Poisson Process - Properties

Properties

As defined above, the stochastic process {N(t)} is a Markov process, or more specifically, a continuous-time Markov process.

To illustrate the exponentially distributed inter-arrival times property, consider a homogeneous Poisson process N(t) with rate parameter λ, and let T_k be the time of the kth arrival, for k = 1, 2, 3, ... . Clearly the number of arrivals before some fixed time t is less than k if and only if the waiting time until the kth arrival is more than t. In symbols, the event occurs if and only if the event occurs. Consequently the probabilities of these events are the same:

In particular, consider the waiting time until the first arrival. Clearly that time is more than t if and only if the number of arrivals before time t is 0. Combining this latter property with the above probability distribution for the number of homogeneous Poisson process events in a fixed interval gives

Consequently, the waiting time until the first arrival T₁ has an exponential distribution, and is thus memoryless. One can similarly show that the other interarrival times T_k − T_k−1 share the same distribution. Hence, they are independent, identically distributed (i.i.d.) random variables with parameter λ > 0; and expected value 1/λ. For example, if the average rate of arrivals is 5 per minute, then the average waiting time between arrivals is 1/5 minute.