Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks
We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sen...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-12-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/25/12/1619 |
| _version_ | 1797381205549645824 |
|---|---|
| author | Kaushlendra Pandey Abhishek K. Gupta Harpreet S. Dhillon Kanaka Raju Perumalla |
| author_facet | Kaushlendra Pandey Abhishek K. Gupta Harpreet S. Dhillon Kanaka Raju Perumalla |
| author_sort | Kaushlendra Pandey |
| collection | DOAJ |
| description | We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a <i>random bipartite associator graph</i> in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an <i>independent</i> PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></semantics></math></inline-formula> as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of <i>k</i>th contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its <i>load</i>. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed. |
| first_indexed | 2024-03-08T20:48:01Z |
| format | Article |
| id | doaj.art-affd1e863f5848e998b695816ef026d6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-08T20:48:01Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-affd1e863f5848e998b695816ef026d62023-12-22T14:07:23ZengMDPI AGEntropy1099-43002023-12-012512161910.3390/e25121619Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular NetworksKaushlendra Pandey0Abhishek K. Gupta1Harpreet S. Dhillon2Kanaka Raju Perumalla3Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, IndiaDepartment of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, IndiaWireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, USADepartment of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, IndiaWe consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a <i>random bipartite associator graph</i> in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an <i>independent</i> PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></semantics></math></inline-formula> as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of <i>k</i>th contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its <i>load</i>. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed.https://www.mdpi.com/1099-4300/25/12/1619Poisson line processPoisson point processCox processload distribution in vehicular communicationvehicular network |
| spellingShingle | Kaushlendra Pandey Abhishek K. Gupta Harpreet S. Dhillon Kanaka Raju Perumalla Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks Entropy Poisson line process Poisson point process Cox process load distribution in vehicular communication vehicular network |
| title | Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks |
| title_full | Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks |
| title_fullStr | Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks |
| title_full_unstemmed | Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks |
| title_short | Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks |
| title_sort | properties of a random bipartite geometric associator graph inspired by vehicular networks |
| topic | Poisson line process Poisson point process Cox process load distribution in vehicular communication vehicular network |
| url | https://www.mdpi.com/1099-4300/25/12/1619 |
| work_keys_str_mv | AT kaushlendrapandey propertiesofarandombipartitegeometricassociatorgraphinspiredbyvehicularnetworks AT abhishekkgupta propertiesofarandombipartitegeometricassociatorgraphinspiredbyvehicularnetworks AT harpreetsdhillon propertiesofarandombipartitegeometricassociatorgraphinspiredbyvehicularnetworks AT kanakarajuperumalla propertiesofarandombipartitegeometricassociatorgraphinspiredbyvehicularnetworks |