In this section, we take a theoretical approach to understanding the impact of intercluster interference on the cellrate performance. where the ZF and LMMSE intracluster interference coordination strategies are adopted on the uplink.
Assuming the BSs have perfect knowledge of the CSI within the cluster, an estimate of the transmitted vector of all K users in cluster c can be obtained for ZFbased MCP as follows,
$$ \hat{\mathbf{x}}_{\text{ZF}}^{(c)} = \left(\mathbf{H}^{(c)^{H}} \mathbf{H}^{(c)}\right)^{1} \mathbf{H}^{(c)} \mathbf{y}^{(c)} $$
((4))
where y
^{(c)} is given by (2).
For LMMSEbased MCP, the estimate of the transmitted vector is given by,
$$ \hat{\mathbf{x}}_{\text{LMMSE}}^{(c)} = \mathbf{H}^{(c)^{H}} \left(\mathbf{H}^{(c)} \mathbf{H}^{(c)^{H}} + {\sigma_{0}^{2}} \mathbf{I}_{BN_{\mathrm{R}}} \right)^{1} \mathbf{y}^{(c)} $$
((5))
where \({\sigma _{0}^{2}}\) is the inverse of the received SNR at the BSs in cluster c and \(\mathbf {I}_{BN_{\mathrm {R}}}\) is the identity matrix of dimension B
N
_{
R
}. In the following subsections, we derive closedform capacity equations for ZF and LMMSE for a modified planar Wyner network.
Modified planar Wyner model
We modify the widely studied planar Wyner network [2, 28–32] to a threecell clustered cooperative network. In [28], a twodimensional hexagonal array of cells is considered, as shown in Fig. 1. This hexagonal array is represented as a checkerboard lattice, and to make it more tractable, it is converted to a rectangular array. In the planar Wyner model, the signal transmitted by a user in the home cell does not experience any attenuation. The signals from neighbouring cells interfere with the home user and the signals are attenuated by a factor of α, where α∈[0,1]. We modify this rectangular array lattice representation to factor in intercluster interference as well. Figure 2 shows the lattice representation for a network with a cluster size of three, and it is obtained by rotating the hexagonal lattice by 45° and scaling by a factor of \(\frac {1}{\sqrt {2}}\) [28]. Interference from only the firsttier neighbouring clusters is considered. This intracluster interference intensity is denoted as α, as in [28]. We introduce a new parameter for intercluster interference and denote it as β, where β∈[0,1]. Since the interference intensities are scalar and deterministic parameters, the scaling of the lattice does not affect the interference pattern.
Without a loss of generality, the following simplifying assumptions are made in this section. An intracell TDMA transmission strategy is considered where only a single user per cell is transmitting at a given time instance. Thus, the number of active users in a threecell clustered network is K=3. The additive white Gaussian noise (AWGN) channel model is assumed with 0 mean and variance \({\sigma _{n}^{2}}\). All the BSs have perfect knowledge of the CSI as well as each other’s codebooks. The BSs within a cluster are connected using an ideal backhaul with infinite capacity. The BSs are perfectly synchronised to ensure that all signals are received at the same time. BPSK modulation is assumed and the power with which the users transmit the antipodal signals are the same, i.e. no power control is performed across cell sites. Assuming that the received amplitude of each user’s signal is A, then the received power of each user is A
^{2}. Since a cluster size of B=3 is assumed, there are nine interfering cells from the firsttier neighbouring clusters as can be seen in Fig. 2. Finally, the MSs and BSs are equipped with single, omnidirectional antennas, i.e. N
_{T}=N
_{R}=1.
