## 2 Related Work
: _Let \(R\) represent a given road network, and \(V\) be the set of vehicles traveling \(R\). Let \(C\subset V\) be the set of vehicles experiencing connection during, at least, \(\rho_{1}\) percent of the trip duration. A deployment is \(\Delta_{\rho_{2}}^{\rho_{1}}\) whenever \(\frac{|C|}{|V|}\geq\rho_{2}\)._  
In addition, the Delta Network can be used to plan the vehicular network in order to meet predefined QoS levels for reaching requirements from vehicular applications that the network administrator intends to make available in the network. There is a wide range of vehicular applications already proposed in the literature, and many others still to be developed, where each application tends to demand a specific level of QoS from the vehicular network. In this sense, Delta Network can also be used to support network planning, defining the expected connectivity of vehicles in terms of their travel time. _Using the Delta Network to Reach a Specific Performance Target_  
In previous works [29; 52], we have turned our attention on developing solutions for Delta in order to reach one single target performance. Basically, our goal was to minimize the number of roadside units in order to achieve a given combination of \(\rho_{2}\) percent of vehicles connected to roadside units during \(\rho_{1}\) percent of the trip duration, i.e., the optimal layout of roadside units for one single point composing the Delta curve (such as the red point in Figure 1). Solving this problem is interesting because it allows the network designer to build the network infrastructure guaranteeing such performance level in order to deploy a set of specific vehicular applications demanding the aforementioned connectivity. Before continuing the discussion on the Delta Network, we describe a strategy for representing road networks with arbitrary topology. We model road networks as a grid-like structure. Basically, we divide the city into a set of same-sized urban cells (road partitions), as depicted in Figure 2. The dimensions of the urban cell may vary according to the desired accuracy, and the expected range of coverage of roadside units. An Integer Linear Program formulation for this problem is presented in [52]. Since obtaining the exact solution requires too much computational effort, we can only solve small instances using this formulation. In order to solve large instances, we propose a greedy strategy named _Delta-g_ in [53]. When we apply the _Delta-g_ heuristic considering the Vehicular Mobility Trace of the city of Cologne Vehicular Mobility Trace of the city of Cologne, Germany, available at: [[http://kolntrace.project.citi-lab.fr/](http://kolntrace.project.citi-lab.fr/)]([http://kolntrace.project.citi-lab.fr/](http://kolntrace.project.citi-lab.fr/))) [54] composed of \(10,000\) seconds of traffic and \(75,515\) vehicles, we obtain the following area to be covered according to pairs of \(\rho_{1}\) (\(y\)-axis) and \(\rho_{2}\) (\(x\)-axis). Figure 1: Delta governs the combinations of \(\rho_{1}\) (share of trip under coverage) and \(\rho_{2}\) (share of vehicles meeting coverage). Source: [PERSON] et al. [29]. In order obtain this result, we run the heuristic for each pair of \(\{\rho_{1}\),\(\rho_{2}\}\) presented above. In the following section, we turn our attention to finding global solutions (instead of solutions for one single pair). Complimentary to this, we highlight that Figure 3 is an adaption of the result presented by our team in [53]. However, in [53] we present the percentage area to be covered in terms of the whole city of Cologne, while in Figure 3 we present the percentage area to be covered considering only areas presenting traffic.