For the last few weeks I’ve been working to wrap my head around something called Poisson processes. A Poisson process is a simple and widely used stochastic process for modeling the time at which arrivals enter a system. If y’all have taken an intro probability course, y’all might’ve hear of something called the Poisson distribution — which is simply the probability of getting exactly n successes in N amount of trials. Like the Poisson distribution, a Poisson process is essentially thought of a the continuous-time version of the Bernoulli process (not trying to imply the Poisson dist. is continuous, but it is the limit of the Bernoulli dist. taken to infinity!)
For a Poisson process, arrivals may occur at arbitrary positive times, and the probability of an arrival at any particular instant is zero. This means that there’s no very clean way of describing a Poisson process in terms of the probability of an arrival at any given instant. Instead, it’s much easier to define a Poisson process in terms of the sequence of interarrival times, that is, the time between each successive arrival.
This chapter has been taking me forever to get through, primarily because it’s soooo large; this book really breaks it down into manageable (and all equally important) subsections. The chapter starts by giving you a Poisson process, then describing more generally what an arrival process is, and from there, talking about the important properties of Poisson processes which make it the most elegantly simple renewal process (which are also a special type of arrival process).
There really isn’t much to know about what exactly a Poisson process is; they are characterized, perhaps predictably, by having exponentially distributed inter arrival times. Explicitly, we say that
- a renewal process is an arrival process for which the sequence of inter arrival times is a sequence of positive identical and independently distributed (IID) random variables.
- A Poisson process is a renewal process in which the inter arrival times have an exponential cumulative distribution function; i.e.; for some real λ > 0, each X_i has a density specified as f_X(x) = λexp(-λx) for all x greater than or equal to 0.
The parameter λ is the rate of the process, and remains constant despite the size of the interval.
Probably the coolest thing about a Poisson process is that, by nature of it’s exponential distribution, it has a memoryless property. This means that, if you suppose X is a rv denoting the waiting time until some arrival, and the arrival occurs at time t, then the distribution of X is the same for all times x < t. That is to say, that the distribution of the remaining time until an arrival, is exactly the same as the distribution of the original waiting time. This is denoted as:
We can use this memoryless property to extrapolate certain ideas about how the distribution of the first arrival in a Poisson process behaves, as well as how this first arrival (after time t) is independent of all arrivals up to and including time t.
Following this idea of a memoryless property that is attributed to exponential processes, we can show a lot more things as well like the idea of stationary and independent increment properties (though I’m not going to go into detail just because it’d make this post very lengthy).
There are other ways to define a Poisson process which might be more intuitive than this, but I like this way of describing what exactly it is and how we can use it to model simplistic stochastic processes. I might write up a post later on how to build up what a Poisson process is from its properties (as opposed to being given a Poisson process, and describing how it behaves from there) because I think a lot of the mathematical nuance is lost in this definition, but it’s way more practical.