# Generating 'bursty' traffic using probability distributions beyond Poisson

I'm trying to develop a more realistic vehicle generation model for populating a traffic microsimulator. I'm trying to model a real-world intersection from which I have historical flow data. Currently my vehicle generation model is a simple Poisson process, every second I perform the following to determine if a vehicle should be generated (released) into the network:

if Uniform(0,1) < $1-e^{-\lambda*1}$:
generate vehicle

Where $\lambda$ represents the flow of vehicles in units vehicles/second.

This method works well, as it generates vehicles at random seconds and gives me the correct flow over long simulation durations (i.e. if I'm simulating for an hour and $\lambda = 0.5$ veh/s, at the end of the simulation hour, ~1800 vehicles are generated).

The problem is that my Poisson vehicle generation model does not replicate the 'bursty' nature of traffic, I've read research papers that support this and they've recommended using a long-tail distribution, such as a Pareto distribution. I've had difficulty finding vehicle traffic generation models, but there is ample research detailing how to develop simulation models for internet/web traffic using long tail.

I've consulted the Pareto wiki page and understand that I can perform inverse transform sampling to generate vehicle release times, slightly different from my above Poisson method. I understand that instead of every second deciding if a vehicle should be released using exponential CDF, I could generate a sequence of vehicle release times following a Pareto distribution, however I don't know how to incorporate the $\lambda$ rate as before, as I need the total number of vehicles at the end of an hour simulation to be proportional to $\lambda$. I understand a Pareto is defined by its shape and scale parameters, but I don't know how to relate these to $\lambda$.

I've also come across Poisson Pareto Burst Processes, where the arrivals of events are Poisson but the event durations are Pareto, which I think could work well for my model, as I'm capable of generating multiple vehicles at the same time, but again I don't know how to ensure the total number of vehicles at the end of an hour is proportional to $\lambda$.

Basically, is there any way to incorporate a rate $\lambda$ from a Poisson process into a long-tail distribution?

Thanks for any ideas.