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Coverage Analysis of a Vehicular Network Modeled as Cox Process Driven by Poisson Line Process

Vishnu Vardhan Chetlur, Harpreet S. Dhillon

arXiv:1709.08577v1cs.ITcs.NImath.IT

TL;DR

Roads are predominantly linear and randomly oriented. The analysis handles serving-node cases separately and uses the PLP definition’s one-to-one correspondence. Line density and node density have conflicting effects on coverage probability, and the analysis compares results with a homogeneous 2D PPP at the same node density.

  • Problem

    Roads are predominantly linear and randomly oriented.

    Source: §I · p. 2

  • Method

    The analysis handles serving-node cases separately and uses the PLP definition’s one-to-one correspondence.

    Source: B. Contributions · p. 4, §IV · p. 9, A. Preliminary Results · p. 10

  • Results

    Line density and node density have conflicting effects on coverage probability, and the analysis compares results with a homogeneous 2D PPP at the same node density.

    Source: §VI · p. 23

  • Takeaways & Limitations

    The reported opposing density effects offer insights for RSU design and deployment in the modeled network.

    Source: §VI · p. 23

    The authors note that models can be made more accurate, often at the cost of reduced tractability.

    Source: §VI · p. 24

Abstract

from arXiv · show

In this paper, we consider a vehicular network in which the wireless nodes are located on a system of roads. We model the roadways, which are predominantly straight and randomly oriented, by a Poisson line process (PLP) and the locations of nodes on each road as a homogeneous 1D Poisson point process (PPP). Assuming that each node transmits independently, the locations of transmitting and receiving nodes are given by two Cox processes driven by the same PLP. For this setup, we derive the coverage probability of a typical receiver, which is an arbitrarily chosen receiving node, assuming independent Nakagami-$m$ fading over all wireless channels. Assuming that the typical receiver connects to its closest transmitting node in the network, we first derive the distribution of the distance between the typical receiver and the serving node to characterize the desired signal power. We then characterize coverage probability for this setup, which involves two key technical challenges. First, we need to handle several cases as the serving node can possibly be located on any line in the network and the corresponding interference experienced at the typical receiver is different in each case. Second, conditioning on the serving node imposes constraints on the spatial configuration of lines, which require careful analysis of the conditional distribution of the lines. We address these challenges in order to accurately characterize the interference experienced at the typical receiver. We then derive an exact expression for coverage probability in terms of the derivative of Laplace transform of interference power distribution. We analyze the trends in coverage probability as a function of the network parameters: line density and node density. We also study the asymptotic behavior of this model and compare the coverage performance with that of a homogeneous 2D PPP model with the same node density.

I. INTRODUCTION

A homogeneous 2D PPP does not capture the coupling between vehicular nodes and their predominantly linear, randomly oriented roads. This paper addresses the open problem of coverage analysis for a road-based Cox-process model, where roads follow a Poisson line process and nodes on roads follow a 1D PPP.

I. INTRODUCTION

  • The 2D PPP model does not capture the coupling between vehicular nodes and the predominantly linear, randomly oriented road infrastructure.Vehicular nodes and RSUs are restricted to roadways, motivating models that represent randomness in both roads and node locations.Source: §I · p. 2
  • Coverage analysis for receivers and transmitters modeled as Cox processes driven by the same PLP is identified as an open problem and the paper’s focus.The analysis characterizes serving distance and conditional interference power distributions to obtain coverage probability.Source: §I · pp. 2–3

A. Related Work

  • Earlier coverage studies considered simpler settings such as a single road or two-road intersection, while the cited PLP-driven uplink study used a PPP receiver.The paper identifies its setting as coverage for both receiver and transmitter Cox processes driven by the same PLP.Source: A. Related Work · pp. 3–4
  • The paper presents its analysis as the first derivation of coverage probability for a vehicular node on a PLP connecting to another vehicular node on that PLP.The technical challenge arises from spatial coupling induced by the underlying PLP.Source: A. Related Work · p. 4

B. Contributions

The paper develops a coverage-analysis procedure for a vehicular network with roads modeled by a PLP and nodes by 1D PPPs. It derives coverage while accounting for serving-node cases and the conditional line distribution, and reports contrasting effects of node and line density.

