Outage Analysis of Train-to-Train Communication Model over Nakagami-𝑚 Channel in High-Speed Railway

Abstract

This paper analyzes the end-to-end outage performance of high-speed-railway train-to-train communication model in high-speed railway over independent identical and nonidentical Nakagami-𝑚 channels. The train-to-train communication is inter-train communication without an aid of infrastructure (for base station). Source train uses trains on other rail tracks as relays to transmit signals to destination train on the same track. The mechanism of such communication among trains can be divided into three cases based on occurrence of possible-occurrence relay trains. We first present a new closed form for the sum of squared independent Nakagami-𝑚 variates and then derive an expression for the outage probability of the identical and non-identical Nakagami-𝑚 channels in three cases. In particular, the problem is improved by the proposed formulation that statistic for sum of squared Nakagami-𝑚 variates with identical 𝑚 tends to be infinite. Numerical analysis indicates that the derived analytic results are reasonable and the outage performance is better over Nakagami-𝑚 channel in high-speed railway scenarios.

1. Introduction

Railway has played a significant role in helping transport passengers and goods. Accidents in railway always result in loss of lives and property. Safety issues have drawn increasing research attention due to the personal and property security.

A part of railway accidents is brought by malfunction of control center system. The train-to-train communication will serve as an assisting role which coordinates operations among trains based on multihop, when control center system is broken. It aims at detecting a potential collision and then broadcasting prewarning messages concerning this emergency to other trains on the same and neighboring tracks.

The multihop train-to-train communication model in physical layer lies in the thoughts that the source train uses the trains operating on other tracks as relays to transmit signals to destination train on the same track. Such mechanism can be divided into three cases based on occurrence of other relay trains. These relay trains occur following the Poisson Process [1], and therefore the arrival procedure follows the distribution of negative exponent. The proposed train-to-train communication model, introducing OFDM and MIMO technique, realizes the intertrain adhoc communication based on the Poisson Process in high-speed railway scenarios.

Since 2006, train-to-train communication has been researched by several organizations, such as the German Aerospace Center (DLR). Reference [2] discusses the RCAS approach consisting only of mobile adhoc components without the necessity of extensions of the railway infrastructure, while [3] describes an overview of the state of the art in collision avoidance related with transportation systems for maritime transportation, aircraft, and road transportation, and the RCAS is introduced. Reference [4] proposes a channel model for direct train-to-train communication appropriate for the 400 MHz band, and [5] presents an infrastructure-less cross-layer train-to-train communication system exploiting all characteristics of a pervasive computing system, like direct communication in mobile adhoc networks. Reference [6] designs an infrastructure-less adhoc inter-vehicle communication system that fulfills these requirements with respect to the boundary conditions in the railway environment. Reference [7] presents analysis and results of a comprehensive measurement campaign investigating the propagation channel in case of direct communication between railway vehicles.

Despite that the fact the RCAS designed by the DLR has progressed tremendously in physical layer, it only work well for the train operation velocity lower than 200 Km/h, which is not able to function in the high-speed railway. Generally speaking, the velocity of high-speed railway train is up to 360 Km/h. In this case, safety distance among trains is 10 Km [8]. The proposed train-to-train communication model in physical layer has been evaluated by BER previously. In this paper, we continue to analyze the proposed multihop train-to-train communication model using outage probability. We first present a new closed-form for the sum of squared independent Nakagami-𝑚 variates with identical 𝑚 [9], the proposed formulation improves the problem that statistic for sum of squared Nakagami-𝑚 variables with identical 𝑚 is infinite. Then we derive an expression for the outage probability of the identical 𝑚 (𝑚=1 or 𝑚=2) and nonidentical Nakagami-𝑚 channels [10] in three cases. Such outage analysis is first applied to the multihop train-to-train communication model. The previous research indicates that the BER of train-to-train communication model reaches 10⁻⁶ when receiving SNR is 10 dB, which meets the requirements of the International Union of Railway (UIC). Therefore the threshold should be set to 10 dB. If the receiving signal SNR is below that value, the signal quality at receiver cannot realize normal communication among trains and the train-to-train communication is regarded as outage. The maximum distance of train-to-train communication is set to 6 Km and it is probable that 𝑚 value of Nakagami-𝑚 channel in two receiving path at receiver is different, for example, one path is 𝑚=1 and the other path is 𝑚=2. This paper considers the Nakagami-𝑚 channel not only with identical 𝑚 but also with nonidentical 𝑚 in receiver’s receiving path.

