Counting Process Simulation

Counting Process Simulation

Simulation idea

We want to simulate a counting process over a time interval T, where events happen:

  • Independently
  • Uniformly in time
  • With constant average rate λ

We approximate this by dividing T into n small intervals. For each interval:

$$\text{Probability of an event} = \frac{\lambda}{n}$$

This is equivalent to simulating a Bernoulli process in discrete time.

Theoretical Properties

  1. Limiting Process:
    As \(n \to \infty\), this discrete process converges to a Poisson process N(t)with rate λ.
  2. Properties of Poisson Process:
    • N(t) counts the number of events up to time t
    • Number of events in disjoint intervals are independent
    • Number of events in interval of length t follows Poisson distribution: $$P(N(t) = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}$$
    • Inter-event times (time between events) are exponentially distributed with mean \(1/\lambda\)
  1. Interpretation of λ:
    • Average number of events per unit time
    • Higher λ → more frequent events

Events occur independently and uniformly over time with average rate λ.