B. Contributions

  • The setup models roadways as a PLP and node locations on each road as a 1D PPP, with Nakagami-m fading and connection to the closest transmitter.Coverage is for an arbitrarily chosen receiving node.Source: B. Contributions · p. 4
  • The analysis derives an exact coverage expression using the derivative of the interference-power Laplace transform.It first characterizes distance distributions for the desired signal and interference at the typical receiver.Source: B. Contributions · pp. 4–5
  • Because the closest transmitting node may lie on different lines, the analysis treats distinct serving-node cases with their differing interference.Conditioning on a serving node also constrains the line configuration, so the conditional line distribution is determined to characterize interference.Source: B. Contributions · pp. 4–5
  • Coverage probability increases with node density on lines but decreases as line density increases.The authors report that this contrast offers insights for RSU design and deployment.Source: B. Contributions · p. 5
  • The comparison with a homogeneous 2D PPP at the same node density indicates that the 2D PPP may not be a good approximation for this Cox-process model.The comparison is presented as evidence of the value of the paper’s coverage analysis.Source: B. Contributions · p. 5

II. MATHEMATICAL PRELIMINARY: POISSON LINE PROCESS

  • The preliminary PLP model assumes motion invariance and represents each line by its perpendicular distance and angle relative to the origin.Line density is defined as mean line length per unit area.Source: §II · pp. 5–6

III. SYSTEM MODEL

The model places wireless nodes on randomly oriented roads represented by a Poisson line process, with transmitting and receiving nodes forming Cox processes on the same roads. The typical receiver associates with its nearest transmitter, which may lie on its own or another road.

A. Spatial Modeling of Wireless Nodes

  • Roads are modeled by a Poisson line process, with node locations on each road modeled as a homogeneous 1D PPP and independent transmission thinning producing transmitting and receiving Cox processes driven by the same line process.The transmitter and receiver densities are λv = pλn and λr = (1 − p)λn, respectively.Source: A. Spatial Modeling of Wireless Nodes · p. 7

B. Transmitter Association Scheme and Propagation Model

  • The analysis assumes equal transmit powers, isotropic antennas, interference-limited operation, and independent Nakagami-m fading with integer m and path-loss exponent α > 2.Thermal noise is neglected.Source: B. Transmitter Association Scheme and… · p. 8
  • The typical receiver connects to its closest transmitting node, which can lie on any line, including the typical line through the receiver.The model distinguishes event E0, when the serving node lies on the typical line, from Ei for the ith closest other line.Source: B. Transmitter Association Scheme and… · p. 8

IV. COVERAGE PROBABILITY

The coverage analysis treats serving nodes on the typical line separately from those on other lines because the interference differs by serving-line case. It develops preliminary distance distributions used in the coverage derivation.

IV. COVERAGE PROBABILITY

  • Coverage probability is derived separately for a serving node on the typical line and on another line, then combined using the law of total probability.For a serving node on another line, the analysis conditions on that line’s distance Yn and averages over Yn in the final step.Source: §IV · p. 9

A. Preliminary Results

  • The distance of the nth closest line from the origin has the distribution of the nth closest point in a 1D PPP with density 2πλl.This follows by projecting the PLP representation points onto the vertical axis.Source: A. Preliminary Results · p. 10
  • The preliminary results derive conditional distributions for the closest-node distances on the line of interest and among lines nearer to or farther from the origin.These distances provide intermediate quantities for the probabilities of serving-line events E0 and En.Source: A. Preliminary Results · pp. 10–11

B. Probabilities of Events En and E0

The serving-line event probabilities compare the closest-node distance on a candidate line with competing distances from other lines. For the typical line, the event probability is obtained by integrating its closest-node distribution against the nearest-node distribution on farther lines.