The rest of this paper is organized as follows. Section 2 gives a description of the proposed train-to-train communication model based on multihop. Section 3 derives expressions for the outage probability of three conditions. Section 4 shows the outage probability simulation results of this model. In Section 5, this paper is concluded.

2. Proposed Train-to-Train Communication Model

In this section, three cases of train-to-train communication model are presented in Figures 1, 2, and 3, where the S, R1, and D represent source terminal, fixed-occurring relay terminal, and destination terminal, respectively. The R1 is the train that meets S on another rail track, and the R2 and R3 represent the first possible-occurring relay terminal and the third possible-occurring relay terminal. The average operating velocities of the R1 and R2 are 100 m/s, and those of D and R3 are −100 m/s. The rail 1, rail 2, rail 3, and rail 4 stand for four parallel rail tracks. These cases are described below.

Figure 1 

Train-to-train communication model of case I.

Figure 2 

Train-to-train communication model of case II.

Figure 3 

Communication model of case III.

2.1. Case I

The S transmits signals to the R1 on different rail tracks when they meet each other. At the same time, the S searches the potential relay R2 on a neighboring rail track 1. If the R2 is not found, R1 will keep broadcasting messages within its communication coverage (6 Km), and if it receives the response of D, their communication link will be held on and the transmission between them is performed.

2.2. Case II

If the R2 is searched, the S and R1 simultaneously transmit the signals to the R2. The R2 also performs the search of the potential relay R3. If the R3 does not exist, it will operate for some seconds until the distance between R2 and D is within a communication range. Finally, the R2 and R1, acting as two relays of the source, will transmit the signals to the destination terminal.

2.3. Case III

If the R3 exists, it will receive the signals from the R2 and R1 and then combine them. Finally, the R3, R2, and R1, as three relays of the source, will forward the signals to the destination terminal.

Under these cases, this paper respectively analyzes the outage probability of the train-to-train communication model.

3. Outage Analysis of Train-to-Train Communication Model

{𝑋_𝑛,𝑛=1,…,𝑀} is 𝑀 independent Nakagami-𝑚 distributed RVs, with PDF expressed as [11] 𝑓_𝑋𝑙(𝑥)=2𝑥^{2𝑚𝑛−1}𝑚^𝑚𝑛_𝑛𝑥^𝑚𝑛_𝑛()!−𝑚𝑚−1exp^𝑛𝑥²𝑥_𝑛, (1) where 𝑚_𝑛 denotes the Nakagami-𝑚 fading parameter and 𝑥_𝑛 is expectation of 𝑋²_𝑛.

Furthermore, let 𝑌_𝑛=𝑋²_𝑛, the PDF given by 𝑓_𝑌𝑙(𝑥)=𝑥^𝑚𝑛−1𝑚^𝑚𝑛_𝑛𝑥^𝑚𝑛_𝑛()!−𝑚𝑚−1exp^𝑛𝑥𝑥_𝑛 (2) and CDF [9] 𝐹_𝑌𝑙(𝑥)−𝑚=1−exp^𝑛𝑥𝑥_𝑛^𝑚𝑛−1_𝑙𝑥𝑚_𝑛𝑥_𝑛^𝑙. (3)

Before analyzing the outage probability of train-to-train communication model, we first give the close-form of the sum of squared Nakagami-𝑚 variates with identical 𝑚 and nonidentical 𝑚.

