4.1 Definition and properties of Poisson processes

Poisson processes are crucial stochastic models that describe random events occurring over time or space. They're characterized by a constant rate parameter and have independent, stationary increments following a Poisson distribution.

These processes possess unique properties like memorylessness and the ability to be superposed or split. They're widely used in various fields, including queueing theory, reliability engineering, and modeling rare events, making them essential tools for analyzing complex systems.

Definition of Poisson processes

• A Poisson process is a stochastic process that models the occurrence of events over time or space, where the events occur randomly and independently of each other
• The Poisson process is characterized by a constant rate parameter $\lambda$, which represents the average number of events occurring per unit time or space
• The number of events occurring in any interval follows a Poisson distribution with mean $\lambda t$, where $t$ is the length of the interval

Counting process

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• A Poisson process can be viewed as a counting process $\{N(t), t \geq 0\}$, where $N(t)$ represents the number of events that have occurred up to time $t$
• The counting process satisfies the following properties:
  • $N(0) = 0$ (no events at time 0)
  • $N(t)$ is a non-negative integer for all $t \geq 0$
  • $N(t)$ is non-decreasing (events cannot be undone)
• The increments of the counting process, $N(t+s) - N(t)$, represent the number of events occurring in the interval $(t, t+s]$

Independent increments

• A key property of Poisson processes is that the increments over disjoint intervals are independent
• This means that the number of events occurring in one interval does not depend on the number of events occurring in any other non-overlapping interval
• Mathematically, for any $0 \leq t_1 < t_2 \leq t_3 < t_4$, the random variables $N(t_2) - N(t_1)$ and $N(t_4) - N(t_3)$ are independent

Stationary increments

• Poisson processes have stationary increments, meaning that the distribution of the number of events occurring in an interval depends only on the length of the interval and not on its starting time
• For any $s, t \geq 0$, the distribution of $N(t+s) - N(t)$ is the same as the distribution of $N(s)$
• This property implies that the rate of events remains constant over time

Poisson distribution for increments

• The number of events occurring in any interval of length $t$ follows a Poisson distribution with mean $\lambda t$

• The probability mass function (PMF) of the number of events $N(t)$ is given by: $P(N(t) = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}, \quad k = 0, 1, 2, \ldots$

• The mean and variance of $N(t)$ are both equal to $\lambda t$

Properties of Poisson processes

Memoryless property

• Poisson processes possess the memoryless property, which means that the probability of an event occurring in the next small interval of time does not depend on the time that has elapsed since the last event

• Mathematically, for any $s, t \geq 0$: $P(N(t+s) - N(t) = k | N(t) = n) = P(N(s) = k)$

• This property is a consequence of the independent and stationary increments of the Poisson process

Superposition of independent Poisson processes

• If $N_1(t)$ and $N_2(t)$ are two independent Poisson processes with rates $\lambda_1$ and $\lambda_2$, respectively, then their sum $N(t) = N_1(t) + N_2(t)$ is also a Poisson process with rate $\lambda = \lambda_1 + \lambda_2$
• This property allows for the modeling of complex systems by combining simpler Poisson processes
• For example, the arrival of customers at a store from two independent sources (walk-ins and online orders) can be modeled as the superposition of two Poisson processes

Splitting of Poisson processes

• If events in a Poisson process with rate $\lambda$ are independently classified into two types (Type 1 and Type 2) with probabilities $p$ and $1-p$, respectively, then the processes counting the events of each type are independent Poisson processes with rates $\lambda p$ and $\lambda (1-p)$
• This property is useful for modeling systems where events can be categorized into different types
• For example, the arrival of calls at a call center can be split into two independent Poisson processes: one for customer service calls and another for technical support calls

Conditional distribution of arrival times

• Given that $n$ events have occurred in the interval $[0, t]$, the conditional distribution of the arrival times of these events is the same as the distribution of $n$ independent and identically distributed (i.i.d.) random variables with a uniform distribution on $[0, t]$
• This property allows for the analysis of the distribution of inter-arrival times between events in a Poisson process
• For example, if we know that 5 customers arrived at a store during a 1-hour period, the distribution of the times between their arrivals can be modeled as 4 i.i.d. uniform random variables on $[0, 1]$

Poisson process as limit of binomial processes

Binomial process definition

• A binomial process is a discrete-time stochastic process that models the number of successes in a fixed number of independent trials, where each trial has the same probability of success
• Let $X_1, X_2, \ldots, X_n$ be i.i.d. Bernoulli random variables with success probability $p$, and define $S_n = \sum_{i=1}^n X_i$
• The process $\{S_n, n \geq 1\}$ is called a binomial process, and $S_n$ follows a binomial distribution with parameters $n$ and $p$

