Let's say we have a stochastic sequence of events modelled as a Poisson process with a rate of $\lambda$ events per unit interval. A agent tries to take a step of length $\Delta$ units. If an event is encountered, the agent stops in place.
In that case, the probability that the agent takes a full step (no events encountered) or that it stops before completion (i.e. it encounters at least 1 event) along the way are the following: $$ P_{\text{no event}} = e^{-\lambda\Delta}$$ $$ P_{\text{stop}} = P_{\text{1 or more events}} = 1-P_{\text{no event}} = 1-e^{-\lambda\Delta}$$
Now, let's assume that the Poisson process is in fact the combination of $N$ independent processes, each with their respective event rate $\lambda_1$, $\lambda_2$, ...$\lambda_N$, where $\lambda=\sum_{i=1}^N \lambda_i$. Once again, the agent takes a step of length $\Delta$ units, but will stop in place as soon as 1 event is encountered. There are $1+N$ possible outcomes:
- No event is encountered;
- The agent first encounters an event from process 1 and stops;
- The agent first encounters an event from process 2 and stops;
- ...
- The agent first encounters an event from process $N$ and stops;
How can I calculate the probability of each outcome? Intuitively, I would weigh the above probability calculations by the fault rates:
$$ P_{\text{no event}} = \prod_{i=1}^N P_{\text{no event process i}} = e^{-(\sum_{i=1}^N \lambda_i)\Delta} = e^{-\lambda\Delta}$$ $$ P_{\text{stop due to event from process }n} = \frac{\lambda_n}{\sum_{i=1}^N \lambda_i} \left(1-P_{\text{no event}}\right) = \frac{\lambda_n}{\sum_{i=1}^N \lambda_i} \left(1-e^{-(\sum_{i=1}^N \lambda_i)\Delta}\right)$$
Is this correct? Also, is there resource material where I could read further about merged Poisson Processes?