Understanding the Poisson Process

• The Poisson process models events occurring randomly in time.
• It's fundamental in queuing theory, reliability analysis, and risk management.

Definition of a Poisson Process

• A Poisson process ${N(t), t \geq 0}$ counts events in a time interval.
• $N(0) = 0$: The process starts with no events at time zero.
• Independent Increments: Events in disjoint time intervals are independent.
• Poisson Distribution: For $t > 0$, $N(t)$ follows a Poisson distribution with mean $\lambda t$ ($\lambda$ is the rate parameter).
• Formula for Poisson Distribution: $$ P(N(t) = n) = \frac{e^{-\lambda t}(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots $$

Properties of a Poisson Process

• Stationary Increments: Distribution depends on interval length, not location.
• Memoryless Property: Future is independent of the past, given the present state.
• Rare Events: Events occur one at a time; simultaneous events have negligible probability.

Interarrival Times

• $T_i$ is the time between the $(i-1)$-th and $i$-th event.
• Interarrival times $T_1, T_2, \dots$ are independent and identically distributed (i.i.d.) exponential random variables with rate parameter $\lambda$.
• Formula for Interarrival Times: $$ P(T_i > t) = e^{-\lambda t}, \quad t \geq 0 $$

Applications of Poisson Process

• Queuing Theory: Models customer arrivals in a queue.
• Reliability Analysis: Predicts failure rate of components or systems.
• Risk Management: Assesses frequency of insurance claims or financial losses.
• Telecommunications: Analyzes call arrivals in a telephone exchange.

Examples

• Call center receives calls with a rate of $\lambda = 5$ calls per minute.
• One can calculate the probability of recieving a specific number of calls in a time frame
• Also can calculate the probability of the time until the next call

Thinning and Superposition

• Thinning: Events independently marked with probability $p$ form a Poisson process with rate $p\lambda$.
• Superposition: Merged independent Poisson processes result in a Poisson process with a rate equal to the sum of individual rates.

Non-Homogeneous Poisson Process

• The rate parameter $\lambda(t)$ varies with time.
• Events in interval $[t, t+dt]$ follow a Poisson distribution with mean $\int_t^{t+dt} \lambda(s) ds$.

Conclusion

• The Poisson process is a tool for modeling random events in time.
• Understanding its properties and applications provides insights across domains.