# The Impact of Impulsive Traffic on Cellular Internet of Things Network Performance Indicators
## 5 Mathematical Model  
In the cellular Internet of Things, cellular network operators use their infrastructure to transfer the data of devices (things) along with the data of cellular users. This necessarily requires finding an effective way to share resources between these things and users so that the quality of service is guaranteed to be maintained for both types. The most widely used cellular IoT technology is NB-IoT. The 3 GPP proposed the in-band deployment pattern in an attempt to solve the problem of resource sharing between LTE and NB-IoT devices, but no specific method has been defined in order to allocate the resources. When allocating resources between these two types of users, it is important to study the traffic characteristics of each type in order for the solutions applied to be more beneficial and achieve a higher quality of service for each of them. The time intervals between successive arrivals of requests in the Erlang model have an exponential distribution. Because the appearance of requests is random, they may appear more frequently at some intervals and less frequently at others. However, this property of the Poisson flow is insufficient to reflect the impulsive nature of request receipt, which is inherent in the operation of some communication systems. The pulsations caused by the appearance of requests for information transmission resources range in character. These can be concentrations that reflect the specifics of the formation of a stream of requests for \begin{table}
\begin{tabular}{c c} \hline \hline
**Abbreviation** & **Definition** \\ \hline LoS & Line of sight \\ \hline LPWA & Low-Power Wide-Area \\ \hline LPWAN & Low Power, Wide Area Networks \\ \hline LTE & Long Term Evolution \\ \hline MIB & Master information block \\ \hline NB & NarrowBand \\ \hline NB-IoT & Narrowband Internet of Things \\ \hline NOMA & Non-orthogonal multiple access \\ \hline NPBCH & Narrowband physical broadcast channel \\ \hline NPDCCH & Narrowband physical downlink control channel \\ \hline NPDSCH & Narrowband physical downlink shared channel \\ \hline NPRACH & Narrowband physical random access channel \\ \hline NPSS & Narrowband primary synchronization \\ signal \\ \hline NPUSCH & Narrowband Uplink Shared Channel \\ \hline NRS & Narrowband reference signal \\ \hline NSSS & Narrowband secondary synchronization signal \\ \hline NTN & Non-terrestrial network \\ \hline OFDM & Orthogonal frequency-division multiplexing \\ \hline PSM & Power saving mode \\ \hline QoS & Quality of Service \\ \hline QPSK & Quadrature Phase Shift Keying \\ \hline SCFDMA & Single Carrier Frequency Division Multiple Access \\ \hline UE & User Equipment \\ \hline UL & Uplink \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._the transmission of real-time service traffic or the characteristics of the grouping of data traffic information messages. In CIoT scenarios, the sensors measure the required parameters and send them in a session series using cellular infrastructure to the data processing centers to make the appropriate decision. These sensors use concentrators and relays to accumulate and amplify signals. So that files containing measurement information can arrive in bursts simultaneously, this is a critical issue to be aware of. Let us construct a mathematical model describing the allocation of channel resources, considering the impulsive character of request arrival. Unlike the Erlang model, we offer a two-level approach for reproducing the incoming traffic flow here. The analytical model of an LTE cell simultaneously serving IoT users as well as cellular users, considering the impulsive nature of IoT requests, is shown in Figure 5.  
Let \(\upsilon\) denote the total number of resource units that the LTE cell can provide to IoT devices. The arrival of a group of requests obeys the Poissonian model, with group arrival intensity denoted by \(\lambda\). The time required to serve one request is exponentially distributed, with a mean duration equal to \(\mu\).