# Probability of collisions in queues of Poisson process

I have a process whereby objects of width $W$ land on a gene at rate $F$ (per second, poisson process, lets assume), and then start to move along at constant speed $V$. I'm trying to work out the rate at which collisions occur - a preceding object hasn't moved out of the way sufficiently fast that a subsequent object can't fit (given it's got finite width) - does the following logic work?

The 'danger zone' is the width $W$, so the existing object will be able to prevent subsequent dockings for $\tfrac{W}{V}$ seconds. And the number of expected 'arrivals' in $t$ seconds is $F\times t$, so for each existing object we'd expect it to prevent $F\times \frac{W}{V}$ subsequent objects.

So for each success, we get $k=F\times \frac{W}{V}$ 'fails', so the proportion of all objects that fail is $\frac{k}{1+k}$. And I guess I can multiply by $F$ to get the failure rate per second.

Seems suspiciously simplistic, and doesn't agree with a quickly mocked-up simulation, but I'm too close to the problem to see the flaw.

• In this model, what happens when a collision occurs? Do the two objects destroy each other? Do they keep moving? – Math1000 Sep 2 '16 at 5:40