Theorem 1 (PDF of the Sum of Squared Nakgami-𝑚 RVs with identical 𝑚). Let {𝑌_𝑛,𝑛=1,…,𝑀} be a set of RVs following the PDF presented in (2), with 𝑥_𝑖−𝑥_𝑗=𝑜(𝑥), 𝑖≠𝑗 and 𝑚₁=𝑚₂=⋯=𝑚_𝑀=𝑚. The PDF of the sum 𝑍_𝑀=^𝑀_𝑖=1𝑌_𝑖 (4) is given by 𝑓_𝑍𝑀(𝑧)=𝑚𝑧^{(𝑀−1)𝑚}Γ(𝑚)Γ()𝑧𝑀𝑚^{(𝑀−1)𝑚}×𝑧^𝑚−1𝑚^𝑚𝑧^𝑚_𝑀()!−𝑚−1exp𝑚𝑧𝑧_𝑀. (5)

Proof. Step 1. For 𝑀=2, the PDF of 𝑍₂=𝑌₁+𝑌₂ can be evaluated as 𝑓_𝑍2(𝑧)=^𝑧₀𝑓_𝑌1𝑥;𝑚₁,𝑥₁𝑓_𝑌2𝑧−𝑥;𝑚₂,𝑥₂𝑑𝑥. (6) Insert (2) into (6) and use [12, equation (3.383.1)], (6) can be derived as 𝑓_𝑍2(𝑧)=𝐵𝑚₁,𝑚₂𝑧^𝑚1₁𝑧^𝑚2₂Γ𝑚₁Γ𝑚₂𝑧^{−1+𝑚1+𝑚2}−𝑚exp²𝑧𝑧₂∗𝐹₁𝑚₁;𝑚₁+𝑚₂𝑚;−¹𝑧₁−𝑚₂𝑧₂𝑧. (7) Since 𝑚₁/𝑧₁−𝑚₂/𝑧₂=𝑜(𝑥) and using [12, equation (9.210.1)] 𝐹₁𝑚₁;𝑚₁+𝑚₂𝑚;−¹𝑥₁−𝑚₂𝑥₂𝑧=^+∞_𝑘=0𝑧^𝑘𝑚𝑘!¹+𝑘−1!/𝑚₁−1!𝑚₂+𝑚₁+𝑘−1!/𝑚₁+𝑚₂!×−𝑚−1₁𝑧₁−𝑚₂𝑧₂𝑧^𝑘, (8) where [−(𝑚₁/𝑧₁−𝑚₂/𝑧₂)𝑧]^𝑘|_𝑘=0=1 and [−(𝑚₁/𝑧₁−𝑚₂/𝑧₂)𝑧]^𝑘|_𝑘≠0=0, (7) can be denoted as 𝑓_𝑍2(𝑧)=𝐵𝑚₁,𝑚₂𝑥^𝑚1₁𝑥^𝑚2₂Γ𝑚₁Γ𝑚₂𝑧^{−1+𝑚1+𝑚2}−𝑚exp²𝑧𝑧₂=𝑚₁𝑧^𝑚1𝑚₂𝑧^𝑚2Γ𝑚₂Γ𝑚₂+𝑚₁×𝑧^𝑚1𝑧^𝑚2−1𝑚^𝑚2₂𝑧^𝑚2_𝑛𝑚₂!−𝑚−1exp²𝑧𝑧₂=𝑚₁𝑧^𝑚1𝑚₂𝑧^𝑚2Γ𝑚₂Γ𝑚₂+𝑚₁𝑧^𝑚1𝑓_𝑌2(𝑧). (9)Step 2. For 𝑀=3, the PDF of 𝑍₃ can be efficiently expressed as 𝑓_𝑍3(𝑧)=^𝑧₀𝑓_𝑍2𝑥;𝑚₁,𝑥₁𝑓_𝑌3𝑧−𝑥;𝑚₃,𝑥₃𝑑𝑥. (10) Following Step  1 of PDF calculation of 𝑍₂, the PDF of 𝑍₃ can be denoted as: 𝑓_𝑍2(𝑧)=𝑚₁𝑧^𝑚1𝑚₂𝑧^𝑚2𝑚₃𝑧^𝑚3×Γ𝑚₃Γ𝑚₂+𝑚₁+𝑚₃𝑧^𝑚1+𝑚2𝑓_𝑌3(𝑧). (11)Step 3. According to the same procedure as Steps 1 and 3, the sum of 𝑀 Nakagami-𝑚 RVs can be expressed as 𝑓_𝑍𝑀(𝑧)=𝑚𝑧𝑌₁^{𝑚(𝑀−1)}Γ(𝑚)Γ()𝑓𝑀𝑚^𝑌𝑀(𝑧). (12)

