Product form solution for g-networks with dependent service

¹ Department of Probability Theory and Mathematical Statistics Peoples' Friendship University of Russia, Moscow, Russia; pbocharov@sci.pfu.edu.ru.; tropic_mos@rambler.ru.
² Department of Information Engineering and Applied Mathematics, University of Salerno, Italy; dapice@diima.unisa.it.
³ Institute of Informatics Problems Russian Academy of Sciences Moscow, Russia; APechinkin@ipiran.ru.

Abstract

We consider a G-network with Poisson flow of positive customers. Each positive customer entering the network is characterized by a set of stochastic parameters: customer route, the length of customer route, customer volume and his service length at each route stage as well. The following node types are considered: Negative customers arriving at each node also form a Poisson flow. A negative customer entering a node with k customers in service, with probability 1/k chooses one of served positive customer as a “target”. Then, if the node is of a type 0 the negative customer immediately “kills” (displaces from the network) the target customer, and if the node is of types 1–3 the negative customer with given probability depending on parameters of the target customer route kills this customer and with complementary probability he quits the network with no service. A product form for the stationary probabilities of underlying Markov process is obtained.