# Markov-Modulated Poisson Process Modeling for Machine-to-Machine Heterogeneous Traffic
## 3 M2M Heterogeneous Model  
To recall, the conditions and thresholds were discussed in many previous papers that we published before such as [11; 24; 25], especially when we dealt with source traffic modeling that requires having thresholds since packets that replicate real traffic behavior (such as voice, video, and data traffic) at different sizes and intervals are generated. However, during a disaster, it is expected that there will be a huge number of M2M devices that may send their payloads simultaneously. We understand that in these situations, source traffic models become very difficult to execute, which makes the usage of aggregated traffic modeling necessary. Therefore, in this paper, we used aggregated traffic modeling, in which we represent each traffic with an average arrival rate (\(\lambda\)) and completion rate (\(\mu\)) for both traffic M2M and H2H. Then, we represent the whole system as a linear system and mathematical balance equations that overcome the limitation of the number of devices attempting to access the network simultaneously. In our previous paper [24], we focused on a constant average arrival rate (\(\lambda\)), in which we strongly assume to have either one H2H arrival or one M2M arrival in the smallest discrete time period. However, in reality, M2M devices may send their payloads in a synchronization manner. The synchronization behavior of M2M devices leads to different M2M storms that form accumulative traffic over the time cycle, which requires deeper analysis and a more accurate representation of different M2M traffic. Our new contribution is based on the realistic behavior of M2M devices rather than ideal and theoretical behavior by turning the constant arrival rate (\(\lambda\)) into an average arrival rate, which varies over time. We are encouraged in this manuscript to consider a Poisson process modulated rate \(\lambda_{\text{(m)}}\) for M2M traffic modulated with a \(\beta\) distribution over a time-space \(\Delta t\) and determined by a Markovian chain state \(s(m)\) (\(m\) indicates a Markov state index varying from 0 to \(M\), with \(M\) being the total number of states), as shown in Figure 2.  
Figure 2: An interpretation of MMPP for M2M traffic, where \(\lambda_{\text{(m)}}\) is the mean arrival rate of a Poisson process. In Figure 2, an MMPP model is used as a framework to accurately model M2M traffic sources by analyzing an event when a huge number of M2M devices behave in a synchronized manner [26; 27]. For H2H traffic, our MMPP model has a constant average arrival rate \(\lambda_{\text{(h)}}\). As for M2M traffic, it has a variable Poisson process-modulated rate \(\lambda_{\text{(m)}}\); \(0\leq m\leq M\). The overall average rate \(\lambda_{\text{(g)}}\) of the MMPP is given by:  
\[\lambda_{\text{(g)}}=\sum_{\text{m}=0}^{M}\lambda_{\text{(m)}}\tau_{\text{(m)}} \tag{1}\]  
where \(\tau_{\text{(m)}}\) is the probability for the system to be in a certain state (\(m\)) at a given time space \(\Delta t\), with the following constraint:  
\[\sum_{\text{m}=0}^{M}\tau_{\text{(m)}}=1\]  
(M + 1) states would be a simple example of M2M devices modeled using an MMPP model, where the first state, \(S_{\text{(0)}}\), represents a \"normal state\". The remaining states show a varying arrival rate over various \"emergency states\" over time \(\lambda_{\text{(m)}}\) until they arrive at the \"worst-scenario state\" (where m = M). A state space and a transition matrix define the process (i.e., a stochastic or probability matrix). As a result, the likelihood of changing states is contingent upon the rate at which each service is completed. The notion is illustrated in Figure 3, where the transition probabilities between two connected states, say, \(S_{\text{(m)}}\) to \(S_{\text{(m+1)}}\), are represented by the variables \(P_{\text{(m,m+1)}}\). One of the three states listed below applies to the system:  
1. The normal state \(S_{\text{(0)}}\) possesses the following equilibrium relationship and incorporates the initial state: \[p(0,1)\pi(0)-p(1,0)\pi(1)=0\] (2)
2.