Corollary 2 (PDF of the sum of Squared Nakagami-𝑚 RVs with identical 𝑚). The CDF of 𝑍_𝑀 is given by 𝐹_𝑀(𝑍)=𝑚𝑧𝑌₁^{𝑚(𝑀−1)}Γ(𝑚)Γ()×))!𝑀𝑚((𝑀−1𝑚−1𝑚/𝑧_𝑀^𝑀𝑚−𝑒^{−(𝑚/𝑧𝑀)𝑧}×^𝑀𝑚−1_𝑘=0((𝑀−1)𝑚−1)!𝑧𝑘!𝑚/𝑧^𝑘_𝑀^{𝑀𝑚−𝑘}. (13)

Theorem 3 (PDF of the Sum of Squared Nakgami-𝑚 RVs with nonidentical 𝑚). Let {𝑌_𝑛,𝑛=1,…,𝑀} be a set of RVs following the PDF presented in (2), with 𝑚₁≠𝑚₂≠⋯≠𝑚_𝑀≠𝑚. The PDF of the sum is given by [11] 𝑓_𝑍𝑀(𝑧)=^𝑀𝑚_𝑖=1^𝑖_𝑘=1𝑦^𝑚𝑖_𝑖𝑚^𝑚ℎ_ℎ∏^𝑀_ℎ=1𝑦^𝑚ℎ_ℎ𝑚^𝑚𝑖_𝑖×^𝑀_{𝑗=1,𝑗≠𝑖}𝑚_𝑗𝑦_𝑗−𝑚_𝑖𝑦_𝑖^−𝑚𝑗𝑧^𝑚𝑖−1𝑚^𝑚𝑖_𝑖𝑦^𝑚𝑖_𝑖()!−𝑚𝑚−1exp^𝑖𝑧𝑦_𝑖=^𝑀𝑚_𝑖=1^𝑖_𝑘=1𝑧^𝑚𝑖_𝑖∏^𝑀_ℎ=1𝑦^𝑚ℎ_ℎ^𝑀_{𝑗=1,𝑗≠𝑖}1𝑦_𝑗−1𝑦_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑥_𝑖. (14)

Corollary 4 (PDF of the sum of Squared Nakagami-𝑚 RVs with nonidentical 𝑚). The CDF of 𝑍_𝑀 is given by [13] 𝐹_𝑀(𝑍)=^𝑀𝑚_𝑖=1^𝑖_𝑘=1𝑦^𝑚𝑖_𝑖𝑚^𝑚ℎ_ℎ∏^𝑀_ℎ=1𝑦^𝑚ℎ_ℎ𝑚^𝑚𝑖_𝑖^𝑀_{𝑗=1,𝑗≠𝑖}𝑚_𝑗𝑦_𝑗−𝑚_𝑖𝑦_𝑖^−𝑚𝑗×−𝑚1−exp^𝑛𝑧𝑦_𝑛^𝑚𝑛−1_𝑙𝑧𝑚_𝑛𝑦_𝑛^𝑙=^𝑀𝑚_𝑖=1^𝑖_𝑘=1𝑦^𝑚𝑖_𝑖∏^𝑀_ℎ=1𝑦^𝑚ℎ_ℎ^𝑀_{𝑗=1,𝑗≠𝑖}1𝑦_𝑗−1𝑦_𝑖^−𝑚𝑗𝐹_𝑌𝑙(𝑥𝑖). (15)

