Product form solution for g-networks with dependent service

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: (0) an exponential node with cn servers, infinite buffer and FIFO discipline; (1) an infinite-server node; (2) a single-server node with infinite buffer and LIFO PR discipline; (3) a single-server node with infinite buffer and PS discipline. 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. ∗ This work was supported by the Russian Foundation for Basic Research, Grant No 02-0790147. 1 Department of Probability Theory and Mathematical Statistics Peoples’ Friendship University of Russia, Moscow, Russia; e-mail: [email protected]; tropic [email protected] 2 Department of Information Engineering and Applied Mathematics, University of Salerno, Italy; e-mail: [email protected] 3 Institute of Informatics Problems Russian Academy of Sciences Moscow, Russia; e-mail: [email protected] c © EDP Sciences 2004 106 P. BOCHAROV ET AL.