Poisson process with different arrival rates

In summary, the conversation is about a problem involving a Poisson process and the probability of having a certain number of individuals in a room at a given time. The problem involves multiple doors and the rate at which individuals enter through each door changes. The person is asking for help with finding the probability of having exactly n individuals at time t.
  • #1
tamino
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Homework Statement



I cannot figure out this example:

suppose that initially individuals enter a room from one door according to a Poisson process with arrival rate lambda1. Suppose that as soon as one inidividual enters, this door is shut down and a second door is open. The numer of individuals that enter the room from the second door follows a Poisson process with arrival rate lambda2. As soon as a second inidividual enters the room, the second door is shut down and a third door is opened; from that one the number of individuals entering from that door follows a posson with rate lambda3. and so on...

what is probability that at time t I have exactly n individuals? THANKS!

Homework Equations





The Attempt at a Solution

 
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  • #2
I'm not sure how to start this problem. I understand what is being asked, but I'm not sure where to start.
 

FAQ: Poisson process with different arrival rates

What is a Poisson process with different arrival rates?

A Poisson process with different arrival rates is a stochastic process where events occur randomly and independently over time, with the rate of occurrence varying at different points in time. It is named after French mathematician Siméon Denis Poisson, who first described it in the early 19th century.

What are the main characteristics of a Poisson process with different arrival rates?

The main characteristics of a Poisson process with different arrival rates are that events occur randomly and independently, the rate of occurrence can vary over time, and the probability of an event occurring in a given time interval is proportional to the length of the interval.

What are some real-world applications of a Poisson process with different arrival rates?

A Poisson process with different arrival rates can be used to model a variety of real-world phenomena, such as the arrival of customers at a store, the number of phone calls received by a call center, or the occurrence of earthquakes in a given region. It is also commonly used in queuing theory and reliability analysis.

How is the arrival rate determined in a Poisson process with different arrival rates?

The arrival rate, also known as the rate parameter, is determined by the average number of events that occur in a given time interval. It can also be calculated by dividing the total number of events by the total time period. In a Poisson process with different arrival rates, the arrival rate can vary over time, which allows for a more realistic modeling of real-world phenomena.

What is the difference between a Poisson process with constant arrival rate and one with different arrival rates?

The main difference between a Poisson process with constant arrival rate and one with different arrival rates is that in the latter, the arrival rate can vary over time. In a Poisson process with constant arrival rate, the rate parameter remains the same throughout the entire time period. This makes a Poisson process with different arrival rates a more flexible and accurate model for certain real-world applications.

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