In this section, we derive performance bounds for various cases and compare the performance of different schemes operating in the CoMP scenario.

### 5.1 Performance bounds

This subsection presents performance bounds/approximations for the throughput and outage probability of the proposed CoMP-ARQ approach. Theorem 1 shows that the performance of the considered protocols becomes bounded if the number of retransmission rounds is finite.

#### Theorem 1

With a fixed number of retransmissions, the throughput achieved by the considered CoMP-HARQ schemes is upper bounded even if the transmission power goes to infinity.

#### Proof

The theorem is proved for the INR protocol which, as stated in the following, results in higher throughput compared to the RTD. The throughput upper bound is found based on the following (in)equalities:

\phantom{\rule{-17.0pt}{0ex}}\begin{array}{l}{\eta}^{\text{INR}}\stackrel{\left(b\right)}{=}N\sum _{m=1}^{M}\frac{R}{m(m+1)}Pr\left\{\sum _{n=1}^{m}log(1+{\gamma}^{n})\ge R\right\}\\ \phantom{\rule{4.5em}{0ex}}+\frac{\mathit{\text{NR}}}{M+1}Pr\left\{\sum _{n=1}^{M+1}log(1+{\gamma}^{n})\ge R\right\}\\ \stackrel{\left(c\right)}{\le}N\phantom{\rule{0.3em}{0ex}}\sum _{m=1}^{M}\phantom{\rule{0.3em}{0ex}}\frac{\mathbf{\text{E}}\{\sum _{n=1}^{m}log(1+{\gamma}^{n}\left)\right\}}{m(m+1)}+\frac{N\mathbf{\text{E}}\{\sum _{n=1}^{M+1}\phantom{\rule{0.3em}{0ex}}log(1\phantom{\rule{0.3em}{0ex}}+{\gamma}^{n}\left)\right\}}{M+1}\\ \stackrel{\left(d\right)}{\le}\frac{N}{N-1}\sum _{m=0}^{M}\frac{1}{m+1}\stackrel{\left(e\right)}{\le}\frac{N}{N-1}\left(1+log(M+1)\right).\end{array}

(13)

Here, (*b*) is obtained by some manipulation on (11) and (*c*) comes from the Markov’s inequality, Pr(X\ge x)\le \frac{\mathbf{\text{E}}\left(X\right)}{x}[55]. Then, (d) follows from

\begin{array}{l}\phantom{\rule{-17.0pt}{0ex}}\begin{array}{l}\Lambda =\mathbf{E}\{\sum _{n=1}^{m}log(1+{\gamma}^{n}\left)\right\}=m\mathbf{E}\{log(1+{\gamma}^{n}\left)\right\}\\ \phantom{\rule{1em}{0ex}}=m{\int}_{0}^{\infty}log(1+x){f}_{{\gamma}^{n}}\left(x\right)\mathrm{d}x\hfill \\ \phantom{\rule{1em}{0ex}}\stackrel{\left(f\right)}{=}m{\int}_{0}^{\infty}\phantom{\rule{0.3em}{0ex}}(1-{F}_{{\gamma}^{n}}({e}^{z}-1\left)\right)\mathrm{d}z\stackrel{\left(g\right)}{\le}m\underset{0}{\overset{\infty}{\int}}\phantom{\rule{0.3em}{0ex}}{e}^{-(N-1)z}\mathrm{d}z\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\frac{m}{N-1},\end{array}\end{array}

(14)

where (*f*) is obtained by variable transform log(1+*x*)=*z* and partial integration and (*g*) is based on (4) and {e}^{-\frac{\lambda}{P}x}\le 1. Finally, (*e*) in (13) follows from \sum _{m=0}^{M}\frac{1}{m+1}\le 1+log(M+1) which is obtained by Riemann integral and the fact that \frac{1}{x} is a decreasing function^{d}. □

