In highspeed environments, the Doppler Effects would lead to irreducible bit error rate (BER) which is called error floor [12, 13]. However, according to technical specifications (TS) of LTE [14, 15], the procedure of triggering handover contains three phases: the user equipments (UEs) measure the RSSI, RSRP or RSRQ, sent the measurement reports to source eNodeB, and then the radio resource control (RRC) of source eNodeB decides whether handover is triggered or not [16]. The 3GPP evaluation documents [17] also point out that the handover measurement and radio link failure (RLF) only depend on the RSSI, RSRP or RSRQ. Though the BER performance would degrade the QoS, if the RSRP remains above a certain threshold for a fixed duration, the wireless link will be reestablished and assured to complete the handover. At most of time, the highspeed train travels through the wide plain and viaduct, the lineofsight (LOS) path experienced freespace loss only between MRS and BSs is available and there are few reflectors or scatterers. The major influence on wireless channel caused by relative motion between transmitter and receiver is Doppler shift instead of Doppler spread. Therefore in highspeed railway scenario, instead of considering Doppler Effects which degrades BER, we only need to consider Doppler shift which would impair handover performance. In [18], Doppler shift in the overlapping region of two neighboring eNodeBs (handover region) is almost unchanged, and can be compensated [19]. In this article we suppose that Train Control Information, such as train's velocity and location, can be shared by the ground eNodeBs, and Doppler shift has been compensated [20, 21].
Without loss of generality, let the train be located at x away from the eNodeB i, the received power in dBm in decibels can be expressed as
R\left(i,x\right)={P}_{t}PL\left(i,x\right)A\left(i,x,\sigma \right)
(1)
where P_{
t
}is the transmit power in dBm, PL(i, x) is the path loss between eNodeB i and location x, and A(i,x,σ) is the shadow component at the location x, generally modeled as a Gaussian random variable with mean zero and standard deviation σ. Typically, σ is 6 or 8dBin [22].
As the railway communication network is the linear coverage topology, we consider a twocell system. The shadow fading losses of the two adjacent eNodeBs are correlated and can be expressed as follow according to [23].
{A}_{i}=a{\xi}_{0}+b{\xi}_{i}
(2)
where a^{2} + b^{2} = 1. Note that i is the link number. ξ_{0} is common to both A_{1} and A_{2}, ξ_{
i
}represents the independent part between the two adjacent eNodeBs. Furthermore, both ξ_{0} and ξ_{
i
}are independent Gaussian random variables with mean zero and standard deviation σ. Meanwhile, because the tail of Gaussian distribution extends to infinity, a fade margin F dB is added to the transmit power.
4.1 Outage in the CoMP handover scheme
In our proposal, the two adjacent eNodeBs communicate with the train simultaneously in the overlapping area. Outage will happen if and only if both signals are of unacceptable quality. So the outage probability using the proposed handover scheme is
{P}_{\text{CoMP\_handover}}=\text{Pr}\left[\text{min}\left(A\left(i,x,\sigma \right),A\left(j,x,\sigma \right)\right)>F\right]
(3)
where, A(i,x,σ) and A(j,x,σ) represent the shadow fading losses of eNodeB i and eNodeB j respectively. F is the fade margin and F = P_{
t
} PL  R_{
s
}. PL is the path loss and R_{
s
}is the receiver sensitivity. By straightforward manipulation, the outage probability can be shown as