For the threecell clustered network, the channel matrix between the users and the BSs in cluster c is given by,
$$ \mathbf{H}^{(c)} = \left[ \begin{array}{ccc} 1 & \alpha & \alpha \\ \alpha & 1 & \alpha \\ \alpha & \alpha & 1 \\ \end{array} \right] $$
((6))
The channel matrix between the users in the neighbouring clusters and the BSs in cluster c is given by,
$$ \mathbf{H}^{(\hat{\underline{c}})} = \left[ \begin{array}{ccccccccc} \beta & \beta & \beta & \beta & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \beta & \beta & \beta & \beta & 0 & 0 & 0 \\ \beta & 0 & 0 & 0 & 0 & 0 & \beta & \beta & \beta \\ \end{array} \right] $$
((7))
Now, (2) can be rewritten as,
$$ \begin{aligned} \mathbf{y}^{(c)} &= \mathbf{H}^{(c)} \mathbf{x}^{(c)} + \mathbf{y}^{(\hat{\underline{c}})} + \mathbf{n}^{(c)} \\ &= \mathbf{H}^{(c)} \mathbf{x}^{(c)} + \tilde{\mathbf{n}}^{(c)} \end{aligned} $$
((8))
where \(\mathbf {y}^{(\hat {\underline {c}})} = \mathbf {H}^{(\hat {\underline {c}})} \mathbf {x}^{(\hat {\underline {c}})}\) and \(\tilde {\mathbf {n}}^{(c)}\) combines the interference from the neighbouring clusters and the noise at the receivers of the home cluster. The covariance matrix of \(\tilde {\mathbf {n}}^{(c)}\) is then given by,
$$\begin{array}{*{20}l} \mathbb{E}\left[\tilde{\mathbf{n}}^{(c)} \tilde{\mathbf{n}}^{(c)^{T}}\right] &= \mathbb{E}\left[\left(\mathbf{y}^{(\hat{\underline{c}})} + \mathbf{n}^{(c)}\right) \left(\mathbf{y}^{(\hat{\underline{c}})} + \mathbf{n}^{(c)}\right)^{T}\right] \\ &= \mathbb{E}\left[\mathbf{y}^{(\hat{\underline{c}})} \mathbf{y}^{(\hat{\underline{c}})^{T}}\right] + \mathbb{E}\left[\mathbf{y}^{(\hat{\underline{c}})} \mathbf{n}^{(c)^{T}}\right] \\ &\quad+ \mathbb{E}\left[\mathbf{n}^{(c)} \mathbf{y}^{(\hat{\underline{c}})^{T}}\right] + \mathbb{E}\left[\mathbf{n}^{(c)} \mathbf{n}^{(c)^{T}}\right] \end{array} $$
((9))
$$\begin{array}{*{20}l} &= \left(4 \beta^{2} A^{2} + {\sigma_{n}^{2}}\right) \mathbf{I}_{3} \end{array} $$
((10))
In (9), the first term equates to the received power of the users in the neighbouring clusters at each cell in cluster c, which is \(4 \beta ^{2} A^{2} \mathbf {I}_{3}\). The second and third terms equate to 0 since the signals transmitted by the users and the noise are uncorrelated. Since the AWGN channel model with 0 mean and variance \({\sigma _{n}^{2}}\) is considered, the final term in (9) equates to \({\sigma _{n}^{2}} \mathbf {I}_{3}\). Combining all these terms gives us (10).
Capacity of ZF
The estimate of the transmitted vector using ZF receivers is given by,
$$\begin{array}{*{20}l} \hat{\mathbf{x}}^{(c)} &= \left(\mathbf{H}^{(c)^{T}} \mathbf{H}^{(c)}\right)^{1} \mathbf{H}^{(c)} \mathbf{y}^{(c)} \end{array} $$
((11))
$$\begin{array}{*{20}l} &= \mathbf{H}^{(c)^{2}} \mathbf{H}^{(c)} \mathbf{y}^{(c)} \end{array} $$
((12))
$$\begin{array}{*{20}l} &= \mathbf{H}^{(c)^{1}} \mathbf{y}^{(c)} \end{array} $$
((13))
$$\begin{array}{*{20}l} &= \mathbf{x}^{(c)} + \mathbf{H}^{(c)^{1}} \tilde{\mathbf{n}}^{(c)} \end{array} $$
((14))
Since the channel matrix is real in the modified Wyner network, we take the transpose of H
^{(c)} in (11) rather than the Hermitian given by (4). Step (12) is obtained since H
^{(c)}, given by (6) is symmetric. Finally, (14) is obtained by substituting (8) in (13).
Theorem
1.
The percell capacity of a threecell clustered network using ZF is given by,
$$ C_{\text{ZF}} = \frac{1}{2} \log_{2} \left[ 1 + \frac{A^{2} (1  \alpha)^{2} (1 + 2 \alpha)^{2}}{\left(4 \beta^{2} A^{2} + {\sigma_{n}^{2}}\right) \left(2 \alpha^{2} + (1 + \alpha)^{2}\right)} \right] $$
((15))
Proof.