The outage event happens when the Y falls below given threshold Y₀ under Nakagami-𝑚 channel and its probability is defined as 𝑃_out(𝑌)=Prob𝑌<𝑌₀−𝑚=1−exp^𝑛𝑌₀𝑦^𝑚𝑛−1_𝑙𝑌₀𝑚_𝑛𝑦^𝑙, (16) where 𝑌 is average receiving signal-to-noise ratio.

Next the outage probabilities of train-to-train communication model under three cases are analyzed.

3.1. Case I

In Figure 4, 𝑌₁ and 𝑌₂ are SNR of receiving signal at R1 and D, respectively. Since each node applies 2∗2 MIMO, 𝑌₁ and 𝑌₂ are sum of 4 squared Nakagami-𝑚 variables at receiver.

Figure 4 

Communication model of case I.

The outage probability 𝐹_outI of case I is expressed as 𝐹_outI=Prob𝑌₁<𝑌₀+Prob𝑌₁>𝑌₀∗Prob𝑌₂<𝑌₀. (17)𝑌₀ is the threshold SNR.

Inserting (13) or (15) into (17), the 𝐹_outI under Nakagami-𝑚 channel with identical 𝑚 and nonidentical 𝑚 can be written as follows.

(1) Identical-𝑚: 𝐹_outI=𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑌₁^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘+𝑚𝑧𝑌₂^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑌₂^3𝑚−𝑚−exp𝑌₂𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₂^4𝑚−𝑘∗1−𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑌₁^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘. (18)

(2) Nonidentical 𝑚: 𝐹_outI=^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏^𝑀_ℎ=1𝑌^𝑚ℎ₁⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀+1−^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁴_ℎ=1𝑌^𝑚ℎ₁×⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₂∏⁴_ℎ=1𝑌^𝑚ℎ₂⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀. (19)

3.2. Case II

In Figure 5, 𝑌₁ are SNR of receiving signal at R1, 𝑌₂, 𝑌₃ are SNR of receiving signal at R2, and 𝑌₄, 𝑌₅ are receiving signal SNR at D.

Figure 5 

Communication model of case II.

The outage probability 𝐹_outI of case I is expressed as 𝐹_outII=Prob𝑌₁<𝑌₀+Prob𝑌₁>𝑌₀∗Prob𝑌₂+𝑌₃<𝑌₀+Prob𝑌₁>𝑌₀∗Prob𝑌₂+𝑌₃>𝑌₀∗Prob𝑌₄+𝑌₅<𝑌₀. (20)

Inserting (13) or (15) into (20), the 𝐹_outII under Nakagami-𝑚 channel with identical 𝑚 and nonidentical 𝑚 can be written as follows.

(1) Identical-𝑚: 𝐹_outII=𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑌₁^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘+𝑚𝑧𝑌₂^7𝑚Γ(𝑚)Γ()×()!7𝑚7𝑚−1𝑚/𝑌₂^7𝑚−𝑚−exp𝑌₂𝑌^𝑘₀×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₂^8𝑚−𝑘∗1−𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑌₁^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘+1−𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑌₁^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘∗1−𝑚𝑧𝑌₂^7𝑚Γ(𝑚)Γ()×()!3𝑚7𝑚−1𝑚/𝑌₂^7𝑚−𝑚−exp𝑌₂𝑌^𝑘₀×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₂^8𝑚−𝑘∗𝑚𝑧𝑌₄^7𝑚Γ(𝑚)Γ()×()!7𝑚7𝑚−1𝑚/𝑌₄^7𝑚−𝑚−exp𝑌₄𝑌^𝑘₀×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₄^8𝑚−𝑘. (21)(2) Nonidentical 𝑚: 𝐹_outII=^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁸_ℎ=1𝑌^𝑚ℎ₁⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀+1−^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁸_ℎ=1𝑌^𝑚ℎ₁×⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₂∏⁸_ℎ=1𝑌^𝑚ℎ₂⁸_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀+1−^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁸_ℎ=1𝑌^𝑚ℎ₁×⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗1−^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₂∏⁸_ℎ=1𝑌^𝑚ℎ₂×⁸_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₄∏⁸_ℎ=1𝑌^𝑚ℎ₄⁸_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀. (22)

3.3. Case III

In Figure 6, 𝑌₁ are SNR of receiving signal at R1, 𝑌₂, 𝑌₃ are SNR of receiving signal at R2, 𝑌₄, 𝑌₅ are receiving signal SNR at R3, and 𝑌₆, 𝑌₇ and 𝑌₈ are receiving signal SNR at D.

Figure 6 

Communication model of case III.

(1) Identical-𝑚: 𝐹_outIII=𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑧₃^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘+𝑚𝑧𝑌₁^7𝑚Γ(𝑚)Γ()×()!3𝑚7𝑚−1𝑚/𝑧₃^7𝑚−𝑚−exp𝑌₂𝑌₀^𝑘×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₂^8𝑚−𝑘∗1−𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑧₃^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘+1−𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑧₃^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘∗1−𝑚𝑧𝑌₁^7𝑚Γ(𝑚)Γ()×()!3𝑚7𝑚−1𝑚/𝑧₃^7𝑚−𝑚−exp𝑌₂𝑌^𝑘₀×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₂^8𝑚−𝑘∗𝑚𝑧𝑌₁^7𝑚Γ(𝑚)Γ()×()!3𝑚7𝑚−1𝑚/𝑧₃^7𝑚−𝑚−exp𝑌₂𝑌^𝑘₀×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₂^8𝑚−𝑘+1−𝑚𝑧𝑌₁^3𝑚Γ(𝑚)Γ()×()!3𝑚3𝑚−1𝑚/𝑌₁^3𝑚−𝑚−exp𝑌₁𝑌^𝑘₀×^3𝑚−1_𝑘=0(3𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₁^4𝑚−𝑘∗1−𝑚𝑧𝑌₂^7𝑚Γ(𝑚)Γ()×()!3𝑚7𝑚−1𝑚/𝑌₂^7𝑚−𝑚−exp𝑌₂𝑌^𝑘₀×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₂^8𝑚−𝑘∗1−𝑚𝑧𝑌₄^7𝑚Γ(𝑚)Γ()×()!3𝑚7𝑚−1𝑚/𝑌₄^7𝑚−𝑚−exp𝑌₄𝑌^𝑘₀×^7𝑚−1_𝑘=0(7𝑚−1)!𝑌𝑘!𝑚/𝑌^𝑘0₄^8𝑚−𝑘∗1−𝑚𝑧𝑌₄^11𝑚Γ(𝑚)Γ()×()!11𝑚11𝑚−1𝑚/𝑌₆^11𝑚−𝑚−exp𝑌₆𝑌^𝑘₀×^11𝑚−1_𝑘=0(11𝑚−1)!𝑌𝑘!𝑚^𝑘0𝑌₆^{12𝑚−𝑘}. (23)

(2) Nonidentical 𝑚: 𝐹_outIII=^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁸_ℎ=1𝑌^𝑚ℎ₁⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀+1−^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁸_ℎ=1𝑌^𝑚ℎ₁⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₂∏⁸_ℎ=1𝑌^𝑚ℎ₂⁸_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀+1−^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁸_ℎ=1𝑌^𝑚ℎ₁⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗1−^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₂∏^𝑀_ℎ=1𝑌^𝑚ℎ₂⁸_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₄∏⁸_ℎ=1𝑌₄^𝑚ℎ8_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀+1−^4𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₁∏⁸_ℎ=1𝑌^𝑚ℎ₁⁴_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗1−^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₂∏^𝑀_ℎ=1𝑌^𝑚ℎ₂⁸_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗1−^8𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₄∏⁸_ℎ=1𝑌^𝑚ℎ₄⁸_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀∗1−^12𝑚_𝑖=1^𝑖_𝑘=1𝑌^𝑚𝑖₆∏¹²_ℎ=1𝑌^𝑚ℎ₆∗¹²_{𝑗=1,𝑗≠𝑖}1𝑌_𝑗−1𝑌_𝑖^−𝑚𝑗𝐹_𝑌𝑙𝑌₀. (24)