Limiting behavior as n → ∞

• As the number of trials $n$ in a binomial process increases and the success probability $p$ decreases in such a way that $np$ remains constant (equal to $\lambda$), the binomial process converges to a Poisson process with rate $\lambda$

• Mathematically, if $n \to \infty$, $p \to 0$, and $np \to \lambda$, then for any fixed $t > 0$: $\lim_{n \to \infty} P(S_{\lfloor nt \rfloor} = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}, \quad k = 0, 1, 2, \ldots$

• This convergence result provides a connection between discrete-time binomial processes and continuous-time Poisson processes

• For example, the number of defective items in a large production batch can be approximated by a Poisson distribution if the probability of a single item being defective is small and the batch size is large

Relationship to other processes

Comparison to renewal processes

• A renewal process is a generalization of a Poisson process, where the inter-arrival times between events are i.i.d. random variables with an arbitrary distribution (not necessarily exponential)
• In a Poisson process, the inter-arrival times are i.i.d. exponential random variables with rate $\lambda$
• Poisson processes are a special case of renewal processes with exponentially distributed inter-arrival times
• Renewal processes are used to model systems where the time between events has a more complex distribution than the exponential distribution

Poisson process as special case of birth-death process

• A birth-death process is a continuous-time Markov chain that models a population where individuals can be born (added to the population) or die (removed from the population) according to certain rates
• In a birth-death process, the birth and death rates can depend on the current population size
• A Poisson process can be viewed as a special case of a birth-death process where the birth rate is constant (equal to $\lambda$) and the death rate is zero
• Birth-death processes are used to model population dynamics, queueing systems, and other applications where the population size can increase or decrease over time

Applications of Poisson processes

Modeling rare events

• Poisson processes are often used to model the occurrence of rare events, such as earthquakes, accidents, or machine failures
• When events are rare and occur independently, the Poisson process provides a good approximation for the distribution of the number of events occurring in a given time interval
• For example, the number of car accidents in a city during a day can be modeled using a Poisson distribution if accidents are rare and occur independently

Queueing theory

• Poisson processes are fundamental in queueing theory, which studies the behavior of waiting lines (queues) in various settings, such as customer service centers, manufacturing systems, or communication networks
• The arrival of customers or jobs to a queueing system is often modeled as a Poisson process, with the rate $\lambda$ representing the average number of arrivals per unit time
• Queueing models based on Poisson processes help analyze system performance measures, such as average waiting times, queue lengths, and server utilization

Reliability engineering

• Poisson processes are used in reliability engineering to model the occurrence of failures in systems or components
• The number of failures occurring in a given time interval can be modeled using a Poisson distribution, with the rate $\lambda$ representing the average number of failures per unit time
• Reliability metrics, such as the mean time between failures (MTBF) and the reliability function, can be derived based on the properties of the Poisson process
• For example, the failure of light bulbs in a large building can be modeled using a Poisson process, helping to plan maintenance schedules and inventory management

Simulating Poisson processes

Inverse transform method

• The inverse transform method is a general technique for simulating random variables with a given cumulative distribution function (CDF)
• To simulate a Poisson process using the inverse transform method:

  1. Generate a sequence of i.i.d. uniform random variables $U_1, U_2, \ldots$ on $[0, 1]$

  2. Set $T_0 = 0$ and compute the event times $T_n$ recursively as:

     $T_n = T_{n-1} - \frac{1}{\lambda} \ln(U_n), \quad n = 1, 2, \ldots$

  3. The sequence $\{T_n, n \geq 1\}$ represents the event times of the simulated Poisson process

• The inter-arrival times $T_n - T_{n-1}$ are i.i.d. exponential random variables with rate $\lambda$, consistent with the properties of a Poisson process

Thinning method

• The thinning method (also known as the acceptance-rejection method) is another technique for simulating Poisson processes, particularly when the rate $\lambda$ is not constant over time
• To simulate a Poisson process with a time-varying rate $\lambda(t)$ using the thinning method:

  1. Find an upper bound $\lambda^*$ such that $\lambda(t) \leq \lambda^*$ for all $t$

  2. Simulate a homogeneous Poisson process with rate $\lambda^*$ using the inverse transform method, obtaining event times $T_1^*, T_2^*, \ldots$

  3. For each event time $T_n^*$, generate a uniform random variable $U_n$ on $[0, 1]$ and accept the event if:

     $U_n \leq \frac{\lambda(T_n^*)}{\lambda^*}$

  4. The accepted event times form a non-homogeneous Poisson process with rate $\lambda(t)$

• The thinning method can be used to simulate Poisson processes with more complex rate functions, such as those encountered in modeling time-varying arrival rates or failure rates