Nonhomogeneous Poisson Processes

Non-homogeneous Poisson process model (NHPP) represents the number of failures experienced up to time t is a non-homogeneous Poisson process {N(t), t ≥ 0}.

The main issue in the NHPP model is to determine an appropriate mean value function to denote the expected number of failures experienced up to a certain time.
With different assumptions, the model will end up with different functional forms of the mean value function. Note that in a renewal process, the exponential assumption for the inter-arrival time between failures is relaxed, and in the NHPP, the stationary assumption is relaxed.

Non-homogeneous Poisson process model is based on the following assumptions:

–>The failure process has an independent increment, i.e. the number of failures during the time interval (t, t + s) depends on the current time t and the length of time interval s, and does not depend on the past history of the process.

–> The failure rate of the process is given by P{exactly one failure in (t, t + ∆t)} = P{N(t, t + ∆t) – N(t)=1} = \lambda(t)∆t + o(∆t) where \lambda(t) is the intensity function.

–> During a small interval ∆t, the probability of more than one failure is negligible, that is, P{two or more failure in (t, t+∆t)} = o(∆t)

–> The initial condition is N(0) = 0.

On the basis of these assumptions, the probability of exactly n failures occurring during the time interval (0, t) for the NHPP is given by
              Pr\begin{Bmatrix}N(t)=n\end{Bmatrix} = \frac{[m(t)]^{n}}{n!}e^{-m(t)}

where  m(t)=E[N(t)]=\int_{0}^{t}\lambda \left ( s\right )ds  and  \lambda (t) is the intensity function. It can be easily shown that the mean value function m(t) is non-decreasing.

Reliability Function:
The reliability R(t), defined as the probability that there are no failures in the time interval (0, t), is given by
              R(t) = P\left \{ N(t)=0 \right \} = e^{-m(t)}

In general, the reliability R(x|t), the probability that there are no failures in the interval (t, t + x), is given by

              R(x|t)=P\left \{ N(t+x)-N(t)=0 \right \} = e^{-[m(t+x)-m(t)]}

and its density is given by
              f(x)=\lambda (t+x)e^{-[m(t+x)-m(t)]}
where \lambda (x)=\frac{\partial [m(x)]}{\partial x}

The variance of the NHPP can be obtained as follows:

              Var[N(t)]=\int_{0}^{t}\lambda (s)ds

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.

My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using or mail your article to See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.