an application of Z-matrix in a mobile radio system

The following is an application of Z-matrix in wireless communication called power balancing problem.

Consider $n$ pairs of mobile users and receiving antennae. For$i=1,\mathrm{\dots },n$, mobile user $i$ transmits radio signal to antenna$i$. Mobile user $i$ transmits at power $P_i$. The radio channelattenuate the signal and user $i$’s signal is received at antenna$i$ with power $G_{ii}{P}_i$, where $G_{ii}$ denote the channelgain. The radio signals also interfere each other. At antenna $i$,the interference due to user $j$ has power $G_{ij}{P}_j$. Thereceiver noise power at antenna $i$ is denoted by $n_i$. Thesignal to interference plus noise at receiver $i$ is

$${\mathrm{\Gamma }}_i=\frac{G_{ii}{P}_i}{{\sum }_{j\ne i}{G}_{ij}{P}_j+n_i}$$

To guarantee the quality of received signal, it is required thatthe signal to interference plus noise ratio ${\mathrm{\Gamma }}_i$ is equal toa predefined constant ${\gamma }_i$ for all $i$. Given ${\gamma }_i$,$i=1,\mathrm{\dots },n$, we want to find $P_1,\mathrm{\dots },P_n$ such that theabove equation holds for $i=1,\mathrm{\dots },n$. Let $A$ be the $n\times n$ matrix with zero diagonal and $(i,j)$-entry$(G_{ij}{\gamma }_i)/G_{ii}$ for $i\ne j$. We want to solve

$$(I-A)𝐩=𝐧$$

where $𝐩={(P_1,\mathrm{\dots },P_n)}^T$ is the power vector and$𝐧={(n_i{\gamma }_i/G_{ii})}_{i=1}^n$. The matrix $I-A$ isa Z-matrix, since all $G_{ij}$ and ${\gamma }_i$ are positiveconstants. The required power vector is ${(I-A)}^{-1}𝐧$ if$I-A$ is invertible. We also required that the components of$𝐩$ to be positive as power cannot be negative. Theresulting power vector ${(I-A)}^{-1}𝐧$ has positivecomponents if ${(I-A)}^{-1}$ is a non-negative matrix. In such case,$I-A$ is an M-matrix.