Intuitively, both the signal and the interference powers grow linearly with *P*. Hence, with high transmission power, the system becomes interference-limited and, according to (4), the users received SINR is bounded. In other words, the theorem states that: (1) the multiplexingz gain of the proposed CoMP scheme, defined as \rho =\underset{P\to \infty}{\mathrm{lim}}\frac{\eta}{log\left(P\right)}[56], is zero if a fixed number of retransmission rounds are considered for the HARQ protocols. (2) To have a positive multiplexing gain, a necessary condition is to scale the number of retransmission rounds with the transmission power. (3) Also, according to (13), the throughput is scaled with the number of retransmission rounds at most logarithmically. Finally, denoting the channel instantaneous capacity by *C*, we can use (*b*) in (13) to write

\begin{array}{l}\begin{array}{c}\phantom{\rule{-14.0pt}{0ex}}{\eta}^{\text{INR}}\stackrel{\left(h\right)}{\le}\mathit{\text{NR}}(\sum _{m=1}^{M}\frac{\phantom{\rule{0.3em}{0ex}}{e}^{-\frac{1}{m}}\mathbf{\text{E}}\left\{{(\frac{1}{m}\sum _{n=1}^{m}log(1+{\gamma}^{n}\left)\right)}^{\frac{1}{R}}\right\}}{m(m+1)}\\ \phantom{\rule{3em}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}+\frac{{e}^{-\frac{1}{M+1}}\mathbf{\text{E}}\left\{{(\frac{1}{M+1}\sum _{n=1}^{M+1}log(1+{\gamma}^{n}\left)\right)}^{\frac{1}{R}}\right\}}{M+1})\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(i\right)}{\le}\mathit{\text{NR}}\mathbf{\text{E}}\left\{{C}^{\frac{1}{R}}\right\}(\sum _{m=1}^{M}\frac{\phantom{\rule{0.3em}{0ex}}{e}^{-\frac{1}{m}}}{m(m+1)}+\frac{{e}^{-\frac{1}{M+1}}}{M+1})\\ \phantom{\rule{4em}{0ex}}\stackrel{\left(j\right)}{\le}\mathit{\text{NR}}\mathbf{\text{E}}\left\{{C}^{\frac{1}{R}}\right\}(\xi +1)\hfill \end{array}\end{array}

(15)

which indicates that for sufficiently high initial transmission rate, where \mathbf{\text{E}}\left\{{C}^{\frac{1}{R}}\right\}\to 1, the throughput scales with the initial rate *R* at most linearly. Note that (*h*) in (15) is obtained with the same procedure as in (*b*) of (13) and then implementation of the exponential Chebyshev’s inequality, Pr(*X*≥*x*)≤*e*^{−tx}E(*e*^{tX}),∀*t*>0 [55]. Also, (*i*) follows from the fact that \frac{1}{m}\sum _{n=1}^{m}log(1+{\gamma}^{n})\le C, i.e., the maximum decodable rates is less than the channel instantaneous capacity *C*, and (*j*) comes from \frac{{e}^{-\frac{1}{M+1}}}{M+1}\le 1 and \sum _{m=1}^{M}\frac{\phantom{\rule{0.3em}{0ex}}{e}^{-\frac{1}{m}}}{m(m+1)}\le \xi where *ξ* is the Euler’s number. Finally, the same qualitative conclusions are valid for the RTD protocol.

#### Theorem 2

For Rayleigh fading channels, the proposed CoMP-HARQ network can be modeled/underestimated by *N* interference-free single-user networks having a modified transmission SNR.

#### Proof

With the same procedure as in (13), the RTD- and INR-based throughput, i.e., (7) and (11), can be rewritten as

\begin{array}{l}{\eta}^{\text{RTD}}=N\sum _{m=1}^{M}\frac{R}{m(m+1)}Pr\left\{log(1+\sum _{n=1}^{m}{\gamma}^{n})\ge R\right\}\\ \phantom{\rule{8.5em}{0ex}}+\frac{\mathit{\text{NR}}}{M+1}Pr\left\{log(1+\sum _{n=1}^{M+1}{\gamma}^{n})\ge R\right\}\end{array}

