# Virtualized MME Design for IoT Support in 5G Systems
## 6 Dimensioning  
The dimensioning of the schemes presented above requires a model that provides an estimation of the service response time as a function of the processing resources. To do that, we assume the vMME model proposed in [32], which is based on the model of a typical cloud processing chain [35]. This proposal uses [PERSON]'s open queuing network (see Figure 9) to model the vMME architecture (explained in Section 3). In this approach, the SDB and the FE tiers are modeled by M/M/1 queues, and the SL tier is modeled by an M/M/m queue. [PERSON]'s theorem states that the numbers of messages in the system's queues are independent of the other queues, and consequently, the service response time of the complete system is equal to the sum of the service response time of the queue in each tier. In the present work, we also assume that the SDB and the FE can be replicated, and therefore, we also model SDB and FE tiers by an M/M/m queue. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scheme** & **Tier** & **Front End (FE)** & **Service Logic (SL)** & **State Database (SDB)** \\ \hline Baseline Scheme (BS) & \(\lambda_{FE}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{smooth}+\lambda_{mM2M}^{smooth}\) & \(\lambda_{SL}=\lambda_{TE}\) & \(\lambda_{SDB}=\lambda_{SL}\) \\ \hline Overdimensioned Scheme (GS) & \(\lambda_{FE}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{peak}+\lambda_{mM2M}^{smooth}\) & \(\lambda_{SL}=\lambda_{TE}\) & \(\lambda_{SDB}=\lambda_{SL}\) \\ \hline Traffic separated & \(\lambda_{FE}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{peak}+\lambda_{mM2M}^{smooth}\) & \
\begin{tabular}{c} MBB \(\lambda_{SL}=\overline{\lambda}_{MBB}\) \\ mM2M \\ \end{tabular} & \(\lambda_{SDB}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{smoothpeak}+\lambda_{mM2M}^{ peak}\) \\ Scheme (TS) & \(\lambda_{FE}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{smooth}+\lambda_{mM2M}^{ peak}\) &
\begin{tabular}{c} MBB \(\lambda_{SL}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{smoothpeak}+\lambda_{mM2M}^{ peak}\) \\ mM2M \\ \end{tabular} \\ \hline Traffic Shaper Scheme (SS) & \(\lambda_{FE}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{peak}+\lambda_{mM2M}^{smooth}\) & \(\lambda_{SL}=\overline{\lambda}_{MBB}+\lambda_{mM2M}^{smooth}+\lambda_{mM2M}^{ peak}\) & \(\lambda_{SDB}=\lambda_{SL}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Target arrival rate at each tier for the proposed schemes. Figure 9: Queue model of the vMME. Given a target arrival rate, the goal of our vMME dimensioning is to determine the minimum number of instances required at each tier \(j\) to guarantee the mean response time budget \(\overline{D}_{j}\). Since we consider Jackson's open queuing network, the mean response time of the system \(\overline{T}\) can be computed as:  
\[\overline{T}=\sum_{j}\overline{T}_{j} \tag{16}\]  
where \(\overline{T}_{j}\) is the mean response time at each tier \(j\in\{FE,SL,SDB\}\). Since we assume that each tier is modeled by an M/M/m queue, it holds that:  
\[\overline{T}_{j}=\frac{1}{\mu_{j}}+\frac{C(m_{j},\rho_{j})}{m_{j}\cdot\mu_{j}- \lambda_{j}} \tag{17}\]  
where \(\rho_{j}=\frac{\lambda_{j}}{\mu_{j}}\), \(\mu_{j}\) is the service rate of one tier instance, \(\lambda_{j}\) is the target arrival rate considered for dimensioning at tier \(j\) (see Table 2), \(m_{j}\) is the number of instances of the tier and \(C(m_{j},\rho_{j})\) is [PERSON]'s C formula. \(C(m_{j},\rho_{j})\) represents the probability that an arriving packet has to wait in the queue of the tier because all of the instances are busy, and it has the following expression:  
\[C(m_{j},\rho_{j})=\frac{\left(\frac{(m_{j},\rho_{j})^{m_{j}}}{m_{j}!}\right) \cdot\left(\frac{1}{1-\rho_{j}}\right)}{\sum_{k=0}^{m_{j}-1}\frac{(m_{j},\rho_ {j})^{k}}{k!}+\left(\frac{(m_{j},\rho_{j})^{m_{j}}}{m_{j}!}\right)\cdot\left( \frac{1}{1-\rho_{j}}\right)} \tag{18}\]  
The processing times of the FE, SDB and output interface are constant. However, the processing time of an SL is different for each control message [32].