# Virtualized MME Design for IoT Support in 5G Systems
## 4 Traffic Models
### Machine to Machine Traffic Model  
We consider two types of M2M devices: massive M2M (mM2M) devices and low latency M2M (IM2M) devices. Let \(p_{h}\), where \(h\in\{mM2M,1M2M\}\), denote the percentage of each type of M2M devices, and let \(r\) be the ratio of M2M devices per MBB UE. Then, the number of M2M devices of each type, \(N_{h}\), is given by:  
\[N_{h}=N_{MBB}\cdot\tau\cdot p_{h} \tag{3}\]  
For both types of M2M devices, we assume a traffic model based on the transmission of reports. Particularly, the traffic model of any M2M device has two possible states: _active_ and _alarm_ (see Figure 4). M2M devices generate small data packets following a Poisson process in both states. In the _active_ state, packets are generated infrequently, with a mean rate \(\overline{\lambda}_{active}\). When an event happens, a percentage of massive or low latency M2M devices, denoted by \(a_{h}\) where \(h\in\{mM2M,1M2M\}\), change to the _alarm_ state. During the _alarm_ state, M2M devices generate packets more frequently, with a mean rate denoted as \(\overline{\lambda}_{alarm}\). We assume that the event has a fixed time duration \(t_{e}\), and the probability that in a certain instant of time an M2M device is in the _alarm_ state is denoted as \(p_{h}^{e}\), where \(h\in\{mM2M,1M2M\}\). After the event, all devices in the _alarm_ state return to the _active_ state. Massive M2M device events occur at certain instants in time caused by a large number of mM2M devices synchronizing their report transmissions (see Section 2.3). Some of these events are periodic, whereas others are not. For simplicity, in our model, we assume that mM2M synchronization events take place at periodic instants of time. On the other hand, IM2M events are caused by alarm situations; that is, they are unpredictable. Consequently, in our model, IM2M events occur at random instants in time. Furthermore, we assume that the percentage of M2M devices participating in a synchronization event, \(a_{h}\), is a discrete random variable. By appropriately choosing \(\overline{a}_{h}\), \(\overline{\lambda}_{active}\) and \(\overline{\lambda}_{alarm}\), we can model the traffic peaks (as explained in Section 2.3). Let us analyze the mean generation rate of the MO-CPCEO procedure per M2M class. As there is one control message per MO-CPCEO procedure (see Section 3.2), the mean generation rate of the MO-CPCEO procedure also defines the mean arrival rate of the MO-CPCEO control messages per M2M class to be processed by the MME. Let us denote this mean arrival rate as \(\overline{\lambda}_{h}^{v}\). The parameter \(v\) defines the time interval used for averaging the arrival rate. Particularly, \(v\) can take three possible values \(\{active,event,maxevent\}\), which corresponds to the three following different traffic situations (see Figure 5):  
* _Active_: intervals of time in which all M2M devices are in the _active_ state; that is, in the absence of events. * _Event_: intervals of time in which an event takes place; that is, intervals of time where \(N_{h}\times a_{h}\) devices are in the _alarm_ state. The averaging is carried out over all possible events, each with a given \(a_{h}\). * _Maxevent_: time interval in which the largest event takes place; that is, the interval of time where \(N_{h}\times max(a_{h})\) devices are in the _alarm_ state. Figure 4: M2M device state transition diagram. Figure 5: Parameters for the M2M traffic model. Then, for each averaging time interval, \(\overline{\lambda}_{h}^{v}\) can be calculated as:  
\[