A Poisson point process of parameter λ is a stochastic process N(t) such that N(O)=0, N(t) is incremented by +1 after a time T distributed according to an exponential law of parameter λ. We are talking about Poisson arrivals if the time between two arrivals is exponential.
Take the value of N(t) for state, then the continuous-time Markov chain associated with the Poisson process λ is:
It is possible to know the probability that N is at number k at time t by the formula:
N(t) is distributed according to a Poisson distribution of parameter λt.
The Poisson process associates and decomposes as follows:
- The union of n Poisson process is a Poisson process whose parameter is the sum of n parameters
- A Poisson process that decomposes into n processes with pi probabilities. These n processes are then respective rate Poisson processes λpi