1 Self-similarity properties in a queuing network model

In this paper a model of subscriber telephone network based on the concept of cellular automata is elaborated [1]. Some fractal properties inherent in the model are revealed that vary depending on parameters assigning its operation rules. The main advantage of the model in question is its compatibility with algorithmic methods- a finite set of formal rules, assigned on a finite set of elements (cells), allows precise realization in the form of algorithms. It should be noted that simulation of queuing networks, in particular, simulation of communication networks by means of cellular automata is a common approach [2, 3], at the same time, revealing self-similarity effects in the simplest model considered below allows to take a fresh view of the fractal nature of teletraffic. A system of cellular automata is a combination of similar cells definitely interconnected. The state of each cell is determined by the state of neighbouring cells, Thus, the state of the j-th cell at time moment 1t + is determined as follows: () ( 1) (), (),,j jy t F y t O j t+ = where F is a certain rule that can be expressed, for example, in terms of the Boolean algebra, and ()O j is a set of neighbouring cells (neighbourhood). Each cell of the system of cellular automata that corresponds to a model of closed queuing network will be considered as a telephone subscriber. It is supposed that each subscriber at random time moments can give a telephone call only to his near acquaintances that correspond to eight cells from the subscriber’s neighbourhood, i.e. Moore’s neighbourhood is used [4] – the cell,i jy will have it