Spectral solutions for QS with distribution laws in the form of probabilistic mixtures
Background. QS are the main mathematical tool for modeling data transmission systems, which are not without reason called queuing networks. The need to regulate such characteristics of mass service systems as waiting time in a queue or queue length is due to the improvement of the quality of operati...
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Povolzhskiy State University of Telecommunications & Informatics
2025-08-01
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| Series: | Физика волновых процессов и радиотехнические системы |
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| Online Access: | https://journals.ssau.ru/pwp/article/viewFile/28790/11405 |
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| author | Veniamin N. Tarasov Nadezhda F. Bakhareva |
| author_facet | Veniamin N. Tarasov Nadezhda F. Bakhareva |
| author_sort | Veniamin N. Tarasov |
| collection | DOAJ |
| description | Background. QS are the main mathematical tool for modeling data transmission systems, which are not without reason called queuing networks. The need to regulate such characteristics of mass service systems as waiting time in a queue or queue length is due to the improvement of the quality of operation of data transmission systems. The ability to regulate these characteristics allows minimizing the waiting time in the queue in the buffers of transmitting devices, as well as the volumes of buffer memory itself. To demonstrate this possibility, the paper examines queuing systems formed by both conventional distribution laws in the form of probability mixtures and time-shifted distribution laws. Aim. In this work, the hyperexponential and hyper-Erlangian distributions of the second order were chosen as components of the QS. Based on these distribution laws, numerical-analytical models were constructed for two queuing systems with normal and shifted distribution laws, with the derivation of a solution for the main characteristic of the queuing system – the average waiting time in the queue. As is known, the remaining characteristics of the QS are derivatives of the average waiting time. Methods. The paper uses a shift of the distribution laws to the right from the zero point. To derive a solution for the average waiting time in a queue, the classical method of spectral solution of the Lindley integral equation is used based on the Laplace transform of the distribution laws that form the considered QS. The obtained calculation formulas for the average waiting time in a queue allow us to calculate the characteristics of such systems for a wide range of changes in teletraffic parameters. Results. The obtained results can be used in modern teletraffic theory in the design and modeling of various promising data transmission systems, including the volumes of buffer memory of transmitting devices. Conclusion. The shift of the distribution laws in time leads to a decrease in their variation coefficients. Due to the quadratic dependence of the average waiting time on the variation coefficients of the arrival and service time intervals, a noticeable decrease in the average waiting time follows in systems with time shifts. |
| format | Article |
| id | doaj-art-2457fc02f55746e98288651642d2ca4e |
| institution | Kabale University |
| issn | 1810-3189 2782-294X |
| language | English |
| publishDate | 2025-08-01 |
| publisher | Povolzhskiy State University of Telecommunications & Informatics |
| record_format | Article |
| series | Физика волновых процессов и радиотехнические системы |
| spelling | doaj-art-2457fc02f55746e98288651642d2ca4e2025-08-20T04:01:09ZengPovolzhskiy State University of Telecommunications & InformaticsФизика волновых процессов и радиотехнические системы1810-31892782-294X2025-08-01282879410.18469/1810-3189.2025.28.2.87-948915Spectral solutions for QS with distribution laws in the form of probabilistic mixturesVeniamin N. Tarasov0https://orcid.org/0000-0002-9318-0797Nadezhda F. Bakhareva1https://orcid.org/0000-0002-9850-7752Povolzhskiy State University of Telecommunications and InformaticsPovolzhskiy State University of Telecommunications and InformaticsBackground. QS are the main mathematical tool for modeling data transmission systems, which are not without reason called queuing networks. The need to regulate such characteristics of mass service systems as waiting time in a queue or queue length is due to the improvement of the quality of operation of data transmission systems. The ability to regulate these characteristics allows minimizing the waiting time in the queue in the buffers of transmitting devices, as well as the volumes of buffer memory itself. To demonstrate this possibility, the paper examines queuing systems formed by both conventional distribution laws in the form of probability mixtures and time-shifted distribution laws. Aim. In this work, the hyperexponential and hyper-Erlangian distributions of the second order were chosen as components of the QS. Based on these distribution laws, numerical-analytical models were constructed for two queuing systems with normal and shifted distribution laws, with the derivation of a solution for the main characteristic of the queuing system – the average waiting time in the queue. As is known, the remaining characteristics of the QS are derivatives of the average waiting time. Methods. The paper uses a shift of the distribution laws to the right from the zero point. To derive a solution for the average waiting time in a queue, the classical method of spectral solution of the Lindley integral equation is used based on the Laplace transform of the distribution laws that form the considered QS. The obtained calculation formulas for the average waiting time in a queue allow us to calculate the characteristics of such systems for a wide range of changes in teletraffic parameters. Results. The obtained results can be used in modern teletraffic theory in the design and modeling of various promising data transmission systems, including the volumes of buffer memory of transmitting devices. Conclusion. The shift of the distribution laws in time leads to a decrease in their variation coefficients. Due to the quadratic dependence of the average waiting time on the variation coefficients of the arrival and service time intervals, a noticeable decrease in the average waiting time follows in systems with time shifts.https://journals.ssau.ru/pwp/article/viewFile/28790/11405ordinary and shifted hyper exponential and hyper-erlang distribution lawslindley integral equationspectral decomposition methodlaplace transform |
| spellingShingle | Veniamin N. Tarasov Nadezhda F. Bakhareva Spectral solutions for QS with distribution laws in the form of probabilistic mixtures Физика волновых процессов и радиотехнические системы ordinary and shifted hyper exponential and hyper-erlang distribution laws lindley integral equation spectral decomposition method laplace transform |
| title | Spectral solutions for QS with distribution laws in the form of probabilistic mixtures |
| title_full | Spectral solutions for QS with distribution laws in the form of probabilistic mixtures |
| title_fullStr | Spectral solutions for QS with distribution laws in the form of probabilistic mixtures |
| title_full_unstemmed | Spectral solutions for QS with distribution laws in the form of probabilistic mixtures |
| title_short | Spectral solutions for QS with distribution laws in the form of probabilistic mixtures |
| title_sort | spectral solutions for qs with distribution laws in the form of probabilistic mixtures |
| topic | ordinary and shifted hyper exponential and hyper-erlang distribution laws lindley integral equation spectral decomposition method laplace transform |
| url | https://journals.ssau.ru/pwp/article/viewFile/28790/11405 |
| work_keys_str_mv | AT veniaminntarasov spectralsolutionsforqswithdistributionlawsintheformofprobabilisticmixtures AT nadezhdafbakhareva spectralsolutionsforqswithdistributionlawsintheformofprobabilisticmixtures |