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A tandem of queueing systems with cyclic control algorithm and with independent conflict input flows is considered. The customers served in the first queueing system are transferred to the second system with random speeds. A cybernetic approach is used to construct a mathematical model in terms of a multidimensional denumerable Markov chain. The classification of the states of this Markov chain is carried out, and a sufficient condition for the existence of the stationary distribution is obtained. 1. The setting of the problem Consider two queueing systems serving the customers under the following assumptions. There are four independent conflict input flows, two flows for each of the systems. Each of these flows is a nonordinary Poisson process. The flows are conflicting in the sense that customers arriving from different sources cannot be served simultaneously in the same queueing system. This means that the problem cannot be reduced to a corresponding problem with fewer input flows by merging the flows together. The customers coming from each flow are placed in a special buffer with an unbounded capacity. The service time of an arbitrary customer is random and service times of different customers are independent random variables with unknown probability distributions. Each system may serve several customers during the same working period. The service algorithm of conflict flows in each of the two systems belongs to the class of cyclic algorithms. We assume that the durations of working periods are commensurable constants. At the moment 0, the service starts in both systems. After a service is completed in the first system for a customer coming from the first flow of the first system, he is transferred to the second system. We assume that the transfer is not instant. We further assume that the speed of the transfer for each of the customers is random and its probability distribution is unknown. Moreover, the speeds of transfer of different customers are varying with time and have different probability distributions. We agree that the transfer of every customer from the first system to the second system is completed with a known probability during a working period. Respectively, the transfer is not completed during a working period and will be continued during the next one with the complementary probability. 2010 Mathematics Subject Classification. Primary 60K25, 90B22.

Stability of a tandem of queueing systems with Bernoulli noninstantaneous transfer of customers