Method of Markovian summation for study the repeated flow in~queueing tandem $\mathrm{M|GI|}\infty \to \mathrm{GI}|\infty$

M. A. Shklennik ; A. N. Moiseev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 1, pp. 125-137

Résumé

The paper presents a mathematical model of queueing tandem $\mathrm{M|GI|}\infty \to \mathrm{GI}|\infty$ with feedback. The service times at the first stage are independent and identically distributed (i.i.d.) with an arbitrary distribution function $B_1(x)$. Service times at the second stage are i.i.d. with an arbitrary distribution function $B_2(x)$. The problem is to determine the probability distribution of the number of repeated customers ($r$-flow) during fixed time period. To solve this problem, the Markov summation method was used, which is based on the consideration of Markov processes and the solution of the Kolmogorov equation. In the course of the solution, the so-called local $r$-flow was studied — the number of $r$-flow calls generated by one incoming customer received by the system. As a result, an expression is obtained for the characteristic probability distribution function of the number of calls in the local $r$-flow, which can be used to study queuing systems with a similar service discipline and non-Markov incoming flows. As a result of the study, an expression is obtained for the characteristic probability distribution function of the number of repeated calls to the system at a given time interval during non-stationary regime, which allows one to obtain the probability distribution of the number of calls in the flow under study, as well as its main probability characteristics.