Assume a clustered process with the typical user is located on the location of the parent (origin) of the representative cluster (which is not an element of the process) and the daughter points are normally/uniformly distributed around the origin of each cluster.
1- What is the distribution of the nearest neighbor (daughter point) distance to the typical user?
2- What is the Laplace transform of the interference on the typical user?
Hi Ali, You can find generating functional of Thomas process in the following reference, you can use that to compute the contact distribution and Laplace transform of interference etc. as in reference:
http://www3.nd.edu/~mhaenggi/pubs/tit09.pdf
I have seen this reference before. But the contribution in this paper as shown is studying the case where the transmitter is located at the origin and considered to be a part of the cluster and the intended receiver is not of the cluster process.
- In my problem formulation the typical receiver is located at the origin of the cluster and hence we seek to make benefit of the clustering gain to leverage the performance.
- Do you think that the generating functional of the reduced palm distribution can model the interference at random point in the space, where the point inside the cluster with deterministic standard deviation around the parent point of the cluster can see different values of the interference and received power!?
The same in this paper, they studied the distribution of the nesrest neighbor in two cases where the reference point is one of the offspring points of the cluster. Also, the simulation of my geometry matchs their cases if the same association is applied. But in my case the association is not the same. Note, both of the aforementioned references in the answers studying for adhoc networks where the links are established between the same process points, not between the orgion and any arbitrary nearest neighbor of the cluster points.
Hi, Ali. Looks like some clarification is in order about your previous comment on https://arxiv.org/abs/1609.08137. This paper does study contact distance distribution (when the reference point is not a part of the point process). Please check Theorem 1. For completeness, note that this paper derives the distribution of both the contact and nearest-neighbor distances. As you stated in your comment, the two cases differ by the way "reference point" is chosen.
Note that Theorem 1 of the above paper is valid whenever the reference point is sampled from a stationary point process independent of the original Poisson cluster process. However, we need slightly more work in order to relax this independence assumption. One simple situation would be when we consider two cluster processes that share the same parent point process (say one is modeling users and the other is modeling BSs). One can sample the reference point from one Poisson cluster process and the "nearest point" from the other PCP. You can find the nearest neighbor distance distribution for such cases in our recent work: https://arxiv.org/abs/1612.07285.
Finally, note that our group at Virginia Tech is actively working on the analysis of wireless networks using Poisson cluster processes. You can find distance distributions for different transmitter and receiver selection policies involving Poisson cluster processes in the following recent publications.
1. https://arxiv.org/abs/1508.02668
2. https://arxiv.org/abs/1509.04747
3. https://arxiv.org/abs/1606.04405
4. https://arxiv.org/abs/1606.06223
More details on our recent work on the analysis of cluster point processes are available on Prof. Harpreet Dhillon's webpage: http://www.dhillon.ece.vt.edu/index.html.
Thanks Mehrnaz for your answer and the comprehensive illustration. Actually the second reference of you above is the most useful one for my case with slight modification on the system model and BS association due to different applications applied. Really thank you, you saved my time by your recent work :).