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

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