# A LoRa-Based Linear Sensor Network for Location Data in Underground Mining
## 4 Analysis  
Although implementation and field trials gives us some confidence that the system will be effective, determining how scalable it is, is difficult with only a few nodes. Consequently, to explore the sizing of the system we turn to mathematical and simulation models.  
We can model this system using conventional Markov methods. Each relay receives messages from tags. We make the simplifying assumption that arrivals are uncorrelated and the time between them is exponentially distributed. Each relay, upon receipt of a message from a tag, in order to avoid collisions, waits a random amount of time before broadcasting the message to its neighbouring relays to forward. During this time the relay is unable to receive other messages. The time includes both the transmission time and an additional randomly distributed interval to reduce the risk of collisions with other messages. With these assumptions we can model the system as a Markov chain as shown in Figure 2.  
The number of relays in the system is denoted by \(n\). Each relay \(k\) receives messages from tags in its locality. We assume each tag generates messages with a mean rate of \(\lambda_{t}\). The number of tags at each relay is \(N_{k}\). The aggregate traffic into relay \(k\) is \(\lambda_{k}=N_{t}\lambda_{t}\). In this analysis we assume that the number of tags is the same for each relay and consequently, the rate at which messages enter  
Figure 1: Field Trials Probability of Successful Delivery.  
Figure 2: Markov chain model of system.  
each relay is also the same, which we denote by \(\lambda\). We make the simplifying assumption that \(1/\lambda\) is exponentially distributed.  
We denote the rate of traffic between relays \(k\) and \(k+1\) by \(\gamma_{k}\). We can determine the following recursive relationship defining the traffic leaving each relay :  
\[\gamma_{k+1}=P_{a}(\gamma_{k}+\lambda) \tag{1}\]  
where \(P_{a}\) is the probability a relay forwards a message and \(\gamma_{0}=0\).  
Any traffic that is transmitted by either a neighbouring relay or a tag will be lost if the relay is currently transmitting or waiting to transmit. We can model this behaviour as an M/M/1/1 queue. In this queue the probability of blocking is:  
\[P_{b}=\frac{\rho}{1+\rho} \tag{2}\]  
where \(\rho\) is the queue utilisation. Consequently, the probability of admission is:  
\[P_{a}=1-P_{b}=\frac{1}{1+\rho} \tag{3}\]  
We now need to determine \(\rho\). For an M/M/\(1/1\) queue \(\rho\) is:  
\[\rho=\frac{\text{arrival rate}}{\text{service rate}} \tag{4}\]  
The service rate is the inverse of the time spent randomly waiting after the medium goes silent plus the time taken  
\[Pr(\text{success})=(\frac{\mu}{n}-\frac{\mu^{n+1}}{n(\mu+n\lambda)^{n}})/(n\lambda). \tag{7}\]  
Figure 3 shows the predictions from the above model and from a simulation of the system. As expected there is good agreement between the analytical model and the simulation when the load on the network is light but, also as expected, the agreement is less pleasing as the load increases. Nevertheless, the agreement is sufficient to explore the scalability of the system.