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Outage Probability Analysis in Dual-Hop Vehicular Networks with the Assistance of Multiple Access Points and Vehicle Nodes

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Abstract

Vehicular networks represent an interesting research field that encompasses connections between vehicles or connections between a vehicle and an infrastructure. In this paper, we consider dual-hop transmission using cooperative communication in vehicular networks. In our system model, multiple intermediate access points and vehicle nodes are willing to help the source vehicle forward data to the destination vehicle. From these intermediate nodes, we choose an optimal node acting as a relay using three proposed protocols. We consider the channel between a vehicle terminal and an access point node as a Rayleigh fading channel, and the channel between two vehicles as a double Rayleigh fading channel. We derive exact expressions of the end-to-end outage probability for each of the three protocols, which are then verified by Monte-Carlo simulations.

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Authors and Affiliations

  1. University of Ulsan, Ulsan, Republic of Korea

    Sang Quang Nguyen & Hyung Yun Kong

Authors
  1. Sang Quang Nguyen
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  2. Hyung Yun Kong
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Corresponding author

Correspondence to Hyung Yun Kong.

Appendices

Appendix 1: Solving \(\prod \limits _{n = 1}^N {{F_{{\omega _{S,A{P_n}}}}}(\upsilon )} \)

From (10), \({F_{{\omega _{S,A{P_n}}}}}\left( {{x_n}} \right) = 1 - {e^{ - {\lambda _1}{x_n}}}\), we can obtain the result for \(\prod \limits _{n = 1}^N {{F_{{\omega _{S,A{P_n}}}}}(\upsilon )} \) as follows

$$\begin{aligned} \prod \limits _{n = 1}^N {{F_{{\omega _{S,A{P_n}}}}}(\upsilon )}= &\, {F_{{\omega _{S,A{P_1}}}}}(\upsilon ){F_{{\omega _{S,A{P_2}}}}}(\upsilon )\,.\,.\,.\,{F_{{\omega _{S,A{P_N}}}}}(\upsilon )\nonumber \\= &\, \left( {1 - {e^{ - {\lambda _1}\upsilon }}} \right) \left( {1 - {e^{ - {\lambda _1}\upsilon }}} \right) \,.\,.\,.\,\left( {1 - {e^{ - {\lambda _1}\upsilon }}} \right) \nonumber \\= &\, {\left( {1 - {e^{ - {\lambda _1}\upsilon }}} \right) ^N} \end{aligned}$$
(31)

Appendix 2: Solving \(\prod \limits _{m = 1}^M {{F_{{\omega _{S,{V_m}}}}}(\upsilon )} \)

We first calculate \({F_{{\omega _{S,{V_m}}}}}(\upsilon )\),

$$\begin{aligned} {F_{{\omega _{S,{V_m}}}}}(\upsilon )= &\, \Pr \left[ {{\omega _{S,{V_m}}} < \upsilon } \right] \nonumber \\= &\, \Pr \left[ {{\omega _{S,Vm,1}}{\omega _{S,Vm,2}} < \upsilon } \right] \nonumber \\= &\, \Pr \left[ {{\omega _{S,Vm,1}} < \frac{\upsilon }{{{\omega _{S,Vm,2}}}}} \right] \nonumber \\= &\, \int _0^\infty {{f_{{\omega _{S,{V_m},2}}}}\left( {{x_{m,2}}} \right) } \int _0^{\upsilon /{x_{m,2}}} {{f_{{\omega _{S,{V_m},1}}}}\left( {{x_{m,1}}} \right) } \,d{x_{m,1}}d{x_{m,2}} \end{aligned}$$
(32)

where \({f_{{\omega _{S,{V_m},1}}}}\left( {{x_{m,1}}} \right) = {\lambda _2}{e^{ - {\lambda _2}{x_{m,1}}}}\) and \({f_{{\omega _{S,{V_m},2}}}}\left( {{x_{m,2}}} \right) = {\lambda _2}{e^{ - {\lambda _2}{x_{m,2}}}}\) are given in (11) and (12), respectively. Therefore, we obtain the result of (32) as (33):

$$\begin{aligned} {F_{{\omega _{S,{V_m}}}}}(\upsilon )= &\, \int _0^\infty {{\lambda _2}{e^{ - {\lambda _2}{x_{m,2}}}}} \int _0^{\upsilon /{x_{m,2}}} {{\lambda _2}{e^{ - {\lambda _2}{x_{m,1}}}}} \,d{x_{m,1}}d{x_{m,2}}\nonumber \\= &\, \int _0^\infty {{\lambda _2}{e^{ - {\lambda _2}{x_{m,2}}}}} \left( {1 - {e^{ - {\lambda _2}\upsilon /{x_{m,2}}}}} \right) d{x_{m,2}}\nonumber \\= &\, 1 - \int _0^\infty {{\lambda _2}{e^{ - {\lambda _2}{x_{m,2}}}}^{ - {\lambda _2}\upsilon /{x_{m,2}}}} d{x_{m,2}} \end{aligned}$$
(33)

By using equation 3.324 in [27], \(\int _0^\infty {{e^{ - \frac{a}{{4x}} - bx}}dx} = \sqrt{\frac{a}{b}} {K_1}\left( {\sqrt{ab} } \right) \), we obtain the result for (33) as follows

$$\begin{aligned} {F_{{\omega _{S,{V_m}}}}}(\upsilon )= &\, 1 - \int _0^\infty {{\lambda _2}{e^{ - (4{\lambda _2}\upsilon )/(4{x_{m,2}}) - {\lambda _2}{x_{m,2}}}}} d{x_{m,2}}\nonumber \\= &\, 1 - {\lambda _2}\sqrt{\frac{{4{\lambda _2}\upsilon }}{{{\lambda _2}}}} {K_1}\left( {\sqrt{4{\lambda _2}\upsilon {\lambda _2}} } \right) \nonumber \\= &\, 1 - 2{\lambda _2}\sqrt{\upsilon }{K_1}\left( {2{\lambda _2}\sqrt{\upsilon }} \right) \end{aligned}$$
(34)

where \({K_1}(.)\) is a modified Bessel function of the second kind. Then, we can obtain the result for \(\prod \limits _{m = 1}^M {{F_{{\omega _{S,{V_m}}}}}(\upsilon )} \) as follows

$$\begin{aligned} \prod \limits _{m = 1}^M {{F_{{\omega _{S,{V_m}}}}}(\upsilon )}= &\, {F_{{\omega _{S,{V_1}}}}}(\upsilon ){F_{{\omega _{S,{V_2}}}}}(\upsilon )\,\ldots\,{F_{{\omega _{S,{V_M}}}}}(\upsilon )\nonumber \\= &\, {\left( {1 - 2{\lambda _2}\sqrt{\upsilon }{K_1}\left( {2{\lambda _2}\sqrt{\upsilon }} \right) } \right) ^M} \end{aligned}$$
(35)

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Nguyen, S.Q., Kong, H.Y. Outage Probability Analysis in Dual-Hop Vehicular Networks with the Assistance of Multiple Access Points and Vehicle Nodes. Wireless Pers Commun 87, 1175–1190 (2016). https://doi.org/10.1007/s11277-015-3047-1

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