I'm trying to understand how to model certain problems as a Poisson problem. I'm unable to get the right thinking in place to understand such problems.
For example, something like this:
Arrivals at a telephone booth are considered to follow Poisson distribution with an average of 10 minutes between successive arrivals. The length of a phone call is distributed exponentially with mean 3 minutes. What is the probability that an arrival does not have to wait before service?
How exactly am I supposed to use the above facts to arrive at the answer? Please explain the concept.
Best Answer
Let $p_i$ be probability that $i$ people are in the queue including the one on the phone. Then in equilibrium $p_i/10=p_{i+1}/3$. Can solve this to give $p_i=(3/10)^i p_0$. Knowing $\sum p_i=1$ gives $p_0 = 1-3/10=7/10$. But $p_0$ is the probability that there is no one in the queue or using the phone, so now you have the answer.