Birth-Death Processes

Mathematical Modeling · Axiom Academy

LESSON Birth-Death Processes - Mathematical Modeling Unit 4: Probabilistic Models - Mathematical Modeling What is a Birth-Death Process? A birth-death process is a continuous-time Markov chain where transitions occur only between neighboring states. The state represents a population count, and changes happen through "births" (increases by 1) or "deaths" (decreases by 1). "Populations change one individual at a time." Birth-death processes are fundamental models in: Population dynamics and ecology Queueing theory (customers arriving and departing) Epidemiology (individuals becoming infected/recovering) Telecommunications (call arrivals and completions) Reliability engineering (system failures and repairs) From Discrete to Continuous Time While discrete-time Markov chains update at fixed time steps, birth-death processes evolve in continuous time. Events can occur at any instant. Transition probabilities p_ ij Continuous-Time (Birth-Death): The Key Insight: Rates Instead of Probabilities In continuous time, we work with transition rates instead of probabilities: = birth rate when in state n (rate of ) = death rate when in state n (rate of ) A rate means: in a small time interval dt , the probability of a transition is approximately . Structure of a Birth-Death Process The state space is typically (non-negative integers representing population size). In a small time interval [t, t+dt) , given the process is in state n :

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