Research of streams in system M|GI| with repeated references the method of limiting decomposition.

In work the queuing system with unlimited number of serving devices on which input the elementary stream of demands with parameter ƒ arrives is considered. The holding time on eachdevice has arbitrary function of distribution B (x) identical to all devices. The demand which hasfinished service, with probability 1- r leaves system, and with probability r addresses to systemfor repeated service.The mathematical model of change of number of demands in given queuing system is constructedand the task of research of total and two-dimensional streams in considered system is put.For the decision of this task the method of limiting decomposition is offered. The enteringstream shares on N independent elementary streams with parameter ƒ⁄ N, demands of each streamgo for service on the corresponding device. Thus, we receive the aggregate of N independent onelinearqueuing systems with the refusals which total characteristics at N  converge to characteristicsof initial model.The method described above in work conducts research of a total stream of primary and repeatedreferences to system M|GI |  and making function of number of total references in consideredqueuing system is found0( , ) exp ( 1) ( 1) ( , )tG x t x t r x h x s ds ⎧⎪ ⎪⎫ = ⎨ − Ґл - − ⎬⎪⎩ ⎭⎪Ўт .Research of a two-dimensional stream in queuing system with the repeated reference and unlimitednumber of serving devices is similarly conducted and its making function which kind allowsto draw a conclusion on dependence of considered streams of references is found: 0( , , ) exp ( 1) ( 1) ( , , )tG x y t x t r y h x y s ds ⎧⎪ ⎪⎫ = ⎨ − Ґл - − ⎬⎪⎩ ⎭⎪Ўт .It is shown that results of research are the generalization of the known special cases, exactlyfor exponential time of service.The got results can be used during conducting of analysis of streams of the different socioeconomicsystems where the effect of the repeated reference, for example, in the trading companies, is observed.

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