(16)

\begin{array}{l}{\eta}^{\text{INR}}=N\sum _{m=1}^{M}\frac{R}{m(m+1)}Pr\left\{\sum _{n=1}^{m}log(1+{\gamma}^{n})\ge R\right\}\\ \phantom{\rule{8.5em}{0ex}}+\frac{\mathit{\text{NR}}}{M+1}Pr\left\{\sum _{n=1}^{M+1}log(1+{\gamma}^{n})\ge R\right\}.\end{array}

(17)

Moreover, from (4), we can show that the auxiliary random variable *γ*^{n} is dominated^{e} by the random variable *ω*^{n} which follows the cdf^{f}

\begin{array}{l}{F}_{{\omega}^{n}}\left(x\right)=1-{e}^{-(\frac{\lambda}{P}+N-1)x}.\end{array}

(18)

Therefore, using (16), (17), and the fact that *F*_{
γ
}^{n}(*x*)≤*F*_{
ω
}^{n}(*x*),∀*x*, i.e., Pr{*γ*^{n}≥*x*}≥ Pr{*ω*^{n}≥*x*}, ∀*x*, we have

\phantom{\rule{-15.0pt}{0ex}}\begin{array}{l}{\eta}^{\text{RTD}}\ge N\sum _{m=1}^{M}\frac{R}{m(m+1)}Pr\left\{log(1+\sum _{n=1}^{m}{\omega}^{n})\ge R\right\}\\ \phantom{\rule{4em}{0ex}}+\frac{\mathit{\text{NR}}}{M+1}Pr\left\{log(1+\sum _{n=1}^{M+1}{\omega}^{n})\ge R\right\}\\ \phantom{\rule{2em}{0ex}}\stackrel{\left(k\right)}{=}N\sum _{m=1}^{M}\frac{R}{m(m+1)}Pr\left\{log(1+\stackrel{\u0304}{P}\sum _{n=1}^{m}{\stackrel{\u0304}{\omega}}^{n})\ge R\right\}\\ \phantom{\rule{4em}{0ex}}+\frac{\mathit{\text{NR}}}{M+1}Pr\left\{log(1+\stackrel{\u0304}{P}\sum _{n=1}^{M+1}{\stackrel{\u0304}{\omega}}^{n})\ge R\right\}={\eta}^{\text{RTD,SU}}\end{array}

(19)

\phantom{\rule{-15.0pt}{0ex}}\begin{array}{l}{\eta}^{\text{INR}}\ge N\sum _{m=1}^{M}\frac{R}{m(m+1)}Pr\left\{\sum _{n=1}^{m}log(1+{\omega}^{n})\ge R\right\}\\ \phantom{\rule{4em}{0ex}}+\frac{\mathit{\text{NR}}}{M+1}Pr\left\{\sum _{n=1}^{M+1}log(1+{\omega}^{n})\ge R\right\}\\ \phantom{\rule{1em}{0ex}}\stackrel{\left(k\right)}{=}N\sum _{m=1}^{M}\frac{R}{m(m+1)}Pr\left\{\sum _{n=1}^{m}log(1+\stackrel{\u0304}{P}{\stackrel{\u0304}{\omega}}^{n})\ge R\right\}\\ \phantom{\rule{4em}{0ex}}+\frac{\mathit{\text{NR}}}{M+1}Pr\left\{\sum _{n=1}^{M+1}log(1+\stackrel{\u0304}{P}{\stackrel{\u0304}{\omega}}^{n})\ge R\right\}={\eta}^{\text{INR,SU}}.\end{array}

(20)

