# On the Optimisation of Practical Wireless Indoor and Outdoor Microcells Subject to QoS Constraints
## 3 System Model of WIOMs and Network Optimisation Problem  
Figure 1 illustrates a typical WIOM network where the microcells are deployed either within a house/building or in large exterior area of many buildings in a 2D grid. (In this work, only a simplistic network model is considered; however, it can be extended to a more general network with geographical database (or geodatabase) following the same approach with the proposed cell planning algorithm which will be shown in the following section.) From a practical point of view, it is assumed there are  
a total of \(N\) candidate locations that are restricted for placing cell sites in an area under investigation and the maximum number of cell sites that can be employed is \(K_{\max}\left(K_{max}\leq N\right)\). Let \(\mathfrak{S}\) denote the set of coordinates of these candidate cell sites, i.e., \(\mathfrak{S}=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{N},y_{N})\}\).  
To meet the QoS requirements, it is required that \(M\) areas in the WIOM network are all covered. For simplicity, the \(m\)-th area, \(m=1,2,\ldots,M\), is represented by a point \((x_{e_{m}},y_{e_{m}})\) in a set \(\mathfrak{S}_{E}=\{(x_{e_{1}},y_{e_{1}}),(x_{e_{2}},y_{e_{2}}),\ldots,(x_{e_{M }},y_{e_{M}})\}\). The network optimisation problem (NOP) can be formulated as  
\[\min_{\begin{subarray}{c}k\in[1,K_{\max}]\\ \mathfrak{S}_{i_{k}}\subset\mathfrak{S}\end{subarray}}k, \tag{1}\]  
\[\text{s. t. }P(x_{e_{m}},y_{e_{m}})\geq P_{\min},\forall m=1,2\ldots,M, \tag{2}\]  
where \(\mathfrak{S}_{i_{k}}\) is a subset of \(\mathfrak{S}\) consisting of \(k\) elements, \(P(x_{e_{m}},y_{e_{m}})\) is the received power at \((x_{e_{m}},y_{e_{m}})\) and \(P_{\min}\) is the minimum received power requirement in the considered area. Here, the received signal strength indicator is based on the QoS requirements [50]. Note that \(\mathfrak{S}\) has a total of \(2^{N}\) subsets and there are \(\binom{N}{k}\) ways to choose a subset of \(k\) elements.  
The NOP in Equation (1) is to find the minimum number of cell sites, i.e., \(k_{\min}\), in the range \([1,K_{\max}]\) as well as their optimal position, i.e., \(\mathfrak{S}_{i_{k_{\min}}}\) among the available positions in the set \(\mathfrak{S}\). The received power at each point \((x_{e_{m}},y_{e_{m}})\) in the set \(\mathfrak{S}_{\mathfrak{E}}\) needs to meet the QoS requirement in Equation (2), which also reflects the required coverage area.  
In order to solve the above NOP, it is worth mentioning that an exhaustive search can be employed, namely exhaustive-search based cell positioning (ESCP). However, such an ESCP algorithm requires a high computational complexity to perform a number of iterations and simulation runs over all possible candidate locations with different numbers of cell sites, especially when considering a large number of candidate cell positions. Therefore, in the following section, we aim at proposing a novel algorithm for cell site searching, which will be shown in Remark 1 that a lower complexity can be achieved with the proposed algorithm when compared to the ESCP algorithm.  
Figure 1: System model of Wireless indoor and outdoor microcells (WIOMs).