# Markov-Modulated Poisson Process Modeling for Machine-to-Machine Heterogeneous Traffic
## 3 M2M Heterogeneous Model
Emergency states \(S_{\text{(m)}}\) happen when m groups combine to produce an accumulative storm and send their data collectively: \[\lambda_{\text{(m)}}=\sum_{\text{j}=1}^{m}\lambda_{\text{(j)}}\] All emergency states \(S_{\text{(m)}}\) follow the following equilibrium relationship: \[p(m-1,m)\pi(m-1)-[p(m,m-1)+p(m,m+1)]\pi(m)+p(m+1,m)\pi(m+1)=0\] (3) where \(\pi_{\text{(m)}}\) is the probability to be in the state \(S_{\text{(m)}}\), and m \(\in[1,2,3,\ldots,M-1]\). 3. The worst-scenario state \(S_{\text{(M)}}\) occurs when all groups dispatch their data simultaneously: \[\lambda_{\text{(M)}}=\sum_{m=1}^{M}\lambda_{\text{(m)}}\] The final state, which is included in \(S_{\text{(M)}}\), has the following equilibrium relationship: \[p(M-1,M)\pi(M-1)-p(M,M-1)\pi(M)=0\] (4)  
Figure 3: MMPP model where each state \(S_{\text{(m)}}\) in the Markov chain represents a mean arrival rate \(\lambda_{\text{(m)}}\), and m is the total number of states. The transition probabilities between states of the chain, from \(S_{\text{(m)}}\) to \(S_{\text{(m+1)}}\), are represented by the variables \(p_{\text{(m,m-1)}}\). The general equation that follows can be used to summarize the three equations mentioned above:  
Figure 2:0  
where:  
* \(\alpha=0\) in the \"normal state\", otherwise \(\alpha=1\);
* \(\xi=0\) in the \"worst-scenario state\", otherwise \(\xi=1\). The state probability \(\pi_{\text{(m)}}\) resides in the state probability vector \(\Pi\) according to the following:  
\[\Pi=\left( \begin{array}{cccccc}\pi_{(0)}&\pi_{(1)}&.&.&.&\pi_{(M)}\end{array}  \right)^{T} \tag{6}\]  
However, a state transition matrix \(\mathbb{P}\) can be created by condensing the transition probabilities:  
\[= \begin{bmatrix}p_{(0,1)}&-p_{(1,0)}&0&.&.&.&0\\ p_{(0,1)}&-\left[p_{(1,0)}&+p_{(1,2)}\right]&p_{(2,1)}&.&.&.&0\\ 0&p_{(1,2)}&-\left[p_{(2,1)}&+p_{(2,3)}\right]&.&.&p_{(3,2)}\\.&.&.&.&.&.&\\.&.&.&.&.&.&\\.&.&.&.&.&.&\\ 0&0&0&.&.&.&-p_{(M,M-1)}\end{bmatrix} \]  
As a result, the equations the \((M+1)\) states can be represented as a linear system in the following form:  
\[P\Pi=0 \tag{7}\]  
The linear system in (7) should be solved while respecting the following two constraints:  
\[\sum_{m=0}^{M}\pi_{(m)}=1 \tag{8}\]  
\[0<\pi_{(m)}<1 \tag{9}\]  
\[\forall\ m\in\{0,1,2,3,...,M\}\]  
Ultimately, we can compute \(\pi_{(0)}\) to \(\pi_{(\text{M})}\) by resolving the linear system mentioned above in (7). Also using the equation in (1), we can calculate \(\lambda_{(\text{g})}\). On one hand, emergencies such as natural catastrophes, terrorist attacks, or various accidents are not predictable in real life. Based on that, in our earlier work [28], we created a case study based on a few use cases found in [29] and 3 GPP technical reports [30]. This was necessary due to the imprecise scenarios, dearth of data, and behavior analysis for M2M devices in emergency situations. By summarizing our previous case study, in Table 1, we can extend it to build a heterogeneous M2M case study represented with three accumulative storms (\(M=3\)). On the other hand, Markov chains can be used to simulate complex situations and to forecast the likelihood of future events. Markov chains were used to forecast some \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Group \#**} & \multirow{2}{*}{**M2M**} & **Message Size** & **Rate** & **Number of Devices** & **Storm Rate** & **Number of Storms** \\  & & **Device Type** & **(Bytes)** & **(msg/Day)** & **(Kilo)** & **(Klps)** & **(Storm/Day)** \\ \hline
1 & Asset tracking & 50 & 100 & 20 & 1600 & 500 \\
2 & Assisted & 100 & 8 & 20 & 3200 & 40 \\  & Environment & 200 & 24 & 20 & 6400 & 120 \\  & monitoring & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of M2M storms when \(M=3\). upcoming natural disasters in the real world, such as earthquakes, tsunamis, and even the spread of disease [30]. The state transition matrix becomes the following:  
\[P=