On estimates of the mean queue length for single-channel queuing systems in terms of statistical unconditional second-order moments of the modified arrival flow

We consider a mathematical model of the simplest single-channel queuing system (QS) with a deterministic service time in the case of an arbitrarily correlated arrival flow. Various generalizations of the Pollaczek–Khinchine formula for the mean queue length are obtained for this QS. An interval model of the arrival flow is proposed. Within the framework of this model, an expression is obtained for the mean queue length in terms of statistical unconditional moments of the second order. All results are obtained under very general assumptions of ergodicity and stationarity. The results of numerical experiments confirming the theoretical conclusions are presented.