Stationary distribution of an open queueing networkwith non-standard customers transitions

We considered the exponential open queueing network which consists of N service systems.There are two input Poisson flows: the main and the additional, which customers are directed toempty systems. Service is exponential and an intensity of service depends on the number of customersat the system. Groups of random size, which less then the number of customers at systemare taken for service with equal probabilities. Because of the such model description we avoid thesituation, when we choose for service a group, which size is more then the number of customersat system.Markov process X(t)={X1(t), …,ХN(t)}, where Xi (t) - a number of customers at system i atmoment t describes this network model.Researched open network is equivalent to model with an infinite number of service nodes atthe each of the system. After service a group is directed to another system without the change ofits size or leaves the network. The traffic equations system is linear and has its unique solution.Considering time-reversed process, we obtained necessary and sufficient conditions of geometricproduct form stationary distribution existence and offered an algorithm of stationary distributionfinding.

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