Here, (*k*) in (19) and (20) is obtained by defining {\stackrel{\u0304}{\omega}}^{n}:{F}_{{\stackrel{\u0304}{\omega}}^{n}}\left(x\right)=1-{e}^{-x} and the equivalent SNR \stackrel{\u0304}{P}=\frac{P}{\lambda +(N-1)P}, i.e., by appropriate scaling of the fading pdf and the transmission SNR. Also, *η*^{RTD,SU} and *η*^{INR,SU} denote the throughput achieved by the RTD and INR protocols, respectively, in the equivalent channel model of (18). In words, (19) and (20) imply that the CoMP-HARQ network performance can be underestimated by *N* interference-free single-input single-output (SISO) Rayleigh fading channels with fading coefficient h\sim \mathcal{C}\mathcal{N}(0,1),\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0304}{\omega}}^{n}=|h{|}^{2}, and transmission SNR \stackrel{\u0304}{P}=\frac{P}{\lambda +(N-1)P}. Interestingly, the approximation *F*_{
γ
}^{n}≃*F*_{
ω
}^{n} becomes very tight for moderate/high values of *N*. Thus, the collection of SISO channels becomes an accurate model for the proposed CoMP-HARQ network, as stated in the theorem. □

One of the benefits of Theorem 2 is that there are closed-form/approximate expressions for the probability terms of (19) and (20) such as

\begin{array}{l}Pr\left\{log(1+\stackrel{\u0304}{P}\sum _{n=1}^{m}{\stackrel{\u0304}{\omega}}^{n})\ge R\right\}=1-V(\frac{{e}^{R}-1}{\stackrel{\u0304}{P}},m),\end{array}

(21)

where *V*(*x*,*y*) is the normalized incomplete Gamma function (please see [57] for more details about (21)).

### 5.2 Comparisons

This subsection compares the performance of different schemes in terms of the throughput and the outage probability.

#### Remark 2

Let *M*<*N*. The spatial diversity gained in the proposed CoMP-HARQ scheme with slow-fading channel is the same as the time diversity achieved in the non-CoMP-HARQ schemes when the channel is fast-fading, i.e., when the channel gains change in each retransmission round independently.

In order to show this, consider a non-CoMP model where each user is always served by a specific BS and there is no cooperation between the BSs. In this way, the link between the *i*-th user and its corresponding BS is an interference-affected SISO channel. Assuming a fast-fading model, i.e., the channels take new values in each retransmission round, the received SINR of the *i*-th user in round *n* is u\left(n\right)=\frac{P{g}_{i}\left(n\right)}{1+P\sum _{j=1\ne i}^{N}{g}_{j}\left(n\right)} where *g*_{
i
}(*n*) represents the gain realization of the channel between the *i*-th user and its corresponding BS at round *n*. Hence, a user throughput in the presence of the RTD and INR protocols is

\phantom{\rule{-15.0pt}{0ex}}\begin{array}{c}{\eta}_{i}^{\text{RTD,Fast}}=\sum _{m=1}^{M+1}\frac{R}{m}Pr\left\{log\left(1+\sum _{n=1}^{m-1}u\left(n\right)\right)<R\le log\left(1+\sum _{n=1}^{m}u\left(n\right)\right)\right\}\end{array}

(22)

\phantom{\rule{-15.0pt}{0ex}}\begin{array}{c}{\eta}_{i}^{\text{INR,Fast}}=\sum _{m=1}^{M+1}\frac{R}{m}Pr\left\{\sum _{n=1}^{m-1}log\left(1+u\left(n\right)\right)<R\le \sum _{n=1}^{m}log\left(1+u\left(n\right)\right)\right\},\end{array}

(23)

respectively, which are the corresponding throughputs obtained for the *i*-th user in the proposed CoMP model when the channel is slow-fading (see (6) and (10)).

In other words, although each channel remains constant in all retransmission rounds of the slow-fading model, by switching between the BSs, a different SINR realization is observed in each round, the same as in the fast-fading channels. Thus, the CoMP model works well as (1) the slow-fading behavior of the channel gives the opportunity to accurately estimate the channels at the receivers and (2) the same diversity as in the fast-fading channels is gained by a simple cooperation approach. Finally, the spatial diversity exploited by the proposed CoMP scheme will be less than the time diversity achieved in the non-CoMP fast-fading channel if *M*≥*N*. This is because it may occur that we return back to the same BSs when the number of retransmissions exceeds the number of BSs. However, this case is of less interest because the maximum number of possible HARQ-based retransmissions is normally less than the number of cooperative BSs.

