Extension of the Polaczek — Khinchin formula to 3d stochastic networks

An elementary 3d stochastic queueing network is studied where the flow of moving systems is recurrent and the flows of entries into the systems (network accumulators) are primitive. Recurrent equations are provided which describe the state of the accumulator systems in the regeneration points. A theorem on the mean queue length in regeneration points (an analog of the Polaczek — Khinchin formula) and a theorem on the queueing time distribution density are proved. The little formula for an elementary 3d network is proved.