We have shown that if two independent Poisson streams are merged, we still get a Poisson stream. Now

We have shown that if two independent Poisson streams are
merged, we still get a Poisson stream. Now consider two independent renewal
counting processes N1(t) and N2(t) being merged into the process N(t) [SEVC
1977]. Let the underlying distribution functions be FX1 , FX2 , and FX. First
show that, conditioned on the fact that the last event in the output stream was
contributed by the first stream, the time to next event in the output stream is
the minimum of X1 and Y2 = Y (X2), where Y (X2) is the residual time to next
event in input stream 2. Next show that x

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