  • Conditioned on Yn, the probability of En is computed from the conditional distributions of the competing distances and the closest-node distance on line Ln.The calculation uses conditional independence of S0, Un, and Vn.Source: B. Probabilities of Events En and E0 · pp. 12–13
  • For event En, the node on the nth closest line serves only if its distance Sn is less than the competing distances S0, Un, and Vn.The competing nodes are on the typical line, nearer lines, and farther lines, respectively.Source: B. Probabilities of Events En and E0 · p. 13
  • The probability of E0 is the integral of FS0(v0) against fV0(v0), where these functions describe the typical-line and farther-line closest-node distances.The typical-line node serves when its distance is less than the closest-node distance on any other line.Source: B. Probabilities of Events En and E0 · p. 13

C. Serving Distance Distribution

The serving-distance distribution is conditioned on which line contains the serving node. The conditional line-count analysis accounts for the no-node constraint induced by conditioning on the serving node.

C. Serving Distance Distribution

  • Conditioned on En and Yn, the serving distance distribution uses the closest-node distance Sn together with the minimum competing distance Wn = min{S0, Un, Vn}.The CDF and PDF are obtained from the conditional distance distributions and the distribution of Wn.Source: C. Serving Distance Distribution · pp. 13–14
  • Conditioned on E0, the serving-distance CDF and PDF are derived using the typical-line closest-node distance S0 and the closest-node distance V0 on other lines.The typical-line event probability is part of the conditional distribution expressions.Source: C. Serving Distance Distribution · pp. 13–15

D. Conditional Probability Mass Function of Number of Lines

  • Conditioning on a serving node at distance R changes the line count in the disc b(o, R), because intersecting lines must have no nodes on their chord segments inside the disc.The typical line is excluded from the count; for En, at least n lines intersect the disc.Source: D. Conditional Probability Mass… · p. 15
  • The conditional PMF counts lines farther than Yn but closer than R, and the resulting conditional line distribution remains Poisson.These lines intersect b(o, R) but not b(o, Yn).Source: D. Conditional Probability Mass… · pp. 15–16

E. Laplace Transform of Interference Distribution

The analysis conditions on the serving event, serving distance, and relevant line distance, then derives Laplace transforms for interference from distinct spatial components. Conditional independence lets the component transforms combine into the total interference transform.

E. Laplace Transform of Interference Distribution

  • For event E_n, interference is partitioned among five node sets: the typical line, serving line, lines closer than Y_n, lines between Y_n and R, and lines farther than R.The corresponding interference components are I_0, I_n, I_in, I_ann, and I_out.Source: E. Laplace Transform of Interference… · p. 17
  • The typical-line interference transform uses the 1D PPP result, with channel fading handled through the Gamma distribution and the PPP probability generating functional.The node process on the typical line is independent of the distance Y_n to the nth closest line.Source: E. Laplace Transform of Interference… · p. 17
  • Conditioned on E_n, R, and Y_n, the five interference components are mutually independent, so their Laplace transforms multiply to give the total conditional transform.The component transforms are derived separately for the typical line and line groups defined by their distances from the receiver.Source: E. Laplace Transform of Interference… · pp. 17–19
  • For event E_0, the typical line and serving line coincide, reducing the interference partition to typical-line nodes, nodes on lines within R, and nodes on lines beyond R.The E_0 conditional transform follows from the component results for the E_n case with Y_0 = 0.Source: E. Laplace Transform of Interference… · p. 19

F. Coverage Probability

Coverage is the probability that the receiver's SIR exceeds the required threshold. The paper expresses coverage using derivatives of the conditional interference Laplace transform, averaged over the serving-distance and serving-line cases.