4. Numerical Analysis

In this section, we show numerical results of the analytical outage probability of train-to-train communication model in three cases. We plot the performance curves in terms of average signal-to-noise ratio (SNR) and also show computer simulation results for verification.

According to the previous research, when the receiving SNR is 10 dB, the Bit Error rate (BER) is 10⁻⁶ which satisfies the communication requirements of high-speed railway set by UIC. Therefore, outage performances at SNR = 10 dB are crucial to the research of train-to-train communication model. If the wireless communication link in railway is disrupted, the railway safety is threatened severely. Figures 7, 9, and 11 show the numerical and simulation results of outage probability of train-to-train communication model versus SNR in case I, case II, and case III when 𝑚=1 or 𝑚=2. For 𝑚=1 in Nakagami-𝑚 channel, the channel approximates Rayleigh channel, while for 𝑚=2 the channel is Rice channel [14]. Because the condition of propagation channel is increasingly becoming better with the increment of parameter 𝑚, the outage performances of Rayleigh channel are not superior to those of Rice channel. As SNR is 10 dB, in Rayleigh channel, the outage probabilities of case I, case II, and case III are, 0.02, 0, and 0, respectively, while those of case I, case II, and case III are all 0 in Rice channel. AS for case I in Rayleigh channel, the outage probability is 0.02, which is bad for wireless communication. But the 𝑚 parameter in high-speed railway is usually bigger than 𝑚=2 in fact [14]. Figures 8, 10, and 12 show the numerical and simulation results of outage probability of train-to-train communication model versus SNR in case I, case II, and case III when 𝑚=1 and 𝑚=2. The outage probability is 7*10⁻⁴, 0, and 0 for three cases at SNR = 10 dB. Low outage probability contributes to the normal receipt of transmit signal and is rewarding to detection and decision of signal.

Figure 7 

Outage probability of train-to-train communication in case I when 𝑚=1 or 𝑚=2.

Figure 8 

Outage probability of train-to-train communication in case I when 𝑚=3.

Figure 9 

Outage probability of train-to-train communication in case II when 𝑚=1 or 𝑚=2.

Figure 10 

Outage probability of train-to-train communication in case II when 𝑚=3.

Figure 11 

Outage probability of train-to-train communication in case III when 𝑚=1 or 𝑚=2.

Figure 12 

Outage probability of train-to-train communication in case III when 𝑚=1 and 𝑚=2.

5. Conclusion

This paper analyzes the end-to-end outage performance of high-speed-railway train-to-train communication model over independent identical and nonidentical Nakagami-𝑚 channels. The mechanism of such communication among trains can be divided into three cases based on occurrence of possible-occurrence relay trains. Numerical and simulation analysis shows that the outage probability of train-to-train communication model in three cases is approximately 0 over Nakagami-𝑚 (𝑚⩾2) channel in high-speed railway, which ensures the normal receipt of transmit signal.

Acknowledgments

This research was supported by the National Natural Science Foundation of China under Grant no. 61222105 the Program for New Century Excellent Talents in University under Grant NCET-09-0206, Fundamental Research Funds for the Central Universities under Grant nos. 2010JBZ008, 2012YJS017, 2012YJS001, and no. 2011YJS010, Key Project of State Key Lab under Grant no. RCS2011ZZ008, National Natural Science Foundation of China under Grant 60830001, Beijing Natural Science Foundation under Grant 4112048, and Program for Changjiang Scholars and Innovative Research Team in University under Grant no. IRT0949.