Geodesic
Analysis of a queuing system equations with a jump intensity of input stream
Matematičeskie zametki SVFU, Tome 25 (2018) no. 3, pp. 18-32 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the queuing system (QS) with an infinite storage, one service device and exponential service. At the input of QS comes double stochastic Poisson flow whose intensity is a jump-like process with intervals of constancy distributed according to the exponential law. It is assumed that the input flow intensity values at the break points on the left and right are independent. In the earlier published works a sufficient condition of existence and uniqueness of the QS stationary regime was obtained. In this paper, the operator analysis of integral equations is performed with respect to the characteristics of the stationary SMO, the necessary condition of existence of the system of integral equations solution is obtained and the existence and uniqueness of the solution is proved. A stationary generating function of the solution in the form of a convergent series is found. A distinctive feature of this work is the construction of the 2nd model of QS and the use of the shift operator of coefficients of the generating function for stationary distribution of the customers number.
Keywords: jump-like process, double stochastic Poisson stream, queuing system, generating functions method. 
@article{SVFU_2018_25_3_a1,
     author = {O. V. Bondrova and T. {\CYRA}. Zhuk and N. I. Golovko},
     title = {Analysis of a queuing system equations with a jump intensity of input stream},
     journal = {Matemati\v{c}eskie zametki SVFU},
     pages = {18--32},
     year = {2018},
     volume = {25},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVFU_2018_25_3_a1/}
}
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O. V. Bondrova; T. А. Zhuk; N. I. Golovko. Analysis of a queuing system equations with a jump intensity of input stream. Matematičeskie zametki SVFU, Tome 25 (2018) no. 3, pp. 18-32. http://geodesic.mathdoc.fr/item/SVFU_2018_25_3_a1/