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Distributed congestion control method for sending safety messages to vehicles at a set target distance

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Abstract

In the present paper, we propose a method for controlling the interval of safety message transmissions in a fully distributed manner that maximizes the number of successful transmissions to vehicles located a set target distance away. In the proposed method, each vehicle estimates the density of vehicles in its vicinity, and, based on the estimated vehicle density, each vehicle calculates an optimal message transmission interval in order to maximize the number of successful message transmissions to vehicles located a set target distance away. The optimal message transmission interval can be analytically obtained as a simple expression when it is assumed that the vehicles are positioned according to a two-dimensional Poisson point process, which is appropriate for downtown scenarios. In addition, we propose two different methods for a vehicle by which to estimate the density of other vehicles in its vicinity. The first method is based on the measured channel busy ratio, and the second method relies on counting the number of distinct IDs of vehicles in the vicinity. We validate the effectiveness of the proposed methods using several simulations.

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Notes

  1. Here, we use slightly different notations in Definition 1 and Theorem from those in Definition 2.1.1 and Corollary 2.1.2 in [4].

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Funding

This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant numbers JP19H04093 and 22H01480.

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Authors and Affiliations

  1. Graduate School of Engineering, Chiba University, 1-33 Yayoi Inage, Chiba-Shi, 263-8522, Chiba, Japan

    Kai Takahashi & Shigeo Shioda

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  1. Kai Takahashi
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  2. Shigeo Shioda
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Correspondence to Shigeo Shioda.

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Kai Takahashi and Shigeo Shioda contributed equally to this work.

Appendix: Derivation of \(\mathcal {L}_{I_S}(s)\)

Appendix: Derivation of \(\mathcal {L}_{I_S}(s)\)

The Laplace transform of \(I_S\) is given as

$$\mathcal {L}_{I_S}(s)=E\left[ e^{-s \sum _i U_i S_i |X_i|^{-\alpha }}\right] .$$

Observe that a set of nodes that transmit at the 0th transmission period \(\{X_i: U_i=1\}\) is a homogeneous Poisson point process with intensity \(E[U_i]\lambda =\rho \lambda \), where \(\lambda \) is the intensity of the original homogeneous Poisson point process \(\{X_i\}\). This observation yields the following expression of \(\mathcal {L}_{I_S}(s)\)

$$\mathcal {L}_{I_S}(s)=E\left[ e^{-s \sum _i S_i |\tilde{X}_i|^{-\alpha }}\right] ,$$

where \(\Psi {\mathop {=}\limits ^\textrm{def}}\{\tilde{X}_i\}\) is a stationary Poisson point process with intensity \(\rho \lambda \). To further calculate \(\mathcal {L}_{I_S}(s)\), we introduce the following definition and the theoremFootnote 1:

Definition 1

(Definition 2.1.1 in Baccelli and Blaszczyszyn [4]) A marked point process is said to be independently marked if, given the location of the points \(\Psi {\mathop {=}\limits ^\textrm{def}}\{X_i\}\), the marks are mutually random vectors and if the conditional distribution of the mark \(M_i\) of the ith point depends only on its location \(X_i\), i.e., \(P(M_i\le m| \Psi )= P(M_i\le m| X_i)\).

Theorem 1

(Corollary 2.1.2 in Baccelli and Blaszczyszyn [4]) For an independently marked Poisson point process \(\{(X_i, M_i)\}\) with intensity measure \(\Lambda \) and marks with distribution \(F_{M,x}(m)=P(M_i\le m|X_i=x)\), we have

\(\{(\tilde{X}_i, S_i)\}\) is a stationary marked Poisson point process where the distribution of \(S_i\) does not depend on the location of the ith point. Denoting the distribution of \(S_i\) by \(F_S\), from Corollary 2.1.2 in Baccelli and Blaszczyszyn [4], we have the following:

$$\begin{aligned} \mathcal {L}_{I_S}(s)= & {} \exp \left\{ -2\pi \rho \lambda \int _0^\infty r \left( 1-\!\int _0^\infty e^{-s yr^{-\alpha }}F_s(dy)\right) dr\right\} \\= & {} \exp \left\{ -2\pi \rho \lambda \int _0^\infty r \left( 1-\mathcal {L}_S(sr^{-\alpha })\right) dr\right\} , \end{aligned}$$
(A1)

where \(\mathcal {L}_{S}(s)\) denotes the Laplace transform of \(F_S(x)\). In the case of Rayleigh fading, \(F_S(x)\) is the exponential distribution with average S, and thus, we have

$$\begin{aligned} \mathcal {L}_{S}\left( sr^{-\alpha }\right) =\frac{r^{\alpha }}{r^{\alpha }+sS}. \end{aligned}$$
(A2)

Substituting (A2) into Eq. A1 and using the following result:

$$\begin{aligned} \int _0^\infty \frac{rsS}{r^{\alpha }+sS}dr&=\frac{(sS)^\delta }{2}\frac{\pi \delta }{\sin \pi \delta },\quad \delta =\frac{2}{\alpha }, \end{aligned}$$

yields

$$\mathcal {L}_{I_S}(s)=\exp \left\{ -\pi \lambda \rho (sS)^\delta \frac{\pi \delta }{\sin \pi \delta }\right\} .$$

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Takahashi, K., Shioda, S. Distributed congestion control method for sending safety messages to vehicles at a set target distance. Ann. Telecommun. 79, 211–225 (2024). https://doi.org/10.1007/s12243-023-00986-3

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