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Let $S_{1k}$ and $S_{2j}$ denote the epoch of the $k^{th}$ and $j^{th}$ arrival in the first and second poisson counting process, respectively. The first and second poisson counting process is the consequence of subdividing of an original poisson counting process.

To determine the probability that $S_{1k} < S_{2j}$, the physical idea is that if k arrival of the first $k+j-1$ goes to the first process then exactly $j-1$ goes to the second process. If, instead, k or more arrival goes to the first process, then at most $j-1$ or fewer than $j-1$ arrival goes to the second process.

At this point before I move forward, I have a question: why is the subset of arrival $k+j-1$ meaningful? Why not the subset $k+j$?

A corollary to "If, instead, k or more arrival goes to the first process, then at most j-1 or fewer than $j-1$ arrival goes to the second process" is that the $k^{th}$ arrival to the first poisson process precedes the second poisson counting process.

Why is this true (why must the $k$ arrival switch to the first poisson process first?) and how it ties with $S_{1k} < S_{2j}$? I am unable to comprehend this.

Context:

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Any good physical reasoning is appreciated

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  • $\begingroup$ What is exactly meant by "the epoch of the $j^{\mathsf{th}}$ arrival to the process"? $\endgroup$ – Math1000 May 10 '18 at 12:30

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