Analysis of customers flows in the infinite-server queueing tandem with feedback

The paper presents the study of a two-stage infinite-server queueing system with feedback. The arrival process is a stationary Poisson process with the rate equals to X. Service times at the first stage are independent and identically distributed (i.i.d.) with an arbitrary distribution function Bi(x). After a completion of the service at the first stage, the customer may return back to the first stage for a new service with the probability r11 or it may move to the second stage with the probability r12 or it may leave the system with the probability (1 - rii - ri2). Service times at the second stage are i.i.d. with an arbitrary distribution function B2(x). When the service at the second stage is completed, the customer may return to the first stage with the probability r21 or it may get a new service at the second stage with the probability r22 or it may leave the system with the probability (1 - r21 - r22). The problem is to study multidimensional stochastic process describing the flows of requests in the system. The method of limiting decomposition is used for the study. We divide the arrival process in the considered tandem into N independent processes according to a polynomial scheme with identical probabilities. As the rate of the original arrivals was equal to X, then the intensity of each generated Poisson process will be equal to UN. After that we construct a single-line tandem for each of these arrival processes to serve their customers. The considered single-line two-stage queueing tandem is a system with loses, that is, the customers arrived during a period of any stage busyness are not servicing (they are lost). The total probability characteristics of the independent one-line systems constructed in this way coincide with the corresponding characteristics of the original infinite-server system if N ^ да. It is shown that the generating function of multidimensional stochastic process {/(t), nii(t), ni2(t), n2i(t), «22(f)} is as follows: ^G(x.Уп,У12,У21'У22^,/) = ^eXP|(^x-1)^Xt + [^r11⁽У11 ^- 1) + Г2(У12 ^- 1)]J J1^(x'У11' У12=У21'У22 =^s)ds + l 0 t [^r21 ⁽У21 ^{- 1)} + ^r22⁽У22 ^-1)]J^f2^(x'У11'У12'У21'У22'^s)ds 0 where for functions f(x, yii, yi2, yu, yu, t), k = i, 2 analytic expressions of Fourier-Stieltjes transformations are obtained. The obtained results can be used for the analysis of flows in different social and economic systems, where there is a repeated circulation of customers to the system under different conditions, for example, in trading or insurance companies, as well as in technical systems for distributed processing of big data and cloud services.

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