2.12.2 Interarrival TimesWe are now interested in other properties of the Poisson process. For instance, suppose that we start observing the process at time t = 0 and we wish to know the pdf for the time of occurrence of the kth arrival. Define the random variable ![]() Since dx is infinitesimal, we can ignore the o(dx) term, and obtain the kth order interarrival time distribution ![]() We have the fundamental result that the kth-order interarrival time distribution for a Poisson process is a kth-order Erlang pdf. Setting k = 1 in (2.57), we have the first-order arrival distribution ![]() This is our familiar exponential pdf with its now famous "no-memory" property. This result proves that the Poisson process is a no-memory process, that is, that future arrivals do not depend on the number or times of occurrences of previous arrivals. |