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Blue boats are passing on a river and their arrival times are modeled by a renewal point process. Place red boats on the river according to a point process $P$ such that the arrival times of all of the boats (the superposition process) becomes another renewal process. Find $P$.

In other words, find a point process whose superposition with a given renewal process is another renewal process. Note the causality constraint: when you place red boats, you don't have any information about the arrival times of the blue boats in the future.

What I found was not enough:

  1. A solution for a similar question here, without causality
  2. Poisson and alternating renewal processes with superposition a renewal process
  3. Renewal Processes Decomposable into i.i.d. Components
  4. Superposition and Decomposition of Stationary Point Processes
  5. Almost Sure Comparisons of Renewal Processes and Poisson Processes
  6. Superposition of Renewal Processes
  7. On the Superposition of Point Processes
  8. Pairs of renewal processes whose superposition is a renewal process
  9. ON QUEUEING SYSTEMS VITH RENEWAL DEPARTURE PROCESSES
Susan_Math123
  • 2,850
  • To me your casuality constraint is not very clear. We know that blue boats arrive according to a renewal process. Usually, by observing blue boats for some time, we can learn the distribution of inter-arrival times. Once we know the distribution we can calculate the distributuion of the residual time (also called forward recurrence time) until the next arrival of a blue boat. So the future becomes known (probabilistically). – rrv Nov 18 '18 at 11:03

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