Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to solve a simple stochastic problem. Imagine there are n agents who want to access a resource, with a probability p at a given time t. What ist the probability that the resource will be free and not accessed by an agent. How does the result change with 2n and 2 available resources? How do i solve this problem?

share|improve this question
    
Could you walk us through a simple example for the case of n=2, i.e. 2 agents with 1 resource, and 4 agents with 2 resources? –  Marcus Fry Oct 29 '10 at 18:02
    
Huh ok, pfff let's imagin you have 2 roomates and a video game console, you know that the other two players want to play a video game with the probability of 0.1 at a time t (for the porpose of the problem, let's assume that only one person can play at a time). So what is the probability that you can play at time t. The 2n is obvious. Now you have 5 roommates and 2 consoles. What's the probability of accessing any one of them. –  inf.ig.sh Oct 29 '10 at 18:55
add comment

1 Answer

up vote 2 down vote accepted

The resource will be free with probability $(1-p)^n$.

When there are $2n$ agents and $2$ resources, one of the resources will be free with probability $(1-p)^{2n} + 2np(1-p)^{2n-1}$.

In general, you want to calculate the tail of a Binomial distribution. When $p = c/n$ and $n$ is large, the distribution is approximately Poisson.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.