Density function of the mixture of two NHPP

#1
I'd like to know how can I calculate the density function of the mixture of two non-homogeneous Poisson process? I should mention that I have the kernel densities of those NHPP s. I can also describe my question as follows: I have two layers that are connected to each other. the number of defects in each of them has NHPP and I'd like to calculate the overall failure intensity of the system (The system will fail when the density of defects in both layers reach a certain limit).
 

BGM

TS Contributor
#2
So you have two non-homogeneous Poisson Process \( N_1(t) \) and \( N_2(t) \) with rate function \( \lambda_1(t) \) and \( \lambda_2(t) \) respectively.

I'd like to calculate the overall failure intensity of the system (The system will fail when the density of defects in both layers reach a certain limit).
Now I guess we need more elaboration on the term "overall failure intensity" and "density of defects in both layers".
 
#3
Thank you for your reply. I have a chemical composite that consists of two layers. The probability of defect generation in each of those layers follows NHPP which depends on the previous history (conditional NHPP). I calculated the kernel density function for each of the layers. Now I am trying to calculate the overall density function for the whole system. Actually, I am trying to calculate the intensity function for the mixture of them. The whole material will fail when both of the layers fail and each of them will fail when the density of defects reach a certain limit).
 

BGM

TS Contributor
#4
follows NHPP which depends on the previous history (conditional NHPP)
It seems that you mean it is not the ordinary NHPP. Is it a Markov chain/process?

Given the individual failure limit

\( \tau_1 = \min\{t: N_1(t) \geq n_1\} \)

\( \tau_2 = \min\{t: N_2(t) \geq n_2\} \)

And you want to find the distribution the overall system failure time, which is given by

\( \tau = \max\{\tau_1, \tau_2\} \)

Is it what you want?
 
#5
Somehow. suppose that I know the intensity function for each step of two different Markov chains. Therefore how can I calculate the intensity function for each of the whole markov chains and also what is the intensity function for both Markov chains to happen together? (the first Markov chain start from 1 to n1 and the second from 1 to n2) (I am not sure how much it is clear!!)
 

BGM

TS Contributor
#6
From the last post, it seems that you are interested in the sum \( N_1(t) + N_2(t) \). Suppose each of them follows a NHPP as stated before and they are independent, then the sum of them will be also a NHPP with the intensity function as the sum \( \lambda_1(t) + \lambda_2(t) \)
 

BGM

TS Contributor
#8
Whenever you assume they are dependent, you actually also need to state the dependency structure. Different model will have different results, and in general the answer should be no except some specific bivariate Poisson Process model.