## 2 Related Work
Modeling the deployment are a Maximum Coverage Problem is also considered. [PERSON] et al. [48] study the placement of the roadside units into an urban area. The authors use a realistic data set and propose modeling the placement as a Knapsack Problem (KP) and also as a Maximum Coverage Problem (MCP-g). The heuristic MCP-g models the deployment of roadside units as a maximum coverage problem, and assumes previously knowledge of all vehicles trajectories. Complimentary, when we intend to maximize the number of distinct vehicles contacting the infrastructure without identifying individual vehicles, we may rely on migration ratios of vehicles between adjacent locations of the road network as presented in [49]. Analytic studies are also found in the literature. [PERSON] et al. [50] address cellular systems as the most feasible solution in the short term to collect information messages from vehicles to a remote control center. The paper proposes a mathematical model to evaluate the impact of the envisioned service on cellular systems capacity and coverage in simplified scenarios. Results show that the acquisition of small and frequent packets from vehicles is affected by interference more than other services, such as the voice service. Finally, the work presented by [PERSON] et al. [51] serves as inspiration for the Delta Network reviewed in Section 3. Authors propose Alpha Coverage to provide worst-case guarantees on the interconnection gap measured in terms of traveled distance. A deployment of roadside units is considered \(\alpha\)-covered if any path of length \(\alpha\) on the road network meets at least one roadside unit. Delta Network: Measuring the Network Performance Using the Relation between the Connected Time and Trip Duration  
Measuring the connected time is a way of characterizing vehicular networks based on the QoS experienced by users. Users receiving more connection tend to receive higher QoS by the network. Hence, measuring the connection time proportionally to the distance traveled by vehicles seems an interesting way of characterizing vehicular networks. Such measurement is given by the Delta Network [29]. Delta measures the connection time according to the trip duration, and we express it as \(\Delta_{p_{2}}^{\rho_{1}}\). Parameter \(\rho_{1}\) represents the connection duration factor. For instance, should vehicles be connected during 5% of the trip, then \(\rho_{1}\) must be 0.05. Complementary, \(\rho_{2}\) indicates the share of vehicles experiencing the connection duration defined by \(\rho_{1}\). Thus, a vehicular network is \(\Delta_{\rho_{2}}^{\rho_{1}}\) whenever \(\rho_{2}\) percent of vehicles experience connection during \(\rho_{1}\) percent of the trip duration. Figure 1 illustrates the Delta metric. The metric is not represented by a single value. Instead, the Delta Network is represented as a curve in a 2D plan. The \(x\)-axis indicates \(\rho_{1}\) (percentage of trip under coverage), while the \(y\)-axis indicates \(\rho_{2}\) (share of vehicles). In fact, Delta is the relation between \(\rho_{1}\) and \(\rho_{2}\). The shadowed area indicates possible service levels for the network. The maximum achievable performance is indicated by the border of the curve. Complementary, outside the curve we have service levels unable to be achieved by the current network setup. Representing Delta as the relation between \(\rho_{1}\) and \(\rho_{2}\) has the advantage of delimiting the entire range of network operation. Such representation allows us to compare the operation of distinct vehicular networks in order to identify successful design strategies. In fact, we can use the Delta network to measure one dimension of the performance of the vehicular network (connected time in terms of trip duration), and also to plan a new network from scratch, or even update an existing one, using Delta to define the location of future roadside units. The Delta Network is formally introduced in Definition 1. **Definition 1** (Deployment \(\Delta_{\rho_{2}}^{\rho_{1}}\)).