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On Dependent Distributions of Appearance and Servicing of Claims in the M/M/1

The random process of receipt of applications in the QS is characterized by a distribution law that establishes a relationship between the values of a random variable and the probabilities of the appearance of these values. In most cases, such a flow is characterized by a distribution function of time intervals between neighboring claims, and the process of their processing is characterized by a probability distribution function of service time intervals. In the overwhelming number of papers in queuing theory, these random variables are considered not correlated and mutually independent. However, these assumptions for the analysis of packet traffic of multiservice communication networks are completely unacceptable. Classical queuing theory explores time series processing under the assumption of independence sampling. However, the traffic of modern multiservice networks is usually is strongly correlated and the methods of classical theory do not work. We are analyzing the distribution of the waiting time in the M / M / 1 queue, in which the time between arrival between the nth and (n - 1)th customers and the service time of the nth customer are correlated random variables.