# Distributed Management Systems for Infocommunication Networks: A Model Based on TM Forum Framework
## 3 Methods
### Analytical Model of Integration System for PCs  
In accordance with the above results, a simultaneous execution of several logically independent parallel processes on the same resource actually increase the system cost of operational management and reduce the system performance; for a considerable number of processes, it may even cause network resource deficit. The structure of software-defined networking (SDN) and, in particular, the approaches to organize logically centralized control of network elements were described in the ITU-T Recommendations, Y.3300 series [18]; also see Figure 15. The application-control interface is intended to implement program control of abstract network resources. The resource-control interface is intended to implement the functions of logically centralized control of network resources. Figure 14: Execution model for two parallel BPs with arranged capture of resources: simulation results. Assume the transitions of requests between the elements of network resources as well as their withdrawals from the network are described by an irreducible Markov chain; in addition, let the processes occurring in the network be the multidimensional birth and death processes. Then, an SDN controller can be represented as a root node \(M\) of the network while the network nodes--the program components of a physically distributed application--act as the other (\(M\)-1) nodes that execute a corresponding business process. For the requests of the class \(R=(1,r)\), the transition probabilities can be described by a matrix (12)  
\[\|P_{r}\binom{-}{n_{i}^{j}}\|=\left| \begin{array}{cccc}P_{r,1,1}&P_{r,1,2}& \ldots&P_{r,1,M}\\ P_{r,2,1}&P_{r,2,2}&\ldots&P_{r,2,M}\\ \ldots&\ldots&\ldots&\ldots\\ P_{r,M,1}&P_{r,M,2}&\ldots&P_{r,M,M}\\ \end{array} \right|, \tag{12}\]  
where \(P_{r}\binom{-}{n_{i}^{j}}\) denote the probabilities of transition and \(n_{i}^{j}\) is the number of requests passing from node \(i\) to node \(j\). Write the multidimensional random interaction process of PCs as (13)  
\[N(t)=\{n_{1}(t),n_{2}(t),\ldots,n_{R}(t)\}; \tag{13}\]  
the probability that \(k\) requests of the \(r\)th class have to be served as \(P(k)=P_{1}(k_{1})\cdot P_{2}(k_{2})\cdot\ldots P_{n}(k_{n})\), where \(k=(k_{1},k_{2},\ldots,k_{n})\) and \(k_{i}=(k_{i1},k_{i2},\ldots,k_{ir})\). Such a network can be described in the multiplicative form [19]  
\[P(n)=G^{-1}(N_{1},N_{k},\ldots,N_{R})\prod\limits_{i=1}^{M}Z_{i}(n_{i}), \tag{14}\]  
where \(P(n)\) denotes the probability that the network is in state \(n\),  
\[n=(n_{1},n_{2},\ldots,n_{M});\ G(N_{R})=\sum\limits_{k}\prod \limits_{i=2}^{N}\left(\frac{\mu_{1}P_{i}}{\mu_{i}}\right)^{k_{i}};\] \[Z_{i}(n_{i})=\frac{n_{i}!}{\frac{\mu_{i}!}{\frac{\mu_{i}!}{\mu_{i }!}}\prod\limits_{i_{R}=1}^{R}\prod\limits_{n_{ir}!}^{\frac{\mu_{ir}!}{\mu_{ir }!}}},\] \[l_{ir}=\sum\limits_{i=1}^{M}l_{ir}P_{ij}(r),i=\overline{1,M},r= \overline{1,R}.\]  
Figure 15: Structure of software-defined networking for centralized control of network resources [18]. As a result,  
* the SDN controller capacity for a request of the \(r\)th class is (15) \[\lambda_{ir}(N_{R})=\sum_{n_{k}=1}^{N_{R}}P_{i}(n_{R},N_{r})\frac{n_{ir}}{n_{i}} \mu_{i}(n_{i});\] (15)
* the number of served requests of the \(r\)th class is (16) \[L_{ir}(N_{R})=\sum_{n_{R}=1}^{N_{R}}P_{i}(n_{R},N_{R})n_{r};\] (16)
* the mean waiting time for a request of the \(r\)th class is (17) \[T_{ir}(N_{r})=\frac{\left[1+L_{i}(N_{R}-1)\right]}{M_{ir}}.\] (17)  
In practice, the values of these indicators depend on system load at request arrival times. In some state the system may block a successive call.