On a system with retrial queue and unreliable server - Lebedev, E.A.; Sharapov, M.M.; Livinska, H.V. (Reports of the National Academy of Sciences of Ukraine, 2020)
We consider a model of retrial queue with one unreliable server whose lifetime is an exponentially distributed random variable with the known failure rate. A two-dimensional Markov chain defines the service process in the system. Its first component indicates the number of sources of repeated calls, and the second one fixes the status of the server at the current time: the server is busy, free, and ready for maintenance or out of order. The main feature of the considered system is that the input flow rate depends on the size of the queue of repeated calls. Each of the sources of repeated calls can generate a call with the same rate. If a primary or repeated call arrives into the system and finds the server idle, its service begins immediately. If the server is busy, the call is directed to the orbit and becomes a source of retrial calls. For the service process, a condition for the existence of a stationary regime and vector-matrix formulas are found. These formulas express stationary probabilities through the model parameters in the explicit form. To control the accuracy of calculations using these formulas, an estimate of the remainder of the series is obtained, which sets the normalizing constant. The rate of the remainder decreasing to zero has an exponential upper estimation. In the case where the input flow is the Poisson one, the exact expression is obtained for a normalizing constant. The application of the obtained results is demonstrated by numerical examples, which show the dependence of the blocking probability in the stationary regime on the parameters of the system. The obtained results can be used to solve optimization problems in the class of threshold strategies.