

Time Intervals $T_n$

Time between consecutive events:

T_n \sim \text{Exp}(\lambda)

Properties: Independent and identically distributed, memoryless property

Application: Find probability that time between customers ≤ 2 minutes.

Conditional Uniform Distribution

Given exactly one event in $(0, t]$ , the event time $W_1$ is uniformly distributed:

P\{W_1 \leq s \mid N(t) = 1\} = \frac{s}{t}, \quad 0 \leq s \leq t

Meaning: Rare events are uniformly distributed in time intervals.

🔄Synthesis and Decomposition

Synthesis (Superposition)

If $\{N_1(t)\}$ (intensity $\lambda_1$ ) and $\{N_2(t)\}$ (intensity $\lambda_2$ ) are independent Poisson processes, then:

N(t) = N_1(t) + N_2(t) \sim \text{Poisson}(\lambda_1 + \lambda_2)

Example: Two service windows with intensities 2 and 3 customers/hour create a combined process with intensity 5 customers/hour.

Decomposition (Thinning)

If $\{N(t)\}$ has intensity $\lambda$ , and each event is independently classified as type 1 with probability $p$ , then:

N_1(t) \sim \text{Poisson}(\lambda p), \quad N_2(t) \sim \text{Poisson}(\lambda(1-p))

Example: SMS process with intensity 10/day, 20% spam probability, creates independent spam (2/day) and useful (8/day) processes.

⚡Non-Homogeneous Poisson Processes

Time-Varying Intensity

When the event rate varies with time (e.g., rush hour traffic, business hour phone calls), we use a non-homogeneous Poisson process with intensity function $\lambda(t)$ .

Key Properties

Increment distribution: $N(t) - N(s) \sim \text{Poisson}(\Lambda(s,t))$ , where $\Lambda(s,t) = \int_s^t \lambda(u) du$ is the cumulative intensity
Mean function: $E[N(t)] = \Lambda(0,t) = \int_0^t \lambda(u) du$
Variance: $D[N(t)] = \Lambda(0,t)$ (still equal to mean)

Example: Linear Growth

For intensity function $\lambda(t) = 2t$ , the increment $N(3) - N(1)$ follows $\text{Poisson}(4)$ , since $\Lambda(1,3) = \int_1^3 2u du = 4$ .

🎯Typical Applications

Service System Counting

Bank with 2 windows, each with service completion rate $\lambda = 3$ customers/hour. Total service completions follow $\text{Poisson}(6)$ . Probability of 10 completions in 1 hour: $P\{N(1) = 10\} = \frac{6^{10}}{10!}e^{-6}$ .

Communication Systems

SMS reception with intensity $\lambda = 10$ per day, spam probability $p = 0.2$ . Spam messages follow $\text{Poisson}(2)$ per day. Probability of no spam in a day: $P\{N_1(1) = 0\} = e^{-2}$ .

Traffic Flow Statistics

Rush hour (7:00-9:00) with intensity function $\lambda(t) = 100 + 50(t-7)$ . Cumulative intensity 7:00-8:00: $\Lambda(7,8) = \int_7^8 (100 + 50(t-7)) dt = 125$ . Traffic flow $N(8) - N(7) \sim \text{Poisson}(125)$ .