\begin{array}{ll}\hfill {P}_{\text{CoMP\_handover}}& =\frac{1}{{\left(2\pi \sigma \right)}^{3/2}}\underset{\infty}{\overset{\infty}{\int}}{e}^{\frac{{t}^{2}}{2{\sigma}^{2}}}\left(\underset{\frac{Fat}{b}}{\overset{\infty}{\int}}{e}^{\frac{{\xi}_{i}^{2}}{2{\sigma}^{2}}}d{\xi}_{i}\underset{\frac{Fat}{b}}{\overset{\infty}{\int}}{e}^{\frac{{\xi}_{j}^{2}}{2{\sigma}^{2}}}d{\xi}_{j}\right)dt\phantom{\rule{2em}{0ex}}\\ =\frac{1}{\sqrt{2\pi}}\underset{\infty}{\overset{\infty}{\int}}{e}^{\frac{{x}^{2}}{2}}{\left[\Phi \left(\frac{Fa\sigma x}{b\sigma}\right)\right]}^{2}dx\phantom{\rule{2em}{0ex}}\\ =\frac{1}{\sqrt{2\pi}}\underset{\infty}{\overset{\infty}{\int}}{e}^{\frac{{x}^{2}}{2}}{\left[\frac{1}{2}\text{erfc}\left(\frac{Fa\sigma x}{\sqrt{2}b\sigma}\right)\right]}^{2}dx\phantom{\rule{2em}{0ex}}\end{array}
(4)
where
\Phi \left(x\right)=\underset{x}{\overset{\infty}{\int}}\frac{1}{\sqrt{2\pi}}{e}^{\frac{{t}^{2}}{2}}dt,\phantom{\rule{2.77695pt}{0ex}}\text{erfc}\left(x\right)=\frac{2}{\sqrt{\pi}}\underset{x}{\overset{\infty}{\int}}{e}^{{t}^{2}}dt
(5)
4.2 Outage in current hard handover scheme
In the current handover scheme, the MRS can connect to only one eNodeB at one time. For ideal case, the MRS is always switched to the eNodeB with the best signal quality. However, this may lead to the wellknown 'pingpong' effect around the cell boundary. In practical systems, handover will be triggered on the condition that the received power of the source eNodeB is lower than that of the target eNodeB by hysteresis level h.
In most of the existing researches, the outage probability in hard handover system is obtained with the hysteresis level h being assumed infinite so that it is impossible to handover to the neighboring cell. This scenario is called an isolated cell. Let shadow fading loss A = ξ, where ξ is Gaussian random variable with mean zero and standard deviation σ. An outage occurs when the fading component is larger than the fade margin, which is expressed as
\begin{array}{ll}\hfill {P}_{\text{isolated}}& =\text{Pr}\left(\xi >F\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{\sqrt{2\pi}\sigma}\underset{F}{\overset{\infty}{\int}}{e}^{\frac{{\xi}^{2}}{2{\sigma}^{2}}}d\xi \phantom{\rule{2em}{0ex}}\\ =\Phi \left(\frac{F}{\sigma}\right)=\frac{1}{2}\text{erfc}\left(\frac{F}{\sqrt{2}\sigma}\right)\phantom{\rule{2em}{0ex}}\end{array}
(6)
It is well known that the hysteresis is finite in practical scenarios and varied according to different communication scenario. In order to make a fair comparison between conventional scheme and our proposal, a finite hysteresis level is assumed in this article. A handover occurs when the received power of the target eNodeB j is larger than that of the source eNodeB i by h dB, so the handover probability can be expressed as
{P}_{x}\left(i,j\right)=\text{Pr}\left[R\left(j,x\right)R\left(i,x\right)\ge h\right]
(7)
When the train is located exactly at the midpoint of overlapping region, the outage probability is composed of three parts:

(1)
If both signals from the two adjacent eNodeBs can't be received by the train, outage will happen. In such case, the outage probability P_{both_outage} can be expressed as
{P}_{\text{both\_outage}}=\text{Pr}\left[A\left(i,x,\sigma \right)>F\&A\left(j,x,\sigma \right)>h\right]
(8)

(2)
Before handover, if the signal strength of source eNodeB is unacceptable and that of target eNodeB is not large enough to trigger handover, this is obviously an outage. Let ε_{
i
}be the event that the train is connecting to eNodeB i, and h be the hysteresis level. The outage probability P_{before_ HO} can be expressed as
