# Thread: Modelling a Busy Traffic Intersection to Calculate Probability

1. ## Modelling a Busy Traffic Intersection to Calculate Probability

Hello, I wasn't sure if this problem was complex enough to warrant it being in the University Statistics thread. If it should be there, feel free to move it.

The problem is as follows.

Suppose Car X is at a stop sign of a cross intersection going north. The road to the east and west don't have stop signs. Hence traffic to the east and west has the right of way.
Next suppose it is rush hour, hence there is a high flow of cars coming from both east and west.

I want to calculate the probability at a given time t in a time period, (say, an hour for 'rush hour') when it would be safe for Car X to continue forward. ie. When no car from the east or west is within a certain distance from the intersection.

The following are my suppositions:
- My intuition is that this would involve a Poisson distribution, counting number of cars N within the time interval (in this case an hour) to calculate lambda.
- Assuming it'd be best to generalize the problem starting from assuming that traffic only comes from the east, solving that problem first, then extending to multiple traffic flows.

Note: I would consider a method as opposed to a solution sufficient in answering my question. My motivation is to provide some grounding to an argument to install lights that I can send to my local gov't, particularly for safety reasons.

2. ## Re: Modelling a Busy Traffic Intersection to Calculate Probability

what you need is

$P[\text{no cars are "emitted" between }[t,t+\delta] | \text{ a car was emitted at time } t]$

where $\delta$ has to account for the time needed for the car emitted at $t$ to clear the intersection and for the car heading north to cross it from a standstill.

Suppose you fix $\dfrac{D}{v} = t_0$

where $D$ is the distance threshold, and $v$ is the speed of the traffic.

$t_0$ is the time it takes for the oncoming car to clear the intersection.

You can fix the time it takes to cross the intersection with some safe time, call this $t_1$

$P[\text{no cars are "emitted" between }[t,t+t_0+t_1] | \text{ a car was emitted at time } t] = e^{-\lambda(t_0+t_1)}$

Let's throw some realistic numbers at this.

say $t_0 = 6,~t_1=4$

we'll let $\lambda$ vary as an independent variable

I suspect the traffic engineer(s) for your town will use more complicated models that take into account varying speeds of cars.

I'd just report the problem directly, that you find yourself unable to safely cross that intersection during certain times of the day, rather than throw a remedial traffic model at them.