# Joint Pricing, Server Orchestration and Network Slice Deployment in Mobile Edge Computing Networks
## 3 System Model
### Delay Model  
When performing slice deployment, the precise location of each user is unknown. The tenants and InP can only know the number of users expected to be served in the region. To address this, ref. [17] employed a subarea partitioning technique where the region is divided into \(Q\) convex subareas. When performing slice deployment, resources are allocated to each subarea. The region may be divided into rectangles of equal area or into subareas with an equal average number of users. For example, in Figure 1, we divide a service region into four rectangular parts of equal area with dashed lines. We use \(\mathcal{Q}=\{1,\ldots,q,\ldots,Q\}\) to represent the set of subareas. Let \(u_{n,q}\) represent the number of users that slice \(n\) plans to serve in subarea \(q\). We denote the slice \(n\) in subarea \(q\) by \((n,q)\). Moreover, \(w_{n,q}^{m}\), \(e_{n,q}^{v}\) and \(s_{n,q}^{v}\) are used to represent the radio, computational and storage resources allocated to \((n,q)\) by base station \(m\) and server \(v\). Note that the distribution of these resources within \((n,q)\) is a matter addressed by intra-slice resource management, which is beyond the scope of this paper.  
The total data rate of \((n,q)\) can be expressed as [17]:  
\[R_{n,q}=\sum_{m=1}^{M}w_{n,q}^{m}\text{log}_{2}\big{(}1+SNR_{m,q}\big{)}, \tag{1}\]  
where \(SNR_{m,q}=\frac{p_{m,q}^{tx}+C_{m,q}^{tx}+G_{m,q}^{rx}-PL\big{(}D_{m,q}\big{)}} {\sigma^{2}}\) is the signal-to-noise ratio (SNR). \(P_{m,q}^{tx}\), \(G_{m,q}^{tx}\), \(P_{L}\big{(}D_{m,q}\big{)}\) and \(\sigma^{2}\) are the transmitted power, antenna gain of the transmitter and receiver, path loss and noise power, respectively [17]. \(D_{m,q}\) denotes the distance between base station \(m\) and the center point of subarea \(q\). Because this paper deals with the problems faced in the network planning phase, does not involve real-time resource scheduling and does not address the association of specific users with sub-channels, we do not consider the co-channel interference in the wireless network.  
The transmission delay \(d_{n,q}^{tran}\) can be calculated by [35]  
\[d_{n,q}^{tran}=\frac{l_{n}u_{n,q}}{R_{n,q}}, \tag{2}\]  
where \(l_{n}\) is the size of the computational task for the users of slice \(n\). The propagation delay \(d_{n,q}^{prop}\) is positively correlated with the distance between the base station and the server [36], which can be denoted as  
\[d_{n,q}^{prop}=\frac{1}{\sum_{m=1}^{M}y_{n,q}^{m}}\sum_{m=1}^{M}y_{n,q}^{m} \sum_{v=1}^{V}a_{v,m}D_{v,m}t_{0}, \tag{3}\]  
where \(y_{n,q}^{m}\) indicates whether \((n,q)\) has any users associated with base station \(m\). \(D_{m,v}\) is the distance between base station \(m\) and server \(v\). \(t_{0}\) is the propagation delay generated per unit distance. Furthermore, the computational delay can be expressed as [35]\[d_{n,q}^{comp}=\frac{l_{n}b_{n}u_{n,q}}{\sum_{v=1}^{V}e_{n,q}^{v}}\,, \tag{4}\]  
where \(b_{n}\) is the computational demand to process each bit (CPU cycles/bit). The total delay can be expressed as \(d_{n,q}=d_{n,q}^{ttm}+d_{n,q}^{prop}+d_{n,q}^{comp}\)[12].