[Math] Homogenous vs. Non-Homogenous Poisson Processes

Question

I understand that at the main difference between a homogenous vs. non-homogenous Poisson process is that a homogenous Poisson process has a constant rate parameter $\lambda$ while a non-homogenous Poisson process can have a variable rater parameter $\lambda(t)$ that is a function of time.

My Question: Let's say I want to model the number of cars that arrive at an intersection in one hour, and for the first 30 minutes cars come every 5 minutes on average, while in the second 30 minutes cars come every 3 minutes on average. I know that it is certainly possible to model this process with a non-homogenous Poisson process, but could I also model this with 2 homogenous Poisson processes added together or something like that?

Follow-up: One thing that I might want to answer is the probability that no arrivals happen throughout the hour. It seems intuitive to me that I could find the probability of 0 arrivals in the first 30 minutes with a standard Poisson process with rate 5, and then add this probability to the probability of 0 arrivals in the second 30 minutes with a standard Poisson process with rate 3. True?

Answer 1

Let's say the time unit is one hour. Then one car every 5 minutes corresponds to an intensity of $\lambda_1=12$, ($12$ per hour) and one car every 3 minutes corresponds to an intensity of $\lambda_2=20$. Now I see two possibilities:

• Using a non-homogeneous Poisson process, with $\lambda(t)=\lambda_1$ if $0\leq t\leq \frac{1}{2}$ and $\lambda(t)=\lambda_2$ if $\frac{1}{2}\lt t\leq 1$.
• Using the fact that the sum of two independent homogeneous Poisson process with parameters $\lambda$ and $\mu$ respectively is a homogeneous Poisson process with parameter $\lambda+\mu$. So you could have, in the first 30 minutes, a Poisson process $P_1$ with parameter $12$, and in the other half hour, the sum of $P_1$ with $P_2$, a Poisson process with parameter $8$, so that the result is a Poisson process with parameter $20$. But $P_2$ would only start after $30$ minutes. So you'd have several origins for time, which seems to me a bad idea... I mentioned it only to recall the property for the sum of Poisson processes.

No car during the first hour means that there is no car during the first 30 minutes, and no car during the next 30 minutes. So you should multiply the probabilities, not add them.