Poisson Process - Characterisation

Characterisation

In its most general form, the only two conditions for a counting process to be a Poisson process are:

• Orderliness: which roughly means

which implies that arrivals don't occur simultaneously (but this is actually a mathematically stronger statement).

• Memorylessness (also called evolution without after-effects): the number of arrivals occurring in any bounded interval of time after time t is independent of the number of arrivals occurring before time t.

These seemingly unrestrictive conditions actually impose a great deal of structure in the Poisson process. In particular, they imply that the time between consecutive events (called interarrival times) are independent random variables. For the homogeneous Poisson process, these inter-arrival times are exponentially distributed with parameter λ (mean 1/λ).

Also, the memorylessness property entails that the number of events in any time interval is independent of the number of events in any other interval that is disjoint from it. This latter property is known as the independent increments property of the Poisson process.