# Interconnected Government Services: An Approach toward Smart Government
## 4 Results--Model Evaluation
### A Model Based on Weighted Digraphs  
The real-life process currently present in the Republic of Serbia regarding the collection of the documentation necessary for applying for child benefits and the newly proposed virtual procedure for collecting and processing the required data is modeled using weighted digraphs. Nowadays, the theory of graphs and digraphs is widely developed. Many diverse problems from theory and practice are considered using mathematical results from this domain of science. Digraph \(G_{1}\), presented in Figure 3, corresponds to the current process of collecting the documentation in the Republic of Serbia. Digraph \(G_{2}\), presented in Figure 4, serves as a model of the newly proposed virtual procedure. Figure 2: The logical architecture of the Integration mediator. The vertices of \(G_{1}\) and \(G_{2}\) correspond to state administration services and ministries (Appendix A, Table A1), as follows:  
0--The Ministry of Interior;  
1--The Municipal Body of Local Self-Government;  
2--The Municipal Body of Government;  
3--The National Health Insurance Fund;  
4--The Country Level Tax Administration;  
5--The Local Authority Tax Administration;  
6--The Pension and Disability Insurance Fund;  
7--The Republic Geodetic Authority;  
8--The National Employment Office;  
9--A Central Register of the Insurance using Obligatory Social Insurance;  
Figure 4: Digraph \(G_{2}\)—the proposed procedure in the presented example. Figure 3: Digraph \(G_{1}\)— the current procedure in the presented example. S--The beginning of the procedure;  
E--The end of the procedure;  
U--The user action. The (oriented) edges of \(G_{1}\) correspond to the order in which the applicant should visit the state administrative services and municipal bodies to collect the necessary documentation. Precisely, there is an edge from vertex \(i\) to vertex \(j\), where \(i\
eq j\), and \(i,j\in\{0,1,\ldots,9,S,E\}\), if and only if for obtaining the documentation from the state administration service \(j\), the applicant was supposed to visit the state administration service \(i\) first. The (oriented) edges of \(G_{2}\) represent the work of the newly proposed smart-government model from the moment the applicant enters the requested data. The current procedure, i.e., the model given by \(G_{1}\), will be compared with the newly proposed one, presented by \(G_{2}\), using digraph weights. The weight of a digraph is the sum of the weights of its edges. We will consider the case when the weights represent the time, in milliseconds, required to obtain the necessary data. The weight of \(G_{1}\) is equal to  
\[w(G_{1})=\sum\limits_{i=1}^{2}d_{si}+\sum\limits_{j\in\{0,3,6\}}d_{1j}+\sum \limits_{k\in\{4,5,7\}}d_{2k}+d_{75}+\sum\limits_{l=1}^{9}d_{le} \tag{1}\]  
The given weights represent the time in milliseconds needed for collecting necessary documentation. Since the weights \(d_{si}\), for \(i=1,2\); \(d_{1j}\), for \(j\in\{0,3,6\}\); \(d_{2k}\), for \(k\in\{4,5,7\}\); and \(d_{75}\) can vary from a few minutes, say 15, to a few days, say 5 days, we will suppose that \(d_{si}\approx d_{1j}\approx d_{2k}\approx d_{75}\approx d\), where \(d\) is arithmetic meand these weights, and \(d\in\left[9\cdot 10^{5},4.32\cdot 10^{8}\right]\). As the weights \(d_{le}\), for all \(l=1,\ldots,9\), denote time needed for submitting the final application, when the applicant has already collected the required documentation, we can ignore them, i.e., we will suppose that \(\sum\limits_{l=1}^{9}d_{le}=0\). Therefore, equality (1) becomes  
\[w(G_{1})\approx 9d \tag{2}\]  
This means that we can consider that \(w(G_{1})\) approximately belongs to the interval \(\left[8.1\cdot 10^{6},3.89\cdot 10^{9}\right]\). The weight of \(G_{2}\) is equal to  
\[w(G_{2})=t+\sum\limits_{i=1}^{9}(t_{ui}+t_{ie}) \tag{3}\]  
The weight \(t\) represents the time the applicant spends in milliseconds entering the requested data. We can assume that \(t\in\left[1.8\cdot 10^{5},9\cdot 10^{5}\right]\).