\begin{array}{ll}\hfill {P}_{\text{before\_HO}}& =\text{Pr}\left[{\epsilon}_{i};A\left(j,x,\sigma \right)<F<A\left(i,x,\sigma \right)\right]\phantom{\rule{2em}{0ex}}\\ =\text{Pr}\left[A\left(i,x,\sigma \right)h<A\left(j,x,\sigma \right)<F<A\left(i,x,\sigma \right)\right].\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\text{Pr}\left[{\epsilon}_{i}A\left(i,x,\sigma \right)h<A\left(j,x,\sigma \right)<F<A\left(i,x,\sigma \right)\right]\phantom{\rule{2em}{0ex}}\end{array}
(9)
where
\begin{array}{l}\mathrm{Pr}[A(i,x,\sigma )h<A(j,x,\sigma )<F<A(i,x,\sigma )]\\ =\frac{{b}^{2}}{{(2\pi )}^{3/2}}{\displaystyle \underset{\infty}{\overset{\infty}{\int}}{e}^{\frac{{t}^{2}}{2}}}{\displaystyle \underset{\frac{Fa\sigma t}{b\sigma}}{\overset{\frac{F+ha\sigma t}{b\sigma}}{\int}}}{\displaystyle \underset{{x}_{2}\frac{h}{b\sigma}}{\overset{\frac{Fa\sigma t}{b\sigma}}{\int}}}{e}^{\frac{{({x}_{1}at)}^{2}+{({x}_{2}at)}^{2}}{2{b}^{2}}}d{x}_{1}d{x}_{2}dt\end{array}
(10)
In fact, the correlation of shadow fading losses would decrease with the increasing angleofarrival difference [24]. We assume that there is no sitetosite correlation (i.e, a = 0) as the angleofarrival difference is quite large. Thus, the above expression can be simplified as
\frac{1}{2\pi}\underset{\frac{F}{\sigma}}{\overset{\frac{F+h}{\sigma}}{\int}}\underset{{x}_{2}\frac{F}{\sigma}}{\overset{\frac{F}{\sigma}}{\int}}{e}^{\frac{{x}_{1}^{2}+{x}_{2}^{2}}{2}}d{x}_{1}d{x}_{2}
(11)

(3)
After handover, although the train has been successfully switched to the target eNodeB, outage will happen if the signal strength of the target eNodeB is too weak. It should be noted that the target eNodeB is determined by the direction to which the train moves towards, that is, the train can't be switched to the previous source eNodeB though the signal strength of the novel source eNodeB is terrible. Thus, the outage probability P_{after_ HO} can be expressed as
\begin{array}{ll}\hfill {P}_{\text{after\_HO}}& =\text{Pr}\left[{\epsilon}_{j};A\left(i,x,\sigma \right)<F<A\left(j,x,\sigma \right)\right]\phantom{\rule{2em}{0ex}}\\ =\text{Pr}[A\left(i,x,\sigma \right)<F<A\left(j,x,\sigma \right)\cdot \text{Pr}\left[{\epsilon}_{j}A\left(i,x,\sigma \right)<F<A\left(j,x,\sigma \right)\right]\phantom{\rule{2em}{0ex}}\end{array}
(12)
where
\begin{array}{ll}\hfill \text{Pr}\left[A\left(i,x,\sigma \right)<F<A\left(j,x,\sigma \right)\right]& =\text{Pr}\left[a{\xi}_{0}+b{\xi}_{i}<F<a{\xi}_{0}+b{\xi}_{i}\right]\phantom{\rule{2em}{0ex}}\\ =P\left(\frac{{\xi}_{0}}{\sigma}=t\right)\cdot P\left(\frac{{\xi}_{i}}{\sigma}<\frac{Fa{\xi}_{0}}{b\sigma}\right)\cdot P\left(\frac{{\xi}_{j}}{\sigma}>\frac{Fa{\xi}_{0}}{b\sigma}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{{\left(\pi \right)}^{3/2}}\underset{\infty}{\overset{\infty}{\int}}{e}^{\frac{{t}^{2}}{2}}\underset{\frac{Fa\sigma t}{b\sigma}}{\overset{\infty}{\int}}\underset{0}{\overset{\frac{Fa\sigma t}{b\sigma}}{\int}}{e}^{\frac{{x}_{1}^{2}+{x}_{2}^{2}}{2}}d{x}_{1}d{x}_{2}dt\phantom{\rule{2em}{0ex}}\end{array}
(13)
It should be noted that the value of second item in (9) is difficult to obtain analytically. Whether the train is connecting to eNodeB i or not can not be determined by a snapshot of the system. It is generally assumed to be 1/2 in the midpoint of the overlapping region. Since A_{
j
}< A_{
i
}in (9), that is, the attenuation loss to eNodeB j is smaller, and the train has a higher chance to connect to eNodeB j. Following [24], this probability is chosen as 0.6.
From the analysis above, the overall outage probability using hard handover scheme can be expressed as
{P}_{\text{hard\_outage}}={P}_{\text{both\_outage}}+{P}_{\text{before\_HO}}+{P}_{\text{after\_HO}}
(14)