FINDING EXPECTED REVENUES IN G-NETWORK WITH SIGNALS AND CUSTOMERS BATCH REMOVAL

Abstract

The paper provides an analysis of G-network with positive customers and signals when signals arriving to the system move customer to another system or destroy in it a group of customers, reducing their number to a random value that is given by a probability distribution. The signal arriving to the system, in which there are no positive customers, does not exert any influence on the queueing network and immediately disappears from it. Streams of positive customers and signals arriving to each of the network systems are independent. Customer in the transition from one system to another brings the latest some revenue, and the revenue of the first system is reduced by this amount. A method of finding the expected revenues of the systems of such a network has been proposed. The case when the revenues from transitions between network states are deterministic functions depending on its states has been considered. A description of the network is given, all possible transitions between network states, transition probabilities, and revenues from state transitions are indicated. A system of difference-differential equations for the expected revenues of network systems has been obtained. To solve it, we propose a method of successive approximations, combined with the method of series. It is proved that successive approximations converge to the stationary solution of such a system of equations, and the sequence of approximations converges to a unique solution of the system. Each approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which are related by recurrence relations. Therefore, it is convenient to use them for calculations on a PC. The obtained results can be applied in forecasting losses in information and telecommunication systems and networks from the penetration of computer viruses into it and conducting computer attacks.