#### Remark 3

It has been previously shown that the INR outperforms the RTD in terms of outage probability and throughput ([38], lemma 1). However, as log(1+*x*)→*x* for small *x*’s, (7) and (11) can both be rewritten as

\begin{array}{l}\eta =N\sum _{m=1}^{M+1}\frac{R}{m}Pr\{{\gamma}^{(m-1)}<R\le {\gamma}^{\left(m\right)}\}\end{array}

(24)

when *P*→0. That is, the same performance is achieved by the INR and RTD protocols at low transmission powers.

This is interesting when we remember that the superiority of the INR over the RTD is at the cost of complexity; in the INR, the codewords are changed in each retransmission which results in more complex encoders and decoders. Therefore, compared to the INR, the RTD is preferable at low transmission powers, because the same throughput and outage probability are achieved in both schemes while the RTD leads to less implementation complexity.

As demonstrated in [58], the gain of the INR scheme over the RTD increases with the initial transmission rate *R*. Also, [58] has previously shown that the difference between the performance of the RTD and INR protocols decreases with the SINR variation between the retransmissions. Thus, compared to the non-CoMP setup, the gain of the INR over RTD decreases in the CoMP scenario.

Finally, it is worth noting that the considered HARQ protocols, RTD and INR, lead to better system performance compared to (1) the open-loop communication model, (2) basic ARQ protocols, and (3) the case when repetition codes are implemented in a cooperative fashion; the open-loop communication model is a special case of the HARQ-based schemes with no retransmissions, i.e., *M*=0. Hence, setting *M*=0 in, e.g., (8), the open-loop system throughput is

\begin{array}{l}{\eta}^{\text{Open}}=\mathit{\text{NR}}\left(1-{F}_{{\gamma}^{1}}({e}^{R}-1)\right)\end{array}

(25)

which is clearly less than the throughput obtained in (7) and (11). In basic ARQ, on the other hand, the same codeword is transmitted in different retransmission rounds and the users decode the data in each round independently of the previously received signals. In this way, with some manipulations, the throughput of the basic ARQ approach is found as

\phantom{\rule{-11.0pt}{0ex}}\begin{array}{c}{\eta}^{\text{Basic}}=N\sum _{m=1}^{M+1}\left(\frac{R}{m}\left(\prod _{l=1}^{m-1}Pr\left\{log(1+{\gamma}^{l})<R\right\}\right)Pr\left\{R\le log(1+{\gamma}^{m})\right\}\right).\end{array}

(26)

Then, as less information is exploited by the basic ARQ decoder, compared to the RTD, the throughput in the RTD model is obviously higher than the throughput in the basic ARQ (the superiority of the HARQ protocols over the basic ARQ has been previously shown in the literature, e.g., [36, 38]). Here, it is interesting to note that with a slow-fading condition, e.g., [38] has shown that there are no performance gains with basic ARQ and the optimal throughput/outage probability achieved by the basic ARQ is the same as the one in the open-loop communication setup if the channel does not change in the retransmissions. However, the proposed CoMP-HARQ approach makes it possible to utilize the SINR variations and, depending on the channel pdf, increase the throughput by implementation of basic ARQ.

Finally, using repetition codes the same codeword is transmitted by the switching BSs in different rounds, the same as in RTD. However, as opposed to RTD, the MRC-based decoder is not implemented in each round but only when all possible repetition rounds are used. In this case, removing the *m*-th, *m*=1…*M*, probability terms of (7), the system throughput for the repetition codes is obtained by

\begin{array}{l}\phantom{\rule{-13.0pt}{0ex}}{\eta}^{\text{Rept}}=\frac{\mathit{\text{NR}}}{M+1}Pr\{R\le log(1+{\gamma}^{(M+1)}\left)\right\}<{\eta}^{\text{RTD}}\le {\eta}^{\text{INR}}.\end{array}

(27)

The same arguments can be used when comparing the outage probability of these schemes. Here, the only important difference is that the RTD HARQ and the repetition code schemes lead to the same outage probability. This is because in RTD, the outage occurs if and only if the data is not decodable in the last retransmission round, the same as in the repetition codes.