F. Coverage Probability

  • Coverage probability is the probability that the SIR measured at the receiver exceeds the threshold β required for successful communication.Source: F. Coverage Probability · p. 20
  • Theorem 1 expresses coverage probability P_c through derivatives of the conditional interference Laplace transform, averaged over serving distance and the serving-line cases.The expression includes the E_0 case and a sum over E_n, with the corresponding serving-distance and line-distance distributions.Source: F. Coverage Probability · p. 20

V. NUMERICAL RESULTS AND DISCUSSION

The numerical section checks the analytical coverage expressions against simulation and examines line- and node-density trends. It also compares coverage with a homogeneous 2D PPP model at the same node density.

V. NUMERICAL RESULTS AND DISCUSSION

  • The analysis compares coverage with a homogeneous 2D PPP model having the same node density to examine the significance of the spatial model.Source: §V · p. 20

A. Numerical Results

  • The analytical coverage results match the Monte Carlo simulations exactly for the evaluated setup.The section uses simulations to verify the theoretical expressions before examining network-parameter trends.Source: A. Numerical Results · p. 21
  • Coverage probability decreases as line density increases, because more closely spaced other lines raise interference while the typical-line serving distance is unchanged.The stated mechanism considers the case where the receiver connects to a node on its typical line.Source: A. Numerical Results · p. 21
  • Coverage probability increases as node density on the lines increases: nodes move closer along the line, and desired signal power increases faster than interference power for nodes on lines closer to the origin.Source: A. Numerical Results · pp. 21–22

B. Comparison with a Homogeneous 2D PPP Model

The comparison shows that a homogeneous 2D PPP is not a reliable approximation across all road and node density regimes, though it is close when line density is high and node density is low. The conclusion summarizes the model’s exact coverage analysis and reports conflicting effects of line and node density.

B. Comparison with a Homogeneous 2D PPP Model

VI. CONCLUSION

  • The analysis models roads by a PLP and nodes on each road by a homogeneous 1D PPP, then derives distance distributions and conditional line distributions to characterize signal and interference.Source: §VI · p. 23
  • The paper derives an exact SIR-based coverage expression in terms of derivatives of the interference-power Laplace transform.Source: §VI · p. 23
  • Coverage increases with node density on lines but degrades as line density increases.Source: §VI · p. 23
  • Throughput and area spectral efficiency were not analyzed, and obtaining parameters from actual data may improve model accuracy at the cost of tractability.Source: §VI · p. 24

APPENDIX

The appendix proves conditional distributions and line-count results needed for the coverage analysis. The proofs use node-void probabilities and the conditional spatial distribution of lines.

A. Proof of Lemma 4

  • The CDF of U_n conditioned on Y_n is derived by treating two ranges separately: 0 ≤ u_n < y_n and y_n ≤ u_n < ∞; its PDF follows by differentiation.Source: A. Proof of Lemma 4 · pp. 24–25

B. Proof of Lemma 5

  • Lemma 5 derives the conditional CDF of V_n using the Poisson count of lines between y_n and v_n and node-void probabilities on intersecting line segments.Source: B. Proof of Lemma 5 · pp. 25–26

C. Proof of Lemma 10

D. Proof of Lemma 13

These proofs derive conditional Laplace transforms of interference for distinct groups of lines around the receiver. They combine line-distance distributions with independence of node locations and the PLP’s Poisson structure.

D. Proof of Lemma 13

  • The n−1 lines closer than Y_n have distances uniformly distributed on (0, y_n); averaging a line’s distance over this range, subject to no nodes closer than r, yields its conditional interference Laplace transform.Source: D. Proof of Lemma 13 · pp. 27–28

E. Proof of Lemma 14

  • Lemma 14 derives the transform for lines farther than y_n but closer than R, using their uniform distance distribution and independent node locations.Source: E. Proof of Lemma 14 · p. 28

F. Proof of Lemma 15

  • Lemma 15 forms the interference transform across the line process from per-line transforms, then averages using the PGFL of the line-distance PPP with density 2πλ_l.Source: F. Proof of Lemma 15 · p. 29
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