🃏Engineering Probability Unit 15 – Poisson Processes & Continuous-Time Markov Chains

Key Concepts and Definitions

• Poisson process models the occurrence of events over time when the events happen independently at a constant average rate
• Exponential distribution describes the time between events in a Poisson process with parameter $\lambda$ representing the average rate of events
• Markov property states that the future state of a process depends only on the current state, not on the past states
• Continuous-Time Markov Chain (CTMC) is a stochastic process that satisfies the Markov property and has a countable state space and continuous time
• Transition rate $q_{ij}$ is the rate at which the process transitions from state $i$ to state $j$ in a CTMC
• Steady-state probabilities $\pi_j$ represent the long-run proportion of time the CTMC spends in each state $j$
• Chapman-Kolmogorov equations describe the relationship between transition probabilities over different time intervals in a CTMC

Poisson Processes Fundamentals

• Poisson process is characterized by a constant average rate of events $\lambda$ (events per unit time)
• Inter-arrival times between events in a Poisson process follow an exponential distribution with parameter $\lambda$
• Probability of $k$ events occurring in a time interval of length $t$ is given by the Poisson distribution: $P(N(t)=k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}$
• Memoryless property of the exponential distribution implies that the time until the next event is independent of the time since the last event
• Superposition of independent Poisson processes results in another Poisson process with a rate equal to the sum of the individual rates
• Splitting a Poisson process randomly into two or more processes results in independent Poisson processes with rates proportional to the splitting probabilities

Properties of Poisson Processes

• Stationary increments property states that the number of events in any interval depends only on the length of the interval, not on its starting time
• Independent increments property implies that the number of events in disjoint time intervals are independent random variables
• Poisson distribution gives the probability of a specific number of events occurring in a fixed time interval
• Exponential distribution describes the time between consecutive events in a Poisson process
• Lack of memory property of the exponential distribution means that the time until the next event is independent of the time since the last event
• Superposition and splitting properties allow for the combination and decomposition of Poisson processes

Applications of Poisson Processes

• Modeling the arrival of customers in a queue (bank, supermarket, call center)
• Describing the occurrence of rare events (earthquakes, accidents, machine failures)
• Analyzing the number of defects or flaws in a manufacturing process
• Studying the arrival of packets in a communication network or requests to a web server
• Modeling the occurrence of mutations in DNA sequences or the spread of a disease
• Predicting the number of insurance claims or warranty returns for a product
• Investigating the arrival of particles in a physical system (radioactive decay, photon emission)

Introduction to Continuous-Time Markov Chains

• CTMC is a stochastic process with a countable state space and continuous time, satisfying the Markov property
• State space is the set of all possible states the process can occupy
• Transitions between states occur according to exponentially distributed holding times
• Markov property implies that the future state of the process depends only on the current state, not on the past states
• Transition rates $q_{ij}$ determine the rate at which the process moves from state $i$ to state $j$
• Generator matrix $Q$ contains the transition rates, with diagonal elements $q_{ii} = -\sum_{j \neq i} q_{ij}$
• Kolmogorov forward and backward equations describe the evolution of the transition probabilities over time

Transition Probabilities and Rates

• Transition probability $p_{ij}(t)$ is the probability of moving from state $i$ to state $j$ in time $t$
• Transition rate $q_{ij}$ represents the instantaneous rate of transition from state $i$ to state $j$
  • Related to the transition probability by $q_{ij} = \lim_{t \to 0} \frac{p_{ij}(t)}{t}$ for $i \neq j$
• Generator matrix $Q$ contains the transition rates and determines the dynamics of the CTMC
  • Diagonal elements $q_{ii} = -\sum_{j \neq i} q_{ij}$ ensure that rows sum to zero
• Kolmogorov forward equation: $\frac{d}{dt}P(t) = P(t)Q$, where $P(t)$ is the matrix of transition probabilities at time $t$
• Kolmogorov backward equation: $\frac{d}{dt}P(t) = QP(t)$
• Solution to the Kolmogorov equations gives the transition probability matrix $P(t) = e^{Qt}$

Steady-State Analysis

• Steady-state or equilibrium probabilities $\pi_j$ represent the long-run proportion of time the CTMC spends in each state $j$
• Steady-state probabilities satisfy the balance equations: $\sum_{i} \pi_i q_{ij} = 0$ for all $j$
  • Intuition: rate of entering a state equals the rate of leaving the state in the long run
• Normalization condition: $\sum_{j} \pi_j = 1$, ensures that probabilities sum to one
• Steady-state probabilities can be found by solving the system of linear equations formed by the balance equations and the normalization condition
• Existence and uniqueness of steady-state probabilities depend on the properties of the CTMC (irreducibility, positive recurrence)
• Mean time spent in each state and other long-run average quantities can be computed using the steady-state probabilities

Real-World Examples and Problem Solving

• Modeling the reliability of a machine with multiple components (states: working, failed, under repair)
• Analyzing the performance of a multi-server queue (states: number of customers in the system)
• Studying the spread of a rumor or information in a social network (states: informed, uninformed)
• Predicting the behavior of a chemical reaction with different molecular states
• Optimizing the inventory management in a production system with different stock levels
• Investigating the population dynamics of a species with birth, death, and migration processes
• Solving problems involving the computation of transition probabilities, hitting times, and steady-state quantities in various application domains