From (14), the received power of all users in cluster c is given by \(A^{2} \mathbf {I}_{3}\), and the noise power is given by the covariance matrix of the background noise \(\left (\mathbf {H}^{(c)}\right)^{1} \tilde {\mathbf {n}}^{(c)}\), given by
$$ \begin{aligned} \mathbb{E}& \left[ \left(\mathbf{H}^{(c)^{1}} \tilde{\mathbf{n}}^{(c)} \right) \left(\mathbf{H}^{(c)^{1}} \tilde{\mathbf{n}}^{(c)} \right)^{T} \right]\\ &\,\,\,\,= \mathbb{E} \left[ \mathbf{H}^{(c)^{1}} \tilde{\mathbf{n}}^{(c)} \tilde{\mathbf{n}}^{(c)^{T}} \mathbf{H}^{(c)^{T}} \right] \end{aligned} $$
((16))
$$\begin{array}{*{20}l} &= \mathbf{H}^{(c)^{1}} \mathbb{E} \left[ \tilde{\mathbf{n}}^{(c)} \tilde{\mathbf{n}}^{(c)^{T}} \right] \mathbf{H}^{(c)^{T}} \end{array} $$
((17))
$$\begin{array}{*{20}l} &= \mathbf{H}^{(c)^{1}} \left(4 \beta^{2} A^{2} + {\sigma_{n}^{2}} \right) \mathbf{I}_{3} \mathbf{H}^{(c)^{1}} \end{array} $$
((18))
$$\begin{array}{*{20}l} &= \left(4 \beta^{2} A^{2} + {\sigma_{n}^{2}} \right) \mathbf{H}^{(c)^{2}} \end{array} $$
((19))
Since H
^{(c)} is deterministic in the Wyner network, we get (17) from (16). Step (18) is then obtained by substituting (10) in (17). The achievable capacity at cell i is then,
$$ {}\mathrm{C}_{\text{ZF}}^{(i)} = \frac{1}{2} \log_{2} \left[ 1 + \frac{A^{2}}{\left(4 \beta^{2} A^{2} + {\sigma_{n}^{2}} \right) \sum_{j=1}^{3} \left(\left(\mathbf{H}^{(c)} \right)^{1}_{ij} \right)^{2}} \right] $$
((20))
Using the GaussJordan method, we can obtain
$$ \mathbf{H}^{(c)^{1}} = \frac{1}{(1  \alpha)(1 + 2 \alpha)} \left[ \begin{array}{ccc} 1 + \alpha & \alpha & \alpha \\ \alpha & 1 + \alpha & \alpha \\ \alpha & \alpha & 1 + \alpha \end{array} \right] $$
((21))
Thus,
$$ \sum_{j=1}^{3} \left(\left(\mathbf{H}^{(c)} \right)^{1}_{ij} \right)^{2} = \frac{(1 + \alpha)^{2} + 2 \alpha^{2}}{(1  \alpha)^{2} (1 + 2 \alpha)^{2}}\ ;\ \forall i $$
((22))
The theorem is proved by substituting (22) in (20).
It can be observed from (15) that there is a singularity at α=1 which can be overcome by considering fading and multiuser scheduling as shown in [31].
Capacity of LMMSE
For the modified Wyner network, the estimate of the transmitted vector using LMMSE receivers is given by,
$$ \hat{\mathbf{x}}_{\text{LMMSE}}^{(c)} = \mathbf{H}^{(c)^{T}} \left(\mathbf{H}^{(c)} \mathbf{H}^{(c)^{T}} + {\sigma_{0}^{2}} \mathbf{I}_{3} \right)^{1} \mathbf{y}^{(c)} $$
((23))
In (23), \({\sigma _{0}^{2}}\) is the inverse of the received SNR at cluster c, given by
$$ {\sigma_{0}^{2}} = \frac{4 \beta^{2} A^{2} + {\sigma_{n}^{2}}}{A^{2}} $$
((24))
where 4β
^{2}
A
^{2} is the interference power from the users in the neighbouring clusters.
In [30], the capacity of an MCP system using LMMSE receivers is derived for a Wyner model where the cells are arranged in a onedimensional linear array. In this linear model, a finite number of cells N cooperate and only two neighbouring cells interfere with the home cell. The N×N channel matrix for such a system is given by,
$$ \begin{aligned} \mathbf{H} &= \left[ \begin{array}{ccccccc} 1 & \alpha & 0 & 0 & 0 & \cdots & 0 \\ \alpha & 1 & \alpha & 0 & 0 & \cdots & 0 \\ 0 & \alpha & 1 & \alpha & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & \cdots & 0 & \alpha & 1 & \alpha \\ 0 & \cdots & \cdots & \cdots & 0 & \alpha & 1 \\ \end{array} \right] \end{aligned} $$
((25))
The achievable capacity of cell i is then shown in [30] to be,
$$ \begin{aligned} C_{\text{LMMSE}}^{(i)} &=  \frac{1}{2} \log_{2} \left[ 1  \frac{1}{N+1} \sum_{k=1}^{N} \frac{{\mu_{k}^{2}}}{{\mu_{k}^{2}} + {\sigma_{0}^{2}}}\right.\\ &\quad\times\left.\vphantom{\sum_{k=1}^{N}} \left(1  \cos\left(\frac{2ik\pi}{N+1}\right) \right) \right] \end{aligned} $$
((26))
where μ
_{
k
}s are the eigenvalues of H and \({\sigma _{0}^{2}} = \frac {{\sigma _{n}^{2}}}{A^{2}}\). This equation is however not applicable in a clustered MCP system since the channel matrix is different and also since the effects of intercluster interference are not considered. We have to therefore rederive the capacity for a threecell clustered network.
Following the derivation as in [30] and [33], the LMMSE estimator for user i in cluster c can be shown to be,
$$\begin{array}{*{20}l} \hat{x}_{i}^{(c)} &= \mathbf{h}_{i}^{(c)^{T}} \left(\mathbf{H}^{(c)} \mathbf{H}^{(c)^{T}} + {\sigma_{0}^{2}} \mathbf{I}_{3} \right)^{1} \mathbf{y}^{(c)} \end{array} $$
((27))
$$\begin{array}{*{20}l} &= \mathbf{h}_{i}^{(c)^{T}} \left(\mathbf{H}^{(c)^{2}} + {\sigma_{0}^{2}} \mathbf{I}_{3} \right)^{1} \mathbf{y}^{(c)} \end{array} $$
((28))
Since H
^{(c)} is symmetric, we get (28) from (27). Let the error of estimation for user i be \(W_{i} = x_{i}^{(c)}  \hat {x}_{i}^{(c)}\). The meansquared error (MSE) of estimation is then given by,
$$ \bar{W}_{i}^{2} = A^{2} \left(1  \mathbf{h}_{i}^{(c)^{T}} \left(\mathbf{H}^{(c)^{2}} + {\sigma_{0}^{2}} \mathbf{I}_{3} \right)^{1} \mathbf{h}_{i}^{(c)} \right) $$
((29))
The eigendecomposition of the square matrix, \(\mathbf {H}^{(c)^{2}} + {\sigma _{0}^{2}} \mathbf {I}_{3}\), can be expressed as
$$ \mathbf{H}^{(c)^{2}} + {\sigma_{0}^{2}} \mathbf{I}_{3} = \mathbf{Q}^{(c)} \mathbf{\Lambda}^{(c)} \left(\mathbf{Q}^{(c)}\right)^{T} $$
((30))
where \(\mathbf {Q}^{(c)} = \left [\mathbf {q}_{1}^{(c)} \mathbf {q}_{2}^{(c)} \mathbf {q}_{3}^{(c)} \right ]\) is the 3×3 orthogonal matrix consisting of the orthonormal eigenvectors of \(\mathbf {H}^{(c)^{2}} + {\sigma _{0}^{2}} \mathbf {I}_{3}\) and \(\mathbf {\Lambda }^{(c)} = \text {diag} \left \{ \lambda _{1}^{(c)}, \lambda _{2}^{(c)}, \lambda _{3}^{(c)} \right \}\) is the diagonal matrix containing the eigenvalues of \(\mathbf {H}^{(c)^{2}} + {\sigma _{0}^{2}} \mathbf {I}_{3}\).
The eigenvalues \(\lambda _{k}^{(c)}\) are related to the eigenvalues of H
^{(c)}, denoted as \(\mu _{k}^{(c)}\), as follows
$$ \lambda_{k}^{(c)} = \mu_{k}^{(c)^{2}} + {\sigma_{0}^{2}}\ ;\ k = 1,2,3 $$
((31))
On the other hand, the eigenvectors of \(\mathbf {H}^{(c)^{2}} + {\sigma _{0}^{2}} \mathbf {I}_{3}\), denoted as \(\mathbf {q}_{k}^{(c)}\), are also the eigenvectors of H
^{(c)}.
The eigenvalues and eigenvectors of H
^{(c)}, given by (6), exhibit the following properties:

1.
The eigenvalues, \(\mu _{k}^{(c)}\) are:
$$ \mu_{k}^{(c)} = 1 + 2 \alpha \cos \left(\frac{2 (k  1) \pi}{3} \right)\ ;\ k = 1,2,3 $$
((32))
i.e. \(\mu _{1}^{(c)} = (1 + 2 \alpha)\) and \(\mu _{2}^{(c)} = \mu _{3}^{(c)} = (1  \alpha)\).

2.
The eigenvectors, \(\mathbf {q}_{k}^{(c)}\) are independent and orthonormal. i.e.
$$ \sum_{j=1}^{3} q_{kj}^{(c)} = 0\ ;\ k = 1,2,3 $$
((33))
$$ \sum_{k=1}^{3} q_{kj}^{(c)^{2}} = 1\ ;\ j = 1,2,3 $$
((34))

3.
It can be shown that
$$ q_{1j}^{(c)} = \left(\frac{1}{3} \right)^{\frac{1}{2}}\ ;\ j = 1,2,3. $$
((35))
Theorem
2.
The percell capacity of a threecell clustered network using LMMSE is given by,
$$ \begin{aligned} C_{\text{LMMSE}} &=  \frac{1}{2} \log_{2} \left[ 1  \frac{1}{3} \left(\frac{\left(1 + 2 \alpha\right)^{2}}{\left(1 + 2 \alpha\right)^{2} + {\sigma_{0}^{2}}} \right.\right.\\ &\quad+\left.\left. \frac{2 \left(1  \alpha\right)^{2}}{\left(1  \alpha\right)^{2} + {\sigma_{0}^{2}}} \right) \right] \end{aligned} $$
((36))
where \({\sigma _{0}^{2}} = \frac {4 \beta ^{2} A^{2} + {\sigma _{n}^{2}}}{A^{2}}\).
Proof.
Using the eigendecomposition in (30), we can obtain the following,
$$\begin{array}{*{20}l} & \mathbf{h}_{i}^{(c)^{T}} \left(\mathbf{H}^{(c)^{2}} + {\sigma_{0}^{2}} \mathbf{I}_{3} \right)^{1} \mathbf{h}_{i}^{(c)} \end{array} $$
((37))
$$\begin{array}{*{20}l} &= \sum_{k=1}^{3} \frac{1}{\lambda_{k}^{(c)}} \left(\alpha q_{k1}^{(c)} + q_{k2}^{(c)} + \alpha q_{k3}^{(c)} \right)^{2} \notag\\ &= \frac{\left(1 + 2 \alpha \right)^{2}}{3 \lambda_{1}^{(c)}} + \frac{1}{\lambda_{2}^{(c)}} \left[ \left(\alpha q_{21}^{(c)} + q_{22}^{(c)} + \alpha q_{23}^{(c)} \right)^{2}\right. \\ &\quad+\left. \left(\alpha q_{31}^{(c)} + q_{32}^{(c)} + \alpha q_{33}^{(c)} \right)^{2} \right] \end{array} $$
((38))
$$\begin{array}{*{20}l} &= \frac{\left(1 + 2 \alpha \right)^{2}}{3 \lambda_{1}^{(c)}} + \frac{(1\alpha)^{2}}{\lambda_{2}^{(c)}} \left[ q_{22}^{(c)^{2}} + q_{32}^{(c)^{2}} \right] \end{array} $$
((39))
$$\begin{array}{*{20}l} &= \frac{\left(1 + 2 \alpha \right)^{2}}{3 \lambda_{1}^{(c)}} + \frac{(1\alpha)^{2}}{\lambda_{2}^{(c)}} \left[ 1  q_{12}^{(c)^{2}} \right] \end{array} $$
((40))
$$\begin{array}{*{20}l} &= \frac{\left(1 + 2 \alpha \right)^{2}}{3 \lambda_{1}^{(c)}} + \frac{2 (1\alpha)^{2}}{3 \lambda_{2}^{(c)}} \end{array} $$
((41))
$$\begin{array}{*{20}l} &= \frac{1}{3} \left[ \frac{\left(1 + 2 \alpha\right)^{2}}{\left(1 + 2 \alpha\right)^{2} + {\sigma_{0}^{2}}} + \frac{2 \left(1  \alpha\right)^{2}}{\left(1  \alpha\right)^{2} + {\sigma_{0}^{2}}} \right] \end{array} $$
((42))
Step (38) can be obtained by substituting (32) and (35) in (37). Steps (39), (40), (41) and (42) can be obtained from (33), (34), (35) and (32), respectively. By substituting (42) in (29), we get
$$ \bar{W}_{i}^{2} = A^{2} \left[1  \frac{1}{3} \left(\frac{\left(1 + 2 \alpha\right)^{2}}{\left(1 + 2 \alpha\right)^{2} + {\sigma_{0}^{2}}} + \frac{2 \left(1  \alpha\right)^{2}}{\left(1  \alpha\right)^{2} + {\sigma_{0}^{2}}} \right) \right] $$
((43))
The percell capacity can then be obtained by computing the mutual information of \(x_{i}^{(c)}\) and \(\hat {x}_{i}^{(c)}\), denoted as \(I\left (x_{i}^{(c)} ; \hat {x}_{i}^{(c)}\right)\). The MSE of the estimation is given by (43). It is shown in [28] that \(I\left (x_{i}^{(c)} ; \hat {x}_{i}^{(c)}\right) = H\left (x_{i}^{(c)}\right)  H(W_{i})\). Since all the users are transmitting at the highest power, A
^{2}, the percell capacity is given by,
$$\begin{array}{*{20}l} C_{\text{LMMSE}}^{(i)} &= \frac{1}{2} \log_{2} \frac{A^{2}}{\bar{W}_{i}^{2}} \end{array} $$
((44))
$$\begin{array}{*{20}l} &=  \frac{1}{2} \log_{2} \left[ 1  \frac{1}{3} \left(\frac{\left(1 + 2 \alpha\right)^{2}}{\left(1 + 2 \alpha\right)^{2} + {\sigma_{0}^{2}}} \right.\right.\\ &\quad+\left.\left. \frac{2 \left(1  \alpha\right)^{2}}{\left(1  \alpha\right)^{2} + {\sigma_{0}^{2}}} \right) \right] \ ;\ \forall i \end{array} $$
((45))
Equation (45) can be obtained by substituting (43) in (44), thus proving the theorem.
Performance analysis
We plot the capacities of ZF and LMMSE for various interference intensities in Fig. 3 using the equations derived earlier. Two intracluster and four different intercluster interference intensities are considered, i.e. α∈{0.5,0.8} and β∈{0,0.005,0.1,0.2}. It must also be noted that these results have been validated using Monte Carlo simulations.
It can be observed that for all intercluster interference intensities, the performances of ZF and LMMSE are better for a lower intracluster interference intensity. For instance, for β=0 and a very high SNR of 60 dB, the performance of ZF and LMMSE for α=0.5 is 12.5 % higher than that of α=0.8. The same is true even for low to medium SNRs. Also, the gap between ZF and LMMSE for a given α is negligible for β=0 and β=0.005. It can be observed that for a small increase in the intensity of intercluster interference, the performance of ZF and LMMSE deteriorate significantly only at very high SNRs. For instance, when β increases from 0 to 0.005, the performance of ZF and LMMSE deteriorate by 50 % at SNR = 60 dB. When β increases further from 0.005 to 0.1, the performance degradation can be observed even at medium SNRs. For instance, when the SNR is 20 dB the performance degrades by about 72 %. It can also be observed that at lower SNRs and higher intercluster interference intensities, LMMSE decoders perform better than ZF. We can conclude through this theoretical study that even the slightest intercluster interference can seriously deteriorate the performance of a threecell clustered network.