# How to Find the Distribution of T for Competing Poisson Processes

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• hezilap
In summary, the conversation involves a problem where red and blue cars arrive independently on a time interval with given rates. The objective is to find the distribution or expected value of the first red car's arrival time, given that its nearest neighbor is a blue car. The solution may involve simulations or analytic methods, and the questioner has reasons to believe an analytic solution exists. Further clarification is needed on the problem's requirements.
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...

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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).

hezilap and 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.

FactChecker said:
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.

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hezilap said:
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.

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

@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?

How about working with order statistics for the two?

## 1. What is a Poisson process?

A Poisson process is a type of stochastic process that models the occurrence of events over time. It is characterized by the following properties: 1) events occur randomly and independently of each other, 2) the average rate of events is constant, and 3) the probability of an event occurring in a given time interval is proportional to the length of the interval.

## 2. What is a competing Poisson process?

A competing Poisson process is a type of Poisson process that models the occurrence of events that are competing with each other. This means that only one type of event can occur at a given time, and the occurrence of one type of event prevents the occurrence of the other type.

## 3. How is a competing Poisson process different from a regular Poisson process?

The main difference between a competing Poisson process and a regular Poisson process is that in a competing process, the events are mutually exclusive, while in a regular process, the events can occur simultaneously. Additionally, the rate of occurrence of events in a competing process is affected by the occurrence of the other type of event, while in a regular process, the rate is constant.

## 4. What are some real-world applications of competing Poisson processes?

Competing Poisson processes have various applications in different fields, such as epidemiology, finance, and biology. For example, in epidemiology, competing processes can model the spread of multiple diseases in a population. In finance, they can be used to model the competition between different types of investments. In biology, they can be used to study the competition between different species in an ecosystem.

## 5. How are competing Poisson processes analyzed and modeled?

Competing Poisson processes can be analyzed using mathematical techniques, such as probability theory and stochastic processes. They can also be modeled using statistical methods, such as maximum likelihood estimation and Bayesian inference. Additionally, computer simulations can be used to study the behavior of competing processes in different scenarios.

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