## 2 General Model
### Sinks  
Each sink belongs to only one IoT-SP. They are responsible of transmitting all the data collected by sensors in a WSN to their IoT-SP server. They are the clients of the wireless connectivity service offered by the OP. The number of IoT-SP\({}_{i}\) Sinks is \(N_{j}\), where \(N_{j}\gg 1\) (\(j=1,2\)), and \(N_{1}+N_{2}=N\).  
In order to model the utility perceived by the sinks that subscribe to the OP we use a quality function \(Q\) based in [29; 30; 31; 32; 33], which evaluates the service offered by the OP as a latency based service [34]:  
\[Q\equiv c\left(\frac{T}{\tau}\right)^{-1}, \tag{1}\]  
Figure 1: Analyzed scenario with all the actors of the market. Each IoT-SP collects its sensing data through an OP and transmits it to a server (srv) where it is processed in order to offer a service to the Sensor Data Users.  
where \(c>0\) is a conversion factor and \(T/\tau\) is the mean sensing-data-unit service time normalized by the mean sensing-data-unit transmission time \(\tau=\frac{1}{\mu}\), that is the minimum possible value of \(T\). Note that \(Q\) decreases when the service time \(T\) increases, which means that the users perceive a worst quality when the delay of the network increases. We have chosen this function due to its ability to model the congestion in the wireless network, which makes it suitable for many IoT scenarios with delay constraints [35]. This quality function also has the ability to model different kinds of users through the value of \(\tau\) and different queueing systems throught \(T\), however, in this model we consider homogeneous sinks, given that we study the competition in a single service provision. We model the OP service as a M/M/1 system, and compute the mean service time \(T\)[36] as  
\[T=\frac{\tau}{1-\tau\lambda}. \tag{2}\]  
The utility function models the perception that sinks have about the OP connectivity service. We propose a utility function for the sinks that subscribe to the OP as the difference between the quality perceived by the sinks and the price charged by the operator, also called compensated utility, which is a function widely used in economics and telecommunications [37; 38; 39; 40; 29]  
\[U_{s}\equiv Q-p=c\left(1-x_{1}rN\tau\right)-p, \tag{3}\]  
where we have re-written the arrival rates as the traffic generated by all the sinks being served \(\lambda=x_{1}rN\), \(r\) is the sensing-data-unit generation rate of one sink, \(p\) is the price charged by the OP to each IoT-SP\({}_{j}\) (\(j=1,2\)) when its sinks transmit one sensing-data-unit and \(x_{1}\) is the fraction of sinks being served by the OP. The utility must be positive \(U_{s}\geq 0\), or equivalently, the price charged by the OP should not be higher than the service value perceived by the sink, otherwise the sink will not subscribe to the service. Note that all the sinks, whichever IoT-SP they belong to, perceive the same utility, which means that the fraction of sinks served by the OP is the same for all the IoT-SPs. The distribution of sinks in the system is described by the vector \(X_{s}=\left(x_{0},x_{1}\right)\), where \(x_{0}\) and \(x_{1}\), are the fraction of sinks not being served and being served by the OP respectively and \(x_{0}+x_{1}=1\).