How to Find the Distribution of T for Competing Poisson Processes

[hezilap]

Hi all, I've been struggling for days now with this problem. Would appreciate any idea you might have.
Red cars and blue cars arrive as independent Poisson processes on [0, ∞) with respective rates λ_r, λ_b. Let T denote the arrival time of the first red car whose nearest neighbor is a blue car. ("Nearest" in the sense of arrival times.)

How can I find the distribution of T, or at least its expected value? My attempts have led me nowhere...

 

[phyzguy]

Interesting problem. I'm assuming you want the distribution of T as a function of λ_r, λ_b. I would first write a simple program to simulate the problem with different values of λ_r, λ_b. This might be enough to help you answer your question, or it might help guide you to an analytic solution. You might also try thinking about limiting cases. For example, when λ_r << λ_b, then the first red car is almost certainly surrounded by two blue cars, so T should just be 1/λ_r (I think).

 

[FactChecker]

Problems that appear simple may not have simple closed-form analytic solutions at all. And even if they do, the slightest change in the problem can completely destroy that approach. Is there some reason that you think this has such a solution? If not, a simulation may be the best you can do for a practical, easily modified, solution.

 

[hezilap]

Problems that appear simple may not have simple closed-form analytic solutions at all. And even if they do, the slightest change in the problem can completely destroy that approach. Is there some reason that you think this has such a solution? If not, a simulation may be the best you can do for a practical, easily modified, solution.

Thanks for replying. The question came up in a small research project I'm doing, and yes, I have good reasons to believe an analytic solution to this particular problem exists (moreover, one a grad student should be able to handle).
I'll run simulations, for sure, but my focus remains on deriving a mathematically sound closed-form solution. If you guys have any ideas whatsoever, even half-baked ones, I'd love to hear them.

 

[FactChecker]

I have good reasons to believe an analytic solution to this particular problem exists (moreover, one a grad student should be able to handle).
... If you guys have any ideas whatsoever, even half-baked ones, I'd love to hear them.

Ok. Then you should be able to give us a hint about how to do it rather than asking us for half-baked ones.

 

[Hornbein]

A good first step would be finding the expectation and distribution for one of those processes.

 

[phyzguy]

@hezilap. Clarification on the problem. When you say "the arrival time of the first red car whose nearest neighbor is a blue car". are R-B and B-R equivalent? What about B-R-B? Is that required?

 

[WWGD]

